Amandine Aftalion
email: aman-dine.aftalion@ehess.fr

Alberto Farina
email: alberto.farina@u-picardie.fr

Luc Nguyen
email: luc.nguyen@maths.ox.ac.uk

Moving planes for domain walls in a coupled system

come

Moving planes for domain walls in a coupled system

Introduction

We study the monotonicity and uniqueness of a non-cooperative system coming from the physics of two-component Bose-Einstein condensates which displays partial phase transition. We are able to use the moving plane device in a case where a priori bounds do not come from standard techniques. The particularity of this system with respect to classical two-component segregated Bose-Einstein condensates is to couple both linearly and nonlinearly the two equations due to spin coupling of the hyperfine states added to the intercomponent coupling. Our system for the functions u and v is the following:

∆u = -u(1 -u 2 -v 2 ) -v(ω -αuv) =: P (u, v) in R N , ∆v = -v(1 -u 2 -v 2 ) -u(ω -αuv) =: Q(u, v) in R N , (1.1)
where u and v are the wave functions for each component, α is a positive parameter such that 1 + α is the intercomponent coupling, and ω is a positive parameter, which is the Rabi frequency of the one body coupling between the two internal states.

We would like to study the phase transition solutions, so we will impose boundary conditions at infinity in one direction, namely:

u(x ′ , x N ) → b, v(x ′ , x N ) → a, as x N → +∞, u(x ′ , x N ) → a, v(x ′ , x N ) → b, as x N → -∞, (1.2)
the limit being uniform in x ′ ∈ R N -1 , where (a, b) is the solution to

a 2 + b 2 = 1, ab = ω α , 0 < a < b. (1.3)
The existence of such solutions requires, in addition to the positivity of α and ω, the condition ω α < 1 2 .

(1.4)

The study of domain wall solutions in coupled Gross-Pitaevskii equations or segregation patterns has been the subject of many papers concerning existence, uniqueness, monotonicity of asymptotic behaviour [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF][START_REF] Alama | Domain walls in the coupled Gross-Pitaevskii equations[END_REF][START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF][START_REF] Sourdis | On the weak separation limit of a two-component bose-einstein condensate[END_REF]. It corresponds to the case ω = 0 and α > 0.

Here, we would like to address a different physical background, that of a twocomponent Bose-Einstein condensates representing two different hyperfine states, and coupled by their spin, to take into account a one body coherent Rabi coupling, which corresponds to ω > 0. In this case, segregation is not complete and this leads to what is called Rabi oscillations which have been experimentally observed in [START_REF] Matthews | Watching a superfluid untwist itself: Recurrence of Rabi oscillations in a Bose-Einstein condensate[END_REF]. The ground states and excited states have been studied in [START_REF] Aftalion | Rabi-coupled two-component Bose-Einstein condensates: Classification of the ground states, defects, and energy estimates[END_REF][START_REF] Dror | Domain walls and vortices in linearly coupled systems[END_REF][START_REF] Lellouch | Twocomponent Bose gases with one-body and two-body couplings[END_REF][START_REF] Qu | Magnetic solitons in Rabi-coupled Bose-Einstein condensates[END_REF][START_REF] Usui | Rabi-coupled countersuperflow in binary Bose-Einstein condensates[END_REF]. The system (1.1)-(1.2) for N = 1 has been analyzed in [START_REF]Phase transition in a Rabi coupled two-component Bose-Einstein condensate[END_REF] where the existence and asymptotic properties of one dimensional domain wall solutions are derived in the case ω/α of order 1 and α large and small. The properties and structures found in [START_REF]Phase transition in a Rabi coupled two-component Bose-Einstein condensate[END_REF] have led us to investigate the monotonicity and uniqueness of solutions. Let us point out that this system has a heteroclinic structure and derives from the minimization of the Gross-Pitaevskii energy. Understanding the solutions of (1.1)-(1.2) is a first step in a future mathematical analysis of the specific vortex patterns appearing in the Rabi coupled condensates called multidimer bound states [START_REF] Aftalion | Rabi-coupled two-component Bose-Einstein condensates: Classification of the ground states, defects, and energy estimates[END_REF][START_REF] Cipriani | Crossover between integer and fractional vortex lattices in coherently coupled two-component bose-einstein condensates[END_REF][START_REF] Kobayashi | Vortex polygons and their stabilities in boseeinstein condensates and field theory[END_REF]: this is a state where a pair of vortices of different components binds together as a molecule. These molecules then interact in a nontrivial manner with other molecules, with the result that a rich hierarchy of patterns, such as honeycombed, triangular, and square, is formed. It is the possibility to have non segregation at infinity with these positive limits which is at the origin of this specific pattern.

Here is our main result:

Theorem 1.1. Assume α > 0, ω > 0, and (1.4) holds. Then all positive solutions to (1.1)-(1.2) depend only on x N , satisfy ∂ N u > 0 and ∂ N v < 0 and are translations of one another along the x N direction.

The proof relies on the moving plane device of [START_REF] Berestycki | Further qualitative properties for elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains i[END_REF][START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF][START_REF]Symmetry of positive solutions of nonlinear elliptic equations in R n[END_REF][START_REF] Serrin | A symmetry problem in potential theory[END_REF] to prove monotonicity and one-dimensionality of the phase transition solutions and the sliding method [START_REF] Berestycki | Monotonicity for elliptic equations in unbounded Lipschitz domains[END_REF][START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains[END_REF][START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF] to prove uniqueness. We point out that it is not clear a priori that these methods can be applied as our system does not satisfy the usual monotonicity known for cooperative systems. Therefore, as a first step, we need to prove key estimates that will allow us to perform these methods and provide a structure to the system. These estimates also hold for solutions of the system without any conditions at infinity: Proposition 1.2. Assume α > 0, ω > 0, and (1.4) holds. Let (u, v) be a positive regular solution of (1.1), then

u 2 + v 2 ≤ 1 and uv ≥ ω α . (1.5) Moreover, either (u, v) ≡ (a, b) or (u, v) ≡ (b, a) or a < u, v < b. (1.6)
The bounds (1.5) provide a special structure to the problem and come at various stages in our proof. In the case ω = 0, the bound is only u 2 + v 2 ≤ 1 and is obtained by Kato's inequality. Indeed, the lower bound is reduced to uv ≥ 0. In this paper, it becomes uv ≥ ω/α, and the upper and lower bounds have to be proved together in a way which is similar to an attractor set. It is not a simple application of comparison principles or Maximum Principle. Proving these bounds requires some non-trivial extra-work, to which we devote the most part of Section 2.

In order to start the moving plane method, we need a sign for the difference between the solution and its reflection with respect to a hyperplane close to infinity in one direction. Since for our non-cooperative system, we do not have asymptotic estimates of solutions at infinity, we show that the moving plane procedure can be started by obtaining decrease estimates of the L 2 norm of the gradients of these differences on large balls. This is inspired by [START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF] and relies on a Lemma in [START_REF] Farina | Monotonicity and one-dimensional symmetry for solutions of -∆ p u = f (u) in half-spaces[END_REF]. In our case, we need in a crucial way a lower bound on uv and the proof is not a mere adaptation of previous works since the estimates are more involved.

To continue the moving plane procedure, or to apply the sliding method, we need to be able to apply the strong Maximum Principle and the Hopf lemma. This relies again on our lower bound for uv and the special structure it provides to the system. Let us point out that once the bounds (1.5) are known, the equations lead to (1.6), but without a strict sign. Therefore, this provides signs to the right hand side of the system (1.1), namely the system gets a structure of the type

-∆u = uf 1 (u, v) + vf 2 (u, v) in R N , -∆v = vf 1 (v, u) + uf 2 (v, u) in R N , (1.7)
with f 1 positive, symmetric, decreasing in both variables and f 2 negative, symmetric, decreasing in both variables. This structure of the system allows to apply the Strong Maximum Principle and the Hopf Lemma. One has to point out that this structure is not assumed from the beginning but is a consequences of our bounds. What our analysis emphasizes is that in order to obtain Maximum Principle or moving planes for non-cooperative systems, it requires both bounds from above for

u 2 + v 2 or -f 1 (u, v)
and below for uv or -f 2 (u, v).

The uniqueness proof for one dimensional solutions is based on the sliding method [START_REF] Berestycki | Monotonicity for elliptic equations in unbounded Lipschitz domains[END_REF][START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains[END_REF][START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF] and first requires an asymptotic behaviour for the ode's. Then the strong Maximum Principle relying on our lower bounds allows to apply the device and get uniqueness. We also refer to [START_REF] Ignat | Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals[END_REF][START_REF]Instability of point defects in a two-dimensional nematic liquid crystal model[END_REF][START_REF]Stability of point defects of degree ± 1 2 in a two-dimensional nematic liquid crystal model[END_REF] where some related discussions on uniqueness of 1D solutions are made for a related system arising in the Landau-de Gennes theory for nematic liquid crystals.

We note that the existence of solutions to (1.1)-(1.2) can be obtained by approximation by solutions on finite intervals as proved in Proposition 4.1.

In the case ω = 0, then the solution to (1.3) is given by a = 0 and b = 1, and therefore domain wall solutions stay between 0 and 1. In the one-dimensional case the existence of a monotone solution has been derived in [START_REF] Alama | Domain walls in the coupled Gross-Pitaevskii equations[END_REF] while uniqueness has been derived in [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF], based on a continuation argument and estimates of the linearized operator, under the assumption that either u ′ > 0 or v ′ < 0. Asymptotic estimates in the large α limit, still in one dimension, have been also obtained in [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF], and rely on properties for a simpler outer system studied by many authors and for instance by [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF][START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF][START_REF]Some symmetry results for entire solutions of an elliptic system arising in phase separation[END_REF][START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation[END_REF]. The other limit, that of weak segregation where α tends to 0 has been analyzed in [START_REF] Sourdis | On the weak separation limit of a two-component bose-einstein condensate[END_REF]. The proof that N-dimensional solutions are in fact onedimensional and monotone is made in [START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF] and relies on the moving plane device. The combination of the results of [START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF] and [START_REF] Aftalion | Interface layer of a two-component Bose-Einstein condensate[END_REF] then leads to the uniqueness of the solution, up to translations.

We also get interesting results using similar estimates in the case where (1.4) does not hold. We prove that solutions to (1.1) are constant without any boundary condition at infinity. Theorem 1.3. Assume α > 0, and

ω α ≥ 1 2 (1.8)
holds. Let (u, v) be a positive regular solution of (1.1). Then (u, v) is identically equal

to (c, c) with c = 1+ω 2+α .
The paper is organized as follows: firstly, we prove our key a priori bounds, rough upper and lower bounds as well as the refined bounds of Proposition 1.2. Then we make the moving plane device work to get monotonicity and the one dimensional property. Lastly, we study the properties of the one-dimensional solutions: existence, uniqueness up to translations and exponential decay to constant at infinity. The proofs of Proposition 1.2 and Theorem 1.3 follow from the same treatment. Note that the conclusion that u ≡ v ≡ c in Theorem 1.3 is equivalent to the pair of inequalities that u 2 + v 2 ≤ 2c 2 and uv ≥ c 2 , which resembles (1.5).

The rough idea of the proof is to bound the supremum of u 2 + v 2 in terms of the infimum of uv and vice versa in such a way that the bounds self-bootstrap to the desired bounds.

A non-degeneracy estimate

We start with a result which gives positive lower and upper bounds for u + v for positive solutions to (1.1). Lemma 2.1. Let (u, v) be a positive regular solution of (1.1). Then

1 + ω max( 2+α 4 , 1) 1/2 ≤ u + v ≤ 1 + ω min( 2+α 4 , 1) 1/2 in R N . In particular u, v ∈ L ∞ (R N ). Proof. Let w = u + v. Then ∆w = w(u 2 + v 2 + αuv -1 -ω). ( 2
and thus the rescaled function w(x) = max( (2+α) 1/2 2 , 1)(1 + ω) -1/2 w((1 + ω) -1/2 x) is a positive supersolution of the Allen-Cahn equation, i.e., it satisfies

-∆ w ≥ w 1 -w2 in R N . (2.5)
To proceed, we need the following lemma.

Lemma 2.2. There exists R 0 > 0 such that, for R ≥ R 0 , the functional

I[ϕ] = B R 1 2 |∇ϕ| 2 + 1 4 (1 -ϕ 2 ) 2 dx admits a non-trivial minimizer ψ in H 1 0 (B R ). Furthermore, ψ is a smooth function in B R satisfying 0 < ψ < 1 in B R .
Let us assume the above lemma for the moment and continue with the proof of Lemma 2.1. Let R 0 be the constant in Lemma 2.2 and m := min B R 0 w > 0 then, for every ε ∈ (0, min{m, 1}), the function ψ ε := εψ satisfies

-∆ψ ε ≤ ψ ε (1 -ψ 2 ε ) in R N ,
and

ψ ε ≤ w in B R 0 , ψ ε = 0 on ∂B R 0 .
The sliding method (see [START_REF] Berestycki | Monotonicity for elliptic equations in unbounded Lipschitz domains[END_REF]Lemma 3.1]), then gives that w ≥ ε in R N and so w ≥ δ in R N , for some δ > 0. Now, from (2.4) we get

-∆w ≥ w 1 + ω -max( 2 + α 4 , 1)w 2 ≥ δ max( 2 + α 4 , 1)(ϑ 2 -w 2 ) (2.6)
where ϑ = 1+ω max( 2+α 4 ,1)

1/2
. Hence, by Kato's inequality we have

∆(ϑ -w) + ≥ δ max( 2 + α 4 , 1)[(ϑ -w) + ] 2 (2.7)
which implies (ϑw) + ≡ 0 (see [START_REF] Brezis | Semilinear equations in R n without condition at infinity[END_REF]Lemma 2]), i.e.

w ≥ 1 + ω max( 2+α 4 , 1) 1/2 in R N . (2.8)
Proof of Lemma 2.2. We have that

I[0] = 1 4 |B R |.
Suppose that R > 1 and consider the function ϕ(x) = min(R -|x|, 1) for x ∈ B R .

We have

I[ϕ] = B R \B R-1 1 2 |∇ϕ| 2 + 1 4 (1 -ϕ 2 ) 2 dx ≤ 3 4 |B R \ B R-1 |.
Clearly, for R sufficiently large, I[ϕ] < I[0] and so I posseses a non-trivial minimizer ψ in H 1 0 (B R ). Replacing ψ by min{|ψ|, 1}, if necessary, we may assume that 0 ≤ ψ ≤ 1 in B R . Therefore ψ is a weak solution of the Allen-Cahn equation in B R and the remaining part of the claim follows by standard elliptic regularity and by the strong maximum principle.

Proof of Proposition 1.2

Proof. We have u, v > 0, and, by Lemma 2.1, there is some p > 1 such that 1. We prove that

u + v ≥ 1 p . ( 2
m ≤ 1 2 (1 + √ 1 -8s * ) where s * = min t≥e -n * (αt 2 -ωt). (2.11)
In particular, as s * ≥ -ω 2 4α , we have

m ≤ 1 2 1 + 1 + 2ω 2 α < ∞.
(2.12)

We have

∆A ≥ 2uP (u, v) + 2vQ(u, v) = 2u 4 + 2v 4 + 4(α + 1)u 2 v 2 -2u 2 -2v 2 -4ωuv = 2A 2 -2A + 4αe -2B -4ωe -B .
Note that αe -2Bωe -B ≥ s * , and, as

n * ≥ -ln ω α , s * ≤ 0. It follows that ∆A ≥ 2A 2 -2A + 4s * = 2 A - 1 2 (1 + √ 1 -8s * ) A - 1 2 (1 - √ 1 -8s * ) ,
and so, by Kato's inequality,

∆ A - 1 2 (1 + √ 1 -8s * ) + ≥ 2A 2 -2A + 4s * = 2 A - 1 2 (1 + √ 1 -8s * ) 2 + . Estimate (2.11) follows from [12, Lemma 2]. 2. We prove that n ≤ -ln ω α + 2 -2 m * . (2.13) Equivalently, uv ≥ ω α + 2 -2 m * .
(2.14)

We have ∆B ≥ -

1 u P (u, v) - 1 v Q(u, v) = -(α + 2)(u 2 + v 2 ) + 2 + ω(u 2 + v 2 ) uv = -(α + 2)A + 2 + ωAe B = 2(1 - A m * ) + (α + 2 - 2 m * )A( ω α + 2 -2 m * e B -1).
Using Kato's inequality, the inequality e x -1 ≥ 1 2 x 2 for x ≥ 0 and recalling (2.9), we get

∆(B -ln α + 2 -2 m * ω ) + ≥ α + 2 -2 m * 4p 2 (B -ln α + 2 -2 m * ω ) 2 + .
We deduce that (Bln

α+2-2 m * ω
) + ≡ 0, again thanks to [START_REF] Brezis | Semilinear equations in R n without condition at infinity[END_REF]Lemma 2]. This proves (2.13).

3. We prove that m * = 1 or n * =ln ω α . Assume by contradiction that the above does not hold. Then m = m * > 1 and

n = n * > -ln ω α . Let h 1 (t) = 1 2 1 + √ 1 -8αt 2 + 8ωt , (2.15)
h 2 (t) = ωt (α + 2)t -2 . (2.16) We claim that m ≤ h 1 (h 2 (m)).
(2.17)

Case (i): α ≥ 2. From (2.13), we have n ≤ -ln ω α+2 ≤ -ln ω 2α . It follows that s * = min t≥e -n (αt 2 -ωt) = αe -2n -ωe -n .
Plugging this into (2.11) yields

m ≤ 1 2 (1 + √ 1 -8αe -2n + 8ωe -n ) = h 1 (e -n ).
Since h 1 is decreasing in [ ω 2α , ∞), this together with (2.13) implies (2.17). Case (ii): α < 2. As α < 2, ω < α 2 < 2α 2-α and so, by (2.12),

m ≤ 1 2 1 + 1 + 2ω 2 α < 1 2 1 + 1 + 8α (2 -α) 2 = 2 2 -α .
Inserting this into (2.13) yields n ≤ln ω 2α . We can now repeat the proof of Case (i) to reach (2.17).

We now compute

d dt h 1 (h 2 (t)) = h ′ 1 (h 2 (t))h ′ 2 (t) = 2(2αh 2 (t) -ω) 1 -8αh 2 (t) 2 + 8ωh 2 (t) 2ω [(α + 2)t -2] 2 = 1 1 -8αh 2 (t) 2 + 8ωh 2 (t) 4ω 2 (αt -2t + 2) (αt + 2t -2) 3 . (2.18) Note that h 2 is decreasing in ( 2 2+α , ∞) and so h 2 (t) < ω α for t > 1. Hence, for t > 1, we have 1 -8αh 2 (t) 2 + 8ωh 2 (t) > 1, αt -2t + 2 < αt + 2t -2 and 4ω 2 < α 2 < (αt + 2t -2) 2 and so d dt h 1 (h 2 (t)) < 1 for all t > 1.
As h 1 (h 2 (1)) = 1, this implies that the inequality equation t ≤ h 1 (h 2 (t)) has no solution in (1, ∞). Therefore, (2.17) implies that m = 1. This finishes Step 3.

We prove (1.5). By

Step 3, we have m * = 1 or n * =ln ω α . If m * = 1, then, in view of (2.13), n ≤ln ω α , and so A ≤ 1 and B ≥ln ω α , which give (1.5).

On the other hand, if n * =ln ω α , then, by (2.11), m ≤ 1. Again we obtain A ≤ 1 and B ≥ln ω α , which also give (1.5) as desired. 5. Finally, we prove the trichotomy that either (u, v) ≡ (a, b) or (u, v) ≡ (b, a) or (1.6) holds.

Suppose that (u, v) ≡ (a, b) and (u, v) ≡ (b, a). From (1.5), we have

a ≤ u, v ≤ b. (2.19)
We note that

P (b, v) = b(b 2 + v 2 -1) + v(αbv -ω) (2.19) ≥ b(b 2 + a 2 -1) + v(αba -ω) = 0.
Hence the constant function b satisfies ∆b = 0 ≤ P (b, v).

Since u ≤ b, and ∆u = P (u, v), the strong maximum principle implies either u < b or u ≡ b. If the latter case holds, the second equation of (1.1) implies that v ≡ a, which contradicts our assumption that (u, v) ≡ (b, a). We thus have u < b.

The remaining inequalites in (1.6) are shown similarly using

P (a, v) = a(a 2 + v 2 -1) + v(αav -ω) (2.19) ≤ a(a 2 + b 2 -1) + v(αab -ω) = 0, Q(u, a) = a(u 2 + a 2 -1) + u(αua -ω) (2.19) ≤ a(b 2 + a 2 2 -1) + u(αba -ω) = 0, Q(u, b) = b(u 2 + b 2 -1) + u(αub -ω) (2.19) ≥ b(a 2 + a 2 2 -1) + u(αab -ω) = 0. 10
We omit the details.

Proof of Theorem 1.3

Proof. We adapt the proof of Proposition 1.2, as the conclusion is equivalent to the following pair of inequalities:

u 2 + v 2 ≤ 2(1 + ω) 2 + α and uv ≥ 1 + ω 2 + α .
By Lemma 2.1, (2.9) holds.

Let A = u 2 + v 2 , B = -ln(uv) and m = sup A, m * = max(m, 2(1 + ω) 2 + α ), n = sup B, and ñ * = max(n, -ln 1 + ω 2 + α ).
(Note the difference between the definition of m * and ñ * and that of m * and n * in the proof of Proposition 1.2.) 1. We prove that

m ≤ 1 2 min 1 + √ 1 -8s 1 , 1 + ω + (1 + ω) 2 -8αs 2 , (2.20)
where s 1 = min t≥e -ñ * (αt 2ωt) ≤ 0 and

s 2 = e -2ñ * ≤ (1+ω) 2 8α . As ñ * ≥ -ln 1+ω 2+α ≥ -ln ω α (due to ω ≥ 1 2 α), s 1 ≤ 0. Also s 2 ≤ (1+ω) 2 (2+α) 2 ≤ (1+ω) 2 8α . The proof of the inequality m ≤ 1 2 (1 + √ 1 -8s 1 ) follows from the differential inequality ∆A ≥ 2A 2 -2A + 4αu 2 v 2 -4ωuv,
exactly as in the proof of (2.11). To obtain m ≤

1 2 (1 + ω + (1 + ω) 2 -8αs 2 ), we use the inequality v 1 v 2 ≤ 1 2 A in the above differential inequality: ∆A ≥ 2A 2 -2(1 + ω)A + 4αe -2B ≥ 2A 2 -2(1 + ω)A + 4αs 2 = 2 A - 1 2 (1 + ω + (1 + ω) 2 -8αs 2 ) A - 1 2 (1 + ω -(1 + ω) 2 -8αs 2 ) .
By Kato's inequality, this leads to

∆ A - 1 2 (1 + ω + (1 + ω) 2 -8αs 2 ) + ≥ 2 A - 1 2 (1 + ω + (1 + ω) 2 -8αs 2 ) 2 +
and so, by [START_REF] Brezis | Semilinear equations in R n without condition at infinity[END_REF]Lemma 2],

A ≤ 1 2 (1 + ω + (1 + ω) 2 -8αs 2 ).
We have thus proved (2.20).

2. As in the proof of Proposition 1.2, we have

n ≤ -ln ω α + 2 -2 m * .
(2.21)

3. We show that m * = 2(1+ω) 2+α or ñ * =ln 1+ω 2+α . Assume by contradiction that this does not hold, so that m = m * > 2(1+ω) 2+α and n = ñ * >ln 1+ω 2+α . Case (a): 1+ω 2+α ≥ ω 2α (i.e. either α ≥ 2 or 0 < α < 2 and ω ≤ 2α 2-α ). In this case, the argument in Step 3 of the proof of Proposition 1.2 gives

m ≤ h 1 (h 2 (m)), (2.22)
where h 1 and h 2 are defined in (2.15)-(2. [START_REF] Farina | Symmetry for solutions of semilinear elliptic equations in R N and related conjectures[END_REF])

Now note that h 1 (h 2 ( 2(1+ω) 2+α )) = 2(1+ω) 2+α .
Thus in order to obtain a contradiction, it suffices to show that

d dt h 1 (h 2 (t)) < 1 for all t > 2(1 + ω) 2 + α . (2.23)
To see this, recall formula (2.18) for the derivative of h 1 • h 2 :

d dt h 1 (h 2 (t)) = 1 1 -8αh 2 (t) 2 + 8ωh 2 (t) 4ω 2 (αt -2t + 2) [(α + 2)t -2] 3 . Now if t > 2(1+ω) 2+α , then as h 2 is decreasing in ( 2 2+α , ∞), we have ω 2+α = h 2 (∞) < h 2 (t) < h 2 ( 2(1+ω) 2+α ) = 1+ω 2+α < ω α (thanks to α < 2ω), and so 1 -8αh 2 (t) 2 + 8ωh 2 (t) > 1.
Also, for t > where h 2 is defined in (2.16) and h1 is defined by

h1 (t) = 1 2 (1 + ω + (1 + ω) 2 -8αt 2 ).
Indeed, By (2.21), n ≤ln h 2 (m). Plugging this into (2.20), we get m ≤ h1 (e -n ) ≤ h1 (h 2 (m)), as h1 is decreasing in (0, ∞).

Next, a direct computation gives h1 (h 2 ( 2(1+ω) 2+α )) = 2(1+ω) 2+α . Thus, as in Case (a), it suffices to show that

d dt h1 (h 2 (t)) < 1 for all t > 2(1 + ω) 2 + α . (2.25)
We compute

d dt h 1 (h 2 (t)) = h ′ 1 (h 2 (t))h ′ 2 (t) = 4αh 2 (t) (1 + ω) 2 -8αh 2 (t) 2 2ω [(α + 2)t -2] 2 = 1 (1 + ω) 2 -8αh 2 (t) 2 8αωh 2 (t) [(α + 2)t -2] 2 . Now for t > 2(1+ω)
2+α , we have ω 2+α < h 2 (t) < 1+ω 2+α as in the previous case thanks to the monotonicity of h 2 . It follows that

d dt h 1 (h 2 (t)) < 1 (1 + ω) 2 -8α (1+ω) 2 (2+α) 2 8αω 1+ω 2+α 4ω 2 = 2α (2 -α)ω < 1 for all t > 2(1 + ω) 2 + α .
This proves (2.25), and so finishes Case (b).

Step 3 is concluded.

4. Finally, we show that u ≡ v ≡ 1+ω 2+α . By Step 3, we have m * = 2(1+ω) 2+α or ñ * =ln 1+ω 2+α . If m * = 2(1+ω) 2+α , then, in view of (2.13), n ≤ln 1+ω 2+α , and so A ≤ 2(1+ω) 2+α and B ≥ln 1+ω 2+α , which give u ≡ v ≡ 1+ω 2+α . On the other hand, if ñ * =ln 1+ω 2+α , then, by (2.11), m ≤ 2(1+ω) 2+α . Again we obtain A ≤ 2(1+ω) 2+α and B ≥ln 1+ω 2+α , which then give u ≡ v ≡ 1+ω 2+α . We conclude the proof.

3 Moving plane device

∂ N v < 0 in R N . (3.1)
The proof is based on the moving planes method, in a version developed by [START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF]. Nevertheless our system requires some major adjustments which rely on new bounds, and in particular the bounds from below for the product uv, as we will point out.

For λ ∈ R, we set

u λ (x ′ , x N ) := u(x ′ , 2λ -x N ), v λ (x ′ , x N ) := v(x ′ , 2λ -x N ) and Σ λ := {x N > λ}.
We want to prove that

u λ (x) ≤ u(x) and v λ (x) ≥ v(x) ∀x ∈ Σ λ , ∀λ ∈ R. (3.2)
This and the strong Maximum Principle will yield Proposition 3.1.

In order to prove that (3.2) holds, we will show that

Λ := {λ ∈ R : u µ ≤ u and v µ ≥ v in Σ µ for every µ ≥ λ} = R. (3.3)
In order to start the moving plane device, we will start by proving that Λ = ∅: Lemma 3.2. There exists λ ∈ R sufficiently large such that

u ≥ u λ and v ≤ v λ in Σ λ for any λ ≥ λ. In other words, Λ ⊃ [ λ, ∞).
The pair (u λ , v λ ) solves

     -∆u λ = g(u λ , v λ ) + v λ (ω -αu λ v λ ) -∆v λ = g(v λ , u λ ) + u λ (ω -αu λ v λ ) a < u λ , v λ < b (P λ ) where g(u, v) = u(1 -u 2 -v 2
) and where we have used (1.6). Let ϕ R be a standard

C 1 cut-off function on R N such that ϕ R = 1 in B R , ϕ R = 0 outside B 2R and |∇ϕ R | ≤ 2/R on R N .
We subtract the equations for u λ and u, and multiply by the test function

(u λ -u) + ϕ 2 R 1 Σ λ .
We find

Σ λ |∇(u λ -u) + | 2 ϕ 2 R = -2 Σ λ ϕ R (u λ -u) + ∇(u λ -u) + • ∇ϕ R + I 1 + I 2 , (3.4)
where

I 1 = Σ λ g(u λ , v λ ) -g(u, v) (u λ -u) + ϕ 2 R , (3.5)
I 2 = Σ λ v λ (ω -αu λ v λ ) -v(ω -αuv) (u λ -u) + ϕ 2 R . (3.6)
Let us point out that, in the above expressions, the term ωαuv is non-negative by our lower bound on uv. We will soon see that this sign of ωαuv plays an essential role in our argument.

We proceed similarly by subtracting the equations for v and v λ and multiplying by (v

-v λ ) + ϕ 2 R 1 Σ λ and get Σ λ |∇(v -v λ ) + | 2 ϕ 2 R = -2 Σ λ ϕ R (v -v λ ) + ∇(v -v λ ) + • ∇ϕ R + I 3 + I 4 (3.7)
where

I 3 = Σ λ g(v, u) -g(v λ , u λ ) (v -v λ ) + ϕ 2 R , (3.8)
I 4 = Σ λ u(ω -αuv) -u λ (ω -αu λ v λ ) (v -v λ ) + ϕ 2 R . (3.9) Let L λ (R) := Σ λ ∩B R |∇(u λ -u) + | 2 + |∇(v -v λ ) + | 2 , J λ (R) := Σ λ (u λ -u) + 2 + (v -v λ ) + 2 ϕ 2 R .
We deduce from (3.4)-(3.7) that for any ϑ ∈ (0, 1),

L λ (R) ≤ ϑL λ (2R) + 4 ϑR 2 J λ (R) + I 1 + I 2 + I 3 + I 4 . (3.10)
We want to estimate I 1 , I 2 , I 3 , I 4 in terms of J λ (R).

Estimate of I 2 and I 4 : We deduce from (3.6) that

I 2 = Σ λ (v -v λ )(αu(v + v λ ) -ω)(u λ -u) + -αv 2 λ (u λ -u) + 2 ϕ 2 R . (3.11)
We recall from Proposition 1.2 that αuvω ≥ 0, which is a crucial estimate. So that in the first term in the square bracket on the right hand side of (3.11), we can keep only (vv λ ) + in the upper bound and find

I 2 ≤ Σ λ (αuv-ω)(v-v λ ) + (u λ -u) + +αuv λ (v-v λ ) + (u λ -u) + -αv 2 λ (u λ -u) + 2 ϕ 2 R . ( 3
.12) A similar computation for I 4 yields

I 4 ≤ Σ λ (αu λ v λ -ω)(v-v λ ) + (u λ -u) + +αuv λ (v-v λ ) + (u λ -u) + -αu 2 (v-v λ ) + 2 ϕ 2 R . (3.13)
Because of (1.2), for λ large enough, in Σ λ , u tends to b, v tends to a and uv tends to ab = ω/α. Moreover in the support of (vv λ ) + , v ≥ v λ ≥ a, so v λ also tends to a, and in the support of (u λu) + , b ≥ u λ ≥ u, so u λ tends to b. This implies that for λ large enough, in Σ λ , u λ v λ also tends to ab = ω/α. Therefore, for some small ε > 0 which will be fixed later, there exists λ large enough, so that for λ ≥ λ, in Σ λ ∩ S, where S is the intersection of the supports of (u λu) + and (v

-v λ ) + , |(αu λ v λ -ω) + (αuv -ω) + 2(αuv λ -ω)| ≤ ε. (3.14)
Moreover, for λ large enough, and for λ ≥ λ, in Σ λ , α|u 2b 2 | ≤ ε/2. We sum the two estimates (3.12) and (3.13), use this, (3.14) and the fact that v λ ≥ a, to find

I 2 + I 4 ≤ 2εJ λ (R) -α Σ λ a 2 (u λ -u) + 2 + b 2 (v -v λ ) + 2 -2ab(v -v λ ) + (u λ -u) + .
(3.15) Estimate of I 1 and I 3 : By the mean value theorem, there exist ξ 1 (x) ∈ (u(x), u λ (x)) and ξ 2 (x) ∈ (v λ (x), v(x)) such that

g(u λ , v λ ) -g(u, v) = ∂g ∂u (ξ 1 , v λ )(u λ -u) + ∂g ∂v (u, ξ 2 )(v λ -v). Since g(u, v) = u(1 -u 2 -v 2 ), we have ∂g ∂v (u, ξ 2 ) ≤ 0. Hence g(u λ , v λ )-g(u, v) (u λ -u) + ≤ ∂g ∂u (ξ 1 , v λ ) (u λ -u) + 2 + ∂g ∂v (u, ξ 2 ) (v -v λ ) + (u λ -u) + . Moreover, ∂g ∂u (ξ 1 , v λ ) = 1 -3ξ 2 1 -v 2 λ , ∂g ∂v (u, ξ 2 ) = -2uξ 2 .
Next, note that, as λ → ∞, u and u λ tend to b in Σ λ ∩ {u λ > u} and v and v λ tend to a in Σ λ ∩ S. Hence, we have in Σ λ ∩ {u λ > u}, lim sup

λ→∞ ∂g ∂u (ξ 1 , v λ ) ≤ -2b 2 , in Σ λ ∩ S, lim λ→∞ ∂g ∂v (u, ξ 2 ) = -2ab.
Therefore, in view of (3.5) and by enlarging λ if necessary we have for λ ≥ λ that

I 1 ≤ Σ λ (-2b 2 + ε) (u λ -u) + 2 + (2ab + ε)(v -v λ ) + (u λ -u) + ϕ 2 R . (3.16)
We argue similarly to I 1 for I 3 and find

I 1 + I 3 ≤ ε 2 J λ (R) -2 Σ λ b 2 (u λ -u) + 2 + a 2 (v -v λ ) + 2 -2ab(v -v λ ) + (u λ -u) + .
(3.17)

Combining the estimates, we get

I 1 + I 2 + I 3 + I 4 ≤ 5ε 2 J λ (R) - Σ λ (αa 2 + 2b 2 ) (u λ -u) + 2 + (αb 2 + 2a 2 ) (v -v λ ) + 2 -2(2 + α)ab(v -v λ ) + (u λ -u) + .
Note that the matrix

M := αa 2 + 2b 2 -(2 + α)ab -(2 + α)ab αb 2 + 2a 2
has determinant 2α(a 2b 2 ) 2 which is positive as soon as (1.4) holds. Hence M has positive eigenvalues µ 1 > µ 2 > 0. In the sequel, we assume that µ 2 > 3ε. Inserting (3.15) and (3.17) into (3.10), for ε > 0, there exists λ > 0, such that we for λ > λ and large R that

L λ (R) ≤ ϑL λ (2R) + (-µ 2 + 3ε)J λ (R). (3.18)
We fix ϑ := 2 -(N +1) and some ε > 0 such that µ 2 -3ε > 0. Then for λ > λ and large R,

L λ (R) ≤ ϑL λ (2R). (3.19)
The conclusion will follow from a lemma proved and used in [START_REF] Farina | Monotonicity and one-dimensional symmetry for solutions of -∆ p u = f (u) in half-spaces[END_REF] which allows to show that L λ (R) is identically zero.

Lemma 3.3 ([18, Lemma 2.1]

). Let ϑ > 0 and γ > 0 such that ϑ < 2 -γ . Moreover let R 0 > 0, C > 0 and L : (R 0 , +∞) → R be a non-negative and non-decreasing function such that

L(R) ≤ ϑL(2R) + G(R) ∀R > R 0 , L(R) ≤ CR γ ∀R > R 0 , where G : (R 0 , +∞) → R + is such that lim R→+∞ G(R) = 0. Then L(R) = 0 ∀R > R 0 .
We have that L λ (R) ≤ CR N since |∇u|, |∇v| ∈ L ∞ (R N ) by elliptic estimates and the L ∞ bound of Lemma 2.1. Lemma 3.3 then yields that L λ (R) = 0 for all λ > λ and large R.

Recalling that u = u λ and v = v λ on ∂Σ λ , we reach the conclusion of Lemma 3.2.

We now have to prove that λ := inf Λ (with Λ defined in (3.3)) is -∞ to complete the proof of Proposition 3.1: Lemma 3.4. We have λ = -∞.

Proof. Assume by contradiction that λ is finite. Then, Λ = [ λ, +∞), and there exist sequences (λ i ) with λ i ∈ (-∞, λ) and (x i ) with x i ∈ Σ λ i such that λ i → λ as i → ∞, and at least one of the two holds:

u λ i (x i ) > u(x i ) for every i, or (3.20a)
v λ i (x i ) < v(x i ) for every i. (3.20b)
Assume that (3.20a) holds; the other case can be treated similarly. We claim that the sequence (

x i N ) ⊂ R is bounded. If not, as x i N > λ i and λ i is bounded, up to a subsequence x i N → +∞ as i → ∞. It follows that 2λ i -x i N → -∞,
u λ i (x i ) = lim i→∞ u((x i ) ′ , 2λ i -x i N ) = a and lim i→∞ u(x i ) = b,
in contradiction with (3.20a) for i sufficiently large. Hence the claim is proved and, up to a subsequence,

x i N → xN as i → ∞.
Let us set

u i (x) := u((x i ) ′ + x ′ , x N ) and v i (x) := v((x i ) ′ + x ′ , x N ).
Since (u, v) is bounded (in view of (1.5)), by standard elliptic estimates |∇ k u|, |∇ k v| ∈ L ∞ (R N ), for k = 1, 2, . . . Thus, after extracting a subsequence if necessary, (u i , v i ) converges in C 2 loc (R N ) to a limit (ū, v), still solution of (1.1). We wish to show that xN = λ. From equation (3.20a), we obtain

ūλ (0 ′ , xN ) = ū(0 ′ , 2 λ -xN ) = lim i→∞ u((x i ) ′ , 2λ i -x i N ) = lim i→∞ u λ i (x i ) ≥ lim i→∞ u(x i ) = ū(0 ′ , xN ). (3.21)
On the other hand, we observe that ((

x i ) ′ + x ′ , x N ) ∈ Σ λ whenever (x ′ , x N ) ∈ Σ λ,
and by definition u λ ≤ u in Σ λ. Consequently, by the convergence of u i to ū we deduce that

ūλ (x ′ , x N ) = lim i→∞ u i (x ′ , 2 λ -x N ) = lim i→∞ u((x i ) ′ + x ′ , 2 λ -x N ) ≤ lim i→∞ u((x i ) ′ + x ′ , x N ) = lim i→∞ u i (x ′ , x N ) = ū(x ′ , x N ) for every (x ′ , x N ) ∈ Σ λ. Analogously, as v λ ≥ v in Σ λ, we have vλ ≥ v in Σ λ. Now    -∆(ū -ūλ ) + c(x)(ū -ūλ ) = (v λ -v)(αūv -ω + αū λ vλ + ūλ (v λ + v)) ū -ūλ ≥ 0 in Σ λ ū -ūλ = 0 on ∂Σ λ, (3.22) with c ∈ C 0 (Σ λ) defined by c(x) = αv(x)v λ(x) + v2 (x) + ū2 (x) + ūū λ(x) + ū2 λ(x) -1.
Because of (1.5), the right hand side on the first line of (3.22) is nonnegative, hence, the strong Maximum Principle implies that necessarily ūūλ > 0 in Σ λ, and a comparison with (3.21) reveals that xN = λ, as desired.

At this point we are ready to reach a contradiction. On the one hand, by the absurd assumption (3.20a)

0 < u λ i (x i ) -u(x i ) = u i (0 ′ , 2λ i -x i N ) -u i (0 ′ , x N ) = 2∂ N u i (0 ′ , ξ i )(λ i -x i N ) ∀i,
for some ξ i ∈ (2λ ix i N , x i N ). As λ i < x i N for every i this implies ∂ N u i (x ′ , ξ i N ) < 0 for every i, and passing to the limit we infer that

∂ N ū(0 ′ , λ) ≤ 0, (3.23)
where we used the fact that λ i ≤ ξ i ≤ x i N with λ i , x i N → λ. On the other hand, thanks to (3.22) and the fact that ūūλ > 0 in Σ λ, the Hopf Lemma implies that

-2∂ N ū(0 ′ , λ) = ∂ -e N (ū(0 ′ , λ) -ūλ (0 ′ , λ)) < 0,
in contradiction with (3.23).

The above argument establishes that (3.20a) cannot occur. With minor changes, we can show that also (3.20b) cannot be verified, and in conclusion λ cannot be finite.

Proof of Proposition 3.1. By (3.3), we directly deduce that ∂ N u ≥ 0 and

∂ N v ≤ 0 in R N . Since -∆(∂ N u) + (3u 2 + (α + 1)v 2 -1)∂ N u = (ω -2(α + 1)uv)∂ N v ≥ 0 in R N -∆(∂ N v) + (3v 2 + (α + 1)u 2 -1)∂ N v = (ω -2(α + 1)uv)∂ N u ≤ 0 in R N ,
the strict inequality follows by the strong Maximum Principle.

One-dimensional symmetry

We extend the monotonicity in x N to all the directions of the open upper hemisphere S N -1 + := ν ∈ S N -1 : ν, e N > 0 . We follow the structure of proof in [START_REF] Farina | Monotonicity and rigidity of solutions to some elliptic systems with uniform limits[END_REF], introduced in [START_REF] Farina | Symmetry for solutions of semilinear elliptic equations in R N and related conjectures[END_REF] and in [START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation[END_REF], though the specificity of our system requires new estimates. Proposition 3.5. For every ν ∈ S N -1 + , we have

∂ ν u > 0 and ∂ ν v < 0 in R N .
In particular, u and v depend only on x N .

We divide the proof into several steps. Proof. Let σ > 0 be arbitrarily chosen. Firstly, we claim that there exists ε = ε(σ) > 0 such that

∂ N u(x) ≥ ε and ∂ N v(x) ≤ -ε ∀x ∈ S σ . (3.24)
By contradiction, assume that there exists a sequence (x i ), with x i ∈ S σ , such that at least one of the two following equalities holds :

lim i→+∞ ∂ N u(x i ) = 0 (3.25a) or lim i→+∞ ∂ N v(x i ) = 0. (3.25b)
We define

u i (x) := u(x + x i ) and v i (x) := v(x + x i ). The sequence {(u i , v i )} is uniformly bounded in W 1,∞ (R N )
, and hence by elliptic regularity (u i , v i ) → (ū, v) in C 2 loc (R N ) up to a subsequence, where (ū, v) is still a solution to (1.1)-(1.2), which also satisfies ∂ N ū ≥ 0 and ∂ N v ≤ 0 on R N . The strong maximum principle and the condition at infinity (1.2) then imply that ∂ N ū > 0 and ∂ N v < 0 on R N , and this contradicts (3.25a) or (3.25b). This completes the proof of claim (3.24).

Now we claim that

The map ν

→ (∂ ν u, ∂ ν v) is in C 0,1 S N -1 , C 0 (R N ) 2 . (3.26)
This is a simple consequence of the globlal Lipschitz continuity of (u, v), which implies that ∂u ∂ν 1

(x) - ∂u ∂ν 2 (x) + ∂v ∂ν 1 (x) - ∂v ∂ν 2 (x) ≤ 2C|ν 1 -ν 2 |
for every x ∈ R N . Combining (3.24) and (3.26), the conclusion follows.

Lemma 3.7. The function u is strictly increasing and v is strictly decreasing with respect to all unit vectors in an open neighborhood of e N in S N -1 .

Proof. Firstly, we write down the equations satisfied by the directional derivatives

u ν = ∂ ν u and v ν = ∂ ν v: -∆u ν + u ν (3u 2 + (α + 1)v 2 -1) + v ν (2(α + 1)uv -ω) = 0 -∆v ν + v ν (3v 2 + (α + 1)u 2 -1) + u ν (2(α + 1)uv -ω) = 0 in R N . (3.27)
Fix some σ > 0 for the moment and let O e N be the neighborhood of e N given by Lemma 3.6. We will show that u ν ≥ 0 and v ν ≤ 0 for all ν ∈ O e N by applying Lemma 3.3 to the quantity

I R := C R |∇u - ν | 2 + |∇v + ν | 2 ,
where C R := Σ σ ∩ B R and Σ σ := {x N > σ}. The conclusion then follows from the strong maximum principle. We test the first equation in (3.27) with u - ν ϕ 2 R where ϕ R is chosen exactly as in Lemma 3.2. Using the bounds (1.5) and the fact that u ν ≥ 0 on {x N = σ} (due to Lemma 3.6), we obtain

C R |∇u - ν | 2 ≤ -2 C 2R u - ν ϕ R ∇u - ν • ∇ϕ R - C 2R (3u 2 + (α + 1)v 2 -1)(u - ν ϕ R ) 2 + C 2R ϕ 2 R (2(α + 1)uv -ω)u - ν v + ν ≤ ϑ C 2R |∇u - ν | 2 + C 2R (u - ν ϕ R ) 2 4 ϑR 2 + sup Σσ (-3u 2 -(α + 1)v 2 + 1) + C 2R ϕ 2 R (2(α + 1)uv -ω)u - ν v + ν ,
where 0 < ϑ < 2 -N . In a similar way, we find for v

+ ν C R |∇v + ν | 2 ≤ ϑ C 2R |∇v + ν | 2 + C 2R (v + ν ϕ R ) 2 4 ϑR 2 + sup Σσ (-3v 2 -(α + 1)u 2 + 1) + C 2R ϕ 2 R (2(α + 1)uv -ω)u - ν v + ν .
We notice that if σ > 0 is sufficiently large, since u tends to b and v tends to a for x N large, then, in Σ σ ,

2(α + 1)uv -ω → 2(α + 1)ab -ω, 3u 2 + (α + 1)v 2 -1 → 2b 2 + αa 2 ,
and 3v 2 + (α + 1)u 2 -1 → 2a 2 + αb 2 .
Thus, for any small δ > 0, we can choose σ and R sufficiently large so that

I R ≤ ϑI 2R - C 2R (2b 2 + αa 2 -δ)(u - ν ) 2 + (2a 2 + αb 2 -δ)(v + ν ) 2 -2(2(α + 1)ab -ω + δ)|u - ν ||v + ν | ϕ 2 R . (3.28)
We point out that

(2b 2 +αa 2 -δ)(u - ν ) 2 +(2a 2 +αb 2 -δ)(v + ν ) 2 ≥ 2 (2b 2 + αa 2 -δ)(2a 2 + αb 2 -δ)|u - ν ||v + ν |. and that, by (1.3) and (1.4), (2b 2 + αa 2 )(2a 2 + αb 2 ) -(2(α + 1)ab -ω) 2 = 2α(a 2 + b 2 ) 2 -3α(α + 4)a 2 b 2 + 4ω(α + 1)ab -ω 2 = 2α 1 - 4ω 2 α 2 > 0.
Hence, by choosing first small δ and then large σ from the start, we have for all sufficiently large R that the integral on the right hand side of (3.28) is non-negative.

As a consequence, we infer that

I R ≤ ϑI 2R for all R sufficiently large. (3.29)
We can now apply Lemma 3.3 to find that I R = 0 for all large R. It follows that u ν ≥ 0 and v ν ≤ 0 in Σ σ = {x N > σ}. Arguing exactly in the same way, we can show that the same conditions are satisfied in {x N < -σ}. By Lemma 3.6, we deduce that u ν ≥ 0 and v ν ≤ 0 in R N for every ν ∈ O e N , with both u ν ≡ 0 and v ν ≡ 0. In view of (1.5) and (3.27), the conclusion follows from the strong maximum principle.

Proof of Proposition 3.5. Here we can essentially apply the same argument used in step 4 of Proposition 6.1 in [START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation[END_REF]. We report the details for completeness. Let Ω be the set of the directions ν ∈ S N -1

+
for which there exists an open neighborhood O ν ⊂ S N -1

+ of ν such that ∂ µ u > 0 and ∂ µ v < 0 in R N , ∀µ ∈ O ν .
The set Ω is open by definition, and contains e N by Lemma 3.7. Since S N -1

+ is arc- connected, if we can show that ∂Ω ∩ S N -1 + = ∅, then we conclude that Ω = S N -1 + , as desired. Thus, let us suppose by contradiction that ν ∈ ∂Ω ∩ S N -1 + (notice in particular that e N , ν > 0). By definition, there exists (ν n ) ⊂ Ω such that ν n → ν. As ∂ νn u > 0 and ∂ νn v < 0 in R N , ∀n, by continuity ∂ ν u ≥ 0 and ∂ ν v ≤ 0 in R N .
By the strong maximum principle, recalling that (u ν , v ν ) solves (3.27), either u ν ≡ 0 or u ν > 0 in R N , and analogously either v ν ≡ 0 or v ν < 0 in R N . The alternatives u ν ≡ 0 and v ν ≡ 0 are in contradiction with assumption (1.2), since ν is not orthogonal to e N , and hence

∂ ν u > 0 and ∂ ν v < 0 in R N . (3.30)
Having established (3.30), it is possible to adapt the same proof of Lemmas 3.6 and 3.7, with ν instead of e N , to deduce that u ν > 0 and

v ν < 0 in R N in all the directions of an open neighborhood O ν of ν in S N -1 +
. Thus, we have that ν ∈ Ω ∩ ∂Ω, in contradiction with the openness of Ω. This shows that ∂Ω ∩ S N -1 + = ∅ which, as already observed, implies Ω = S N -1 + . Finally, the fact that Ω = S N -1 + implies that both ∂ τ u ≡ 0 and ∂ τ v ≡ 0 for every τ ∈ S N -1 orthogonal to e N , which proves the last assertion.

4 Existence and uniqueness of positive 1D solutions when 2ω < α. Proof of Theorem 1.1.

In this section, we assume that 0 < ω < 1 2 α unless otherwise stated. By Proposition 3.5, positive solutions to (1.1)-(1.2) depend only on x N and are monotone. To conclude the proof of Theorem 1.1, it remains to prove the uniqueness up to translations of such one-dimensional solutions.

We are led to consider on R the system

u ′′ = P (u, v), v ′′ = Q(u, v), (4.1)
subject to (u, v) → (a, b) at -∞ and (u, v) → (b, a) at ∞. (4.2)
The main result of this section is: The proof of the 'uniqueness' part in Proposition 4.1 uses the sliding method (cf. [START_REF] Berestycki | Monotonicity for elliptic equations in unbounded Lipschitz domains[END_REF][START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains[END_REF][START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF]) with the help of the bounds (1.5) as well as the following lemma on the asymptotic behavior of solutions. Let

λ ± := 1 2 (α + 2) ± (α -2) 2 + 32ω 2 α , (4.3)
µ := 2ω(α + 2) α (α -2) 1 -4ω 2 α 2 + (α -2) 2 + 32ω 2 α > 0, (4.4)
which are related to the eigenvalues and eigenvectors of the linearized operator associated with (4.1) near the critical point (a, b). We refer to Appendix A for a brief discussion on the origin of these constants. It should be noted that the asymptotic behavior of solutions changes somewhat when ω = 0. See Lemma A.2 in the appendix.

An easy variational argument gives existence of the positive solutions to (4.1) on finite intervals. The sliding method can be adapted to this case yielding: Lemma 4.3. Suppose that 0 < ω < 1 2 α. For every R ∈ (0, ∞), there exists a unique positive solution to (4.1) in (-R, R) satisfying For n = 1, 2, . . ., let (u n , v n ) be the positive solution to (4.1) obtained in Lemma 4.

(u(-R), v(-R)) = (a, b) and (u(R), v(R)) = (b, a). (4.7) Furthermore, u ′ > 0 and v ′ < 0 in (-R, R).
3 with R = n. Fix x n ∈ (-n, n) such that u n (x n ) = 1 2 (a + b). Define l n = -n -x n , r n = n -x n , and (ũ n (x), ṽn (x)) = (u n (x + x n ), v n (x + x n )) for x ∈ [l n , r n ]. By Lemma 4.3, a ≤ u n , v n ≤ b.
Using elliptic estimates on unit closed subintervals of [l n , r n ], we have

|ũ (k) n | + |ṽ (k) n | ≤ C in [l n , r n ] for k = 0, 1, 2, 3, (4.8)
where C is a positive constant independent of n and k. Then, passing to a subsequence if necessary, we may assume that l n → l * ∈ [-∞, 0], r n → r * ∈ [0, ∞] (where l * and r * cannot be simultaneously finite), (ũ n , ṽn ) converges in C 2 loc (l * , r * ) to some (u * , v * ) satisfying (4.1), u ′ * ≥ 0, v ′ * ≤ 0 in (l * , r * ). Note that for each n, the Hamiltonian

h n := 1 2 (|ũ ′ n | 2 + |ṽ ′ n | 2 ) - 1 4 (1 -ũ2 n -ṽ2 n ) 2 - α 2 (ũ n ṽn - ω α ) 2
is constant in [l n , r n ]. As (ũ(R), ṽ(R)) = (b, a), it follows that h n ≥ 0 and so

h * := 1 2 (|u ′ * | 2 + |v ′ * | 2 ) - 1 4 (1 -u 2 * -v 2 * ) 2 - α 2 (u * v * - ω α ) 2 ≥ 0.
As said above, one has that l * = -∞ or r * = ∞ (or both). We will only treat the case that r * = ∞; the other case can be dealt with similarly. In this case we have u * (x) ≥ 1 2 (a + b) for every x > 0, since x > 0 ≥ l * implies ũn (x) ≥ ũn (0) = a+b

(x), v ′ * (x)) → 0 as x → ∞, 0 ≤ h * = - 1 4 (1 -b2 -ã2 ) 2 - α 2 ( bã - ω α ) 2
and so h * = 0, b2 +ã 2 = 1 and bã = ω α . As

(x), v * (x)) → (b, a) as x → ∞. Now if l * is finite, (4.8) yields |ũ (k) * | + |ṽ (k) * | ≤ C in (l * , r * ) for k = 0, 1, 2, (4.9)
where C is a positive constant independent of k, and so u * extends to a C 1 function in [l * , r * ). Now, since for every x ∈ (l * , r * ), we have x ∈ [l n , r n ] for large n, from (4.8) we also get that

|u * (x) -a| = lim n→∞ |ũ n (x) -a| = lim n→∞ |ũ n (x) -ũn (l n )| ≤ lim sup n→∞ C|x -l n | = C|x -l * |
which leads to u * (l * ) = a. A similar argument gives v * (l * ) = b. Now, by the strong maximum principle and the Hopf lemma (see the argument in Step 5 of the proof of Proposition 1.2), we have u ′ * (x) > 0 and v ′ * (x) < 0 for every x ∈ [l * , r * ), and using those properties with x = l * we get h * > 0, which contradicts the previous conclusion that h * = 0. Hence l * = -∞. As above, this implies that (u * (x), v * (x)) → (a, b) as x → -∞.

We have thus shown that (u * , v * ) is a positive and strictly monotone solution to (4.1)-(4.2), as desired.

2. We use the sliding method to show that positive solutions to (4.1)-(4.2) are translations of one another.

Let (u, v) and (ū, v) be two positive solutions to (4.1)-(4.2). By Lemma 4.2, the limits

ℓ + 1 := lim x→∞ (b -u(x))e -λ -x , l+ 1 := lim x→∞ (b -ū(x))e -λ -x , ℓ + 2 := lim x→∞ (v(x) -a)e -λ -x , l+ 2 := lim x→∞ (v -a)e -λ -x , ℓ - 1 := lim x→-∞ (u(x) -a)e λ -x , l- 1 := lim x→-∞ (ū(x) -a)e λ -x , ℓ - 2 := lim x→-∞ (b -v(x))e λ -x , l- 2 := lim x→-∞ (b -v)e λ -x
exist and are positive. Thus, in view of (1.6), there is some large T 0 such that

u(x -T 0 ) ≤ ū(x) ≤ u(x + T 0 ) and v(x -T 0 ) ≥ v(x) ≥ v(x + T 0 ) for all x ∈ R. Let T = inf{t ∈ [-T 0 , T 0 ] : ū(x) ≤ u(x+s) and v(x) ≥ v(x+s) for all t ≤ s ≤ T 0 , x ∈ R}.
Set ũ(x) = u(x + T ) and ṽ(x) = u(x + T ). Then ũ ≥ ū and ṽ ≤ v. The result will follow once we show that ũ ≡ ū and ṽ ≡ v. Assume by contradiction that this does not hold. a. We show that ũ > ū and ṽ < v. (4.10)

Note that ũ ≥ ū, ṽ ≤ v, and (ũ, ṽ) is also a solution to (4.1) satisfying (4.7). Also, we have

P (ũ, ṽ) -P (ũ, v) = (ṽ -v) (1 + α)ũ(ṽ + v) -ω ≤ (ṽ -v) (1 + α)ũṽ -ω (1.5) ≤ (ṽ -v) (1 + α) ω α -ω ≤ 0, and
Q(ũ, ṽ) -Q(ū, ṽ) = (ũ -ū) (1 + α)ṽ(ũ + ū) -ω ≥ (ũ -ū) (1 + α)ṽũ -ω (1.5) ≥ (ũ -ū) (1 + α) ω α -ω ≥ 0. So we have ũ′′ ≤ P (ũ, v), ū′′ = P (ū, v) ṽ′′ ≥ Q(ū, ṽ), v′′ = Q(ū, v).
Assertion (4.10) thus follows from the strong maximum principle.

b. We proceed to deduce a contradiction. Define

l+ 1 := lim x→∞ (b -ũ(x))e -λ -x , l+ 2 := lim x→∞ (ṽ(x) -a)e -λ -x , l- 1 := lim x→-∞ (ũ(x) -a)e λ -x , l- 2 := lim x→-∞ (b -ṽ(x))e λ -x .
Recall that, by Lemma 4.2, or both. Because of the unique correspondence of linearized solutions and nonlinear solutions near a hyperbolic critical point of ODEs [14, Chapter 13, Theorem 4.5] (with p = 1) -see the argument leading to (A.10) -we then deduce that (ū, v) ≡ (ũ, ṽ), which contradicts (4.10). We conclude the proof.

ℓ + 2 = µℓ + 1 , ℓ - 1 = µℓ -

Proof of Lemma 4.3

The existence in Lemma 4.3 follows from an easy variational argument. As far as we are concerned with the application of Lemma 4.3 to the proof of Proposition 4.1, it is enough to show that u and v are monotone. This can be done as in Subsection 3.1.

Here we provide an alternative proof which yields also uniqueness, which echoes the argument in Step 2 of the proof of Proposition 4.1.

We start with an adaptation of Proposition 1.2 for finite domains.

Lemma 4.4. Suppose that 0 < ω <1 2 α and R ∈ (0, ∞), and let (u, v) be a positive solution of (4.1) in (-R, R) satisfying (4.7). Then (1.5) and (1.6) hold in [-R, R].

Proof. The proof is similar to though easier than that of Proposition 1.2, thanks to the boundary condition (4.7). We will only give a sketch.

Let A = u 2 + v 2 , B =ln(uv) and define m = max A and n = max B. 1 If m is attained at the endpoints, we have m ≤ 1. Otherwise, m = A(x 0 ) for some x 0 ∈ (-R, R). We then have 0 ≥ A ′′ (x 0 ) ≥ 2u(x 0 )P (u(x 0 ), v(x 0 )) + 2v(x 0 )Q(u(x 0 ), v(x 0 )) ≥ 2A 2 (x 0 ) -2A(x 0 ) + 4s * where s * := min t≥e -n (αt 2ωt).

In either case, we obtain We can now follows exactly the arguments in Steps 3 and 4 of the proof of Proposition 1.2 to reach the conclusion. We omit the details.

Proof of Lemma 4.3. Note that (4.1) is the Euler-Lagrange equation for the functional

I[u, v] = R -R 1 2 (|u ′ | 2 + |v ′ | 2 ) + 1 4 (1 -u 2 -v 2 ) 2 + α 2 (uv - ω α ) 2 dx.
The existence of a positive solution to (4.1) satisfying (4.7) follows from a simple variational argument. The uniqueness follows from the sliding method as we have seen earlier. Suppose that (u, v) and (ū, v) are positive solutions of (4.1) in (-R, R) satisfying (4. If c > 0, then we have w 1,c ≥ u in [R, ∞), w 1,c (R) > u(R), lim x→∞ (w 1,cu) > 0, and there is some x 1 ∈ (R, ∞) such that w 1,c (x 1 ) = u(x 1 ), which gives a contradiction to the strong maximum principle, in view of (A.13) and (A.14). We thus have that c = 0, which implies that w 1,0 ≥ u in [R, ∞), which gives (A.12).

2 A

 2 priori bounds. Proofs of Proposition 1.2 and Theorem 1.3.

. 9 )

 9 Let A = u 2 + v 2 , B =ln(uv) and m = sup A, m * = max(m, 1), n = sup B, and n * = max(n,ln ω α ). (2.10)

13 3. 1 Proposition 3 . 1 .

 13131 Monotonicity with respect to x NWe are going to show that if (u, v) is a solution to (1.1)-(1.2), then it is monotone with respect to x N . This relies strongly on the estimates (1.5)-(1.6). Under the assumptions of Theorem 1.1, since (1.5)-(1.6) hold, we have ∂ N u > 0 and

Lemma 3 . 6 .

 36 Let σ > 0 be arbitrarily chosen. There exists an open neighborhood O e N of e N in S N -1 such that ∂u ∂ν (x) > 0 and ∂v ∂ν (x) < 0 ∀x ∈ S σ , ∀ν ∈ O e N , where S σ := R N -1 × (-σ, σ).

Proposition 4 . 1 .

 41 Suppose that 0 < ω < 1 2 α. Then there exist positive solutions to (4.1)-(4.2), and these solutions are translations of one another, i.e. if (u, v) and (ū, v) both satisfy (4.1)-(4.2), then is a constant T such thatū(x) = u(x + T ) and v(x) = v(x + T ).Furthermore, u ′ > 0 and v ′ < 0 in R.Proof of Theorem 1.1. The result is a consequence of Propositions 3.5 and 4.1.

Lemma 4 . 2 .

 42 Suppose that 0 < ω < 1 2 α. Let (u, v) be a positive solution of (4.1)-(4.2) and λ -and µ be defined by (4.3)-(4.4). Then the limits ℓ 1 := lim x→∞ (bu(x))e λ -x and ℓ 2 := lim x→∞ (v(x)a)e λ -x exist,

4. 1

 1 Proof of Proposition 4.1 Let us assume Lemmas 4.2 and 4.3 for the moment and proceed with the proof of Proposition 4.1. Lemma 4.3 will be proved in the next subsection. Lemma 4.2 follows from a routine asymptotic analysis near a hyperbolic critical point for ODEs. Its proof is postponed to Appendix A. Proof. 1. We prove the existence of a solution to (4.1)-(4.2) by sending R → ∞ in Lemma 4.3, where some care is needed to show that the solutions on finite intervals do not flatten to the constant solutions (a, a) or (b, b).

2

  for large n. Then, by the monotonicity of u * and v * , as x → ∞, (u * (x), v * (x)) has a limit, say ( b, ã), which satisfies1 2 (a + b) ≤ b ≤ b and a ≤ ã ≤ b. By (4.1), (u ′′ * (x), v ′′ * (x)) tends to (P ( b, ã), Q( b, ã)) as x → ∞. Applying the mean value theorem to u| [n,n+1/2] , we can find ξ n ∈ (n, n + 1/2) such that u ′ * (ξ n ) → 0. Likewise, there exist η n ∈ (ξ n , ξ n+1 ) such that u ′′ * (η n ) → 0.It follows that P ( b, ã) = 0. This then implies that sup [ξn-2,ξn+2] u ′ * → 0, and so u ′ * (x) → 0 as x → ∞. Similarly, Q( b, ã) = 0 and v ′ * (x) → 0 as x → ∞. Now, note that the equation P (x, y) = Q(x, y) = 0 has three solutions in the positive quadrant, namely (a, b), (b, a) and (c, c) where c = 1+ω 2+α ∈ (a, b). Also, as (u ′ *

1 2 (

 2 a+b) ≤ b ≤ b, we thus have ( b, ã) = (b, a), i.e. (u *

2

 2 and similar relations hold for the counterparts with bar and tilde on top. By the minimality of T and (4.10), we have that l+

≥ u(u 2 + v 2 - 1 )( 1 . 5 ) 3 .

 21153 [START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains i[END_REF]. Extend (u, v) to the whole of R by defining (u, v) ≡ (a, b) on (-∞, -R) and (u, v) ≡ (b, a) on (R, ∞). LetT = inf{t ∈ [0, 2R] : ū(x) ≤ u(x + s) and v(x) ≥ v(x + s) for all x ∈ [-R, R], t ≤ s ≤ 2R}.T is well-defined thanks to Lemma 4.4. To conclude it suffices to show that T = 0. which implies lim sup x→∞ ln(|X| + |Y |) x ≤ -(λ -+ δ). Appealing again to [14, Chapter 13, Theorem 4.3], we thus have lim sup x→∞ ln(|X| + |Y |) x ≤ -λ + . This leads to lim x→∞ (bu(x))e λx = lim x→∞ (v(x)a)e λx = 0 for all 0 < λ < λ + . (A.11)As as b √ 2 < λ + , this gives a contradiction to Lemma A.1 below and so concludes the proof.Lemma A.1. Suppose that 0 ≤ ω < 1 2 α and let (u, v) be a positive solution of (4.1)-(4.2). There is someC > 0 such that bu(x) ≥ u(u 2b 2 ). (A.13)Now, take some R > 0 such that u(R)≥ b √ 3 in (R, ∞). Select x 0 such that b tanh( b √ 2 (Rx 0 )) = u(R).To prove (A.12), it suffices to show thatu(x) ≤ b tanh( b √ 2 (xx 0 )) in (R, ∞).To this end, we note that the functionw 1,c = b tanh( b √ 2 (xx 0 )) + c satisfies for c ≥ 0, w ′′ 1,c = w 1,0 (w 2 1,0b 2 ) ≤ w 1,c (w 2 1,cb 2 ) in (R, ∞), (A.14)where we have used w1,0 (x) ≥ w 1,0 (R) = u(R) ≥ b√ Clearly there is some large c > 0 such that w 1,c ≥ u in [R, ∞). Letc = inf{c ≥ 0 : w 1,c ≥ u in [R, ∞)}.

  2(1+ω) 2+α > 1, we have αt -2t + 2 < αt + 2t -2 and 4ω 2 < (αt + 2t -2) 2 . (2.23) hence follows. This concludes Case (a).

	Case (b): 1+ω 2+α < ω 2α (i.e. 0 < α < 2 and ω > 2α 2-α ).
	We start by showing that	m ≤ h1 (h 2 (m)),	(2.24)

Note that in the notation of (2.10), we have m = m * and n = n * thanks to (4.7).

Set ũ(x) = u(x + T ) and ṽ(x) = u(x + T ). Note that ũ ≥ ū, ṽ ≤ v, (ũ, ṽ) is also a solution to (4.1) in the interval (-R, R -T ), and, in view of (1.5), we have as before that ũ′′ ≤ P (ũ, v), ū′′ = P (ū, v) ṽ′′ ≥ Q(ū, ṽ), v′′ = Q(ū, v).

In particular, if T was positive, it would follow from the strong maximum principle and the Hopf lemma that there would exist some small ε > 0 such that

which would contradict the definition of T . We hence have T = 0, as desired.

A Appendix: proof of Lemma 4.2.

We now prove of the exponential decay of solutions (u, v) to (4.1)-(4.2) to constants. This was needed in the proof of Proposition 4.1. We perform a standard asymptotic analysis near a hyperbolic critical point of ODEs.

We write u = bû and v = a + v. The system (4.1) becomes

The functions P and Q are polynomials and a direct computation gives

In particular, we have, for large x, that

The matrix A has two positive eigenvalues λ 2 ± (see (4.3)). For ω > 0, an Aeigenbasis of R 2 can be chosen as (1, µ) and (-µ, 1) (which correspond to the eigenvalues λ 2 -and λ 2 + , respectively) where µ is defined in (4.4). Note that

This signifies some difference in the asymptotic behavior of (û, v) for ω = 0 and for ω > 0.

Proof of Lemma 4.2. 1. We prove (4.5) and the relation ℓ 2 = µℓ 1 .

As û(x), v(x) → 0 as x → ∞, we have from (A.1)-(A.2) that û′′ (x), v′′ (x) → 0 as x → ∞. By interpolation, this implies û′ (x), v′ (x) → 0 as x → ∞. We now write v3 = û′ , v4 = v′ , v = (û, v, v3 , v4 ) and recast (A.1)-(A.2) as a first order system

where

and f is a polynomial satisfying f(0) = 0 and D f(0) = 0. Note that, as 0 < ω α < 1 2 , M has real and nonzero eigenvalues

Hence the origin is a hyperbolic critical point of (A.5). As v(x) → 0 as x → 0, we thus have that, for all large x, v(x) belongs to the stable manifold of (A.5) at the origin. By the Stable Manifold Theorem (see e.g. [14, Chapter 13, Theorem 4.3]), we then have that v(x) converges exponentially to 0 as x → ∞ and the rate of convergence is O(e -λx ) for any 0

We have

Applying [14, Chapter 13, Theorem 4.5], we can find a constant k and some δ > 0 such that

) and Y = O(e -(λ -+δ)x ). (A.10) Assertion (4.5) and the relation ℓ 2 = µℓ 1 are readily seen.

2. We next show that ℓ 1 and ℓ 2 are positive.

Suppose by contradiction the assertion does not hold. As ℓ 2 = µℓ 1 , one has ℓ 1 = ℓ 2 = 0. Returning to (A.10) we have that

The asymptotic behavior changes somewhat in the case ω = 0, which we record here for comparison. (This is not used in the paper.) Lemma A.2. Suppose α > 0 and ω = 0. Then, a = 0 and b = 1. Let (u, v) be a positive solution of (4.1)-(4.2). Then the following statements hold. x exists and is equal to

Proof. The proof is similar to that of Lemma 4.2 and is omitted.


Carolin Benjamins

Elena Raponi

Anja Jankovic

Koen Van Der Blom

Maria Laura Santoni

Marius Lindauer

Carola Doerr

PI is back! Switching Acquisition Functions in Bayesian Optimization

Bayesian Optimization (BO) is a powerful, sample-efficient technique to optimize expensive-to-evaluate functions. Each of the BO components, such as the surrogate model, the acquisition function (AF), or the initial design, is subject to a wide range of design choices. Selecting the right components for a given optimization task is a challenging task, which can have significant impact on the quality of the obtained results. In this work, we initiate the analysis of which AF to favor for which optimization scenarios. To this end, we benchmark SMAC3 using Expected Improvement (EI) and Probability of Improvement (PI) as acquisition functions on the 24 BBOB functions of the COCO environment. We compare their results with those of schedules switching between AFs. One schedule aims to use EI's explorative behavior in the early optimization steps, and then switches to PI for a better exploitation in the final steps. We also compare this to a random schedule and round-robin selection of EI and PI. We observe that dynamic schedules oftentimes outperform any single static one. Our results suggest that a schedule that allocates the first 25 % of the optimization budget to EI and the last 75 % to PI is a reliable default. However, we also observe considerable performance differences for the 24 functions, suggesting that a per-instance allocation, possibly learned on the fly, could offer significant improvement over the state-of-the-art BO designs.

Introduction

Bayesian Optimization (BO) [START_REF] Mockus | Bayesian Approach to Global Optimization: Theory and Applications[END_REF] is a technique to find the global optimum of blackbox problems when each evaluation of the objective function is expensive or otherwise resourceintensive, e.g., problems requiring elaborated physical experiments or computationally-heavy simulations. BO is widely known as one of the most efficient black-box approaches in terms of number of function evaluations required, hence a promising tool for the optimization of low-budget problems.

BO is defined as a modular framework which consists of (i) defining an initial set of points where the true (unknown, expensive) objective function is evaluated, (ii) approximating the objective function based on the initial design by placing a prior probability distribution on it (i.e., building a surrogate model), (iii) selecting where to sample next by optimizing an acquisition function (also known as infill criterion) that takes care of the trade-off between exploration and exploitation, (iv) evaluating new samples on the true objective function, and finally (v) computing the posterior probability distribution on the true objective function based on new knowledge, hence updating the surrogate model. Steps (iii) to (v) are repeated iteratively until the evaluation budget is exhausted.

2022 NeurIPS Workshop on Gaussian Processes, Spatiotemporal Modeling, and Decision-making Systems. Figure 1: Switching AFs is beneficial. Scaled log-regret averaged over all 24 BBOB functions for 5 dimensions with the 95 % confidence interval for the schedules from Table 1.

Name Schedule static (EI) static (PI) random round robin explore-exploit (0.25) explore-exploit (0.5) explore-exploit (0.75)

Table 1: Schedule portfolio of PI, EI and their combinations.

Each component of the sequential BO pipeline is subject to a wide range of design choices, which in turn affects the overall performance of the BO procedure [START_REF] Lindauer | Towards assessing the impact of bayesian optimization's own hyperparameters[END_REF][START_REF] Cowen-Rivers | An empirical study of assumptions in Bayesian optimisation[END_REF][START_REF] Bossek | Initial design strategies and their effects on sequential model-based optimization: an exploratory case study based on BBOB[END_REF]. For example, each acquisition function (AF) exhibits different behavior, with some being more explorative and others more exploitative. To date, design choices have been predominantly static during BO execution. However, other areas of optimization indicate that dynamic choices can lead to performance gains [START_REF] Karafotias | Parameter control in evolutionary algorithms: Trends and challenges[END_REF]; [START_REF] Doerr | Theory of parameter control mechanisms for discrete blackbox optimization: Provable performance gains through dynamic parameter choices[END_REF]; [START_REF] Adriaensen | Automated dynamic algorithm configuration[END_REF].

In this work, we investigate whether the same applies to BO. Specifically, we propose a dynamic schedule that switches between two AFs, namely Probability of Improvement (PI) and Expected Improvement (EI) Forrester et al. [2008], during the BO procedure. This simple approach is already quite effective and outperforms any single static schedule on most problems. We assess our method on the BBOB test suite from the COCO library [START_REF] Hansen | COCO: A platform for comparing continuous optimizers in a black-box setting[END_REF], which is a standard benchmark in black-box optimization. We purposely opt for such a benchmark in order to shift focus from a purely performance-oriented view towards understanding the problem characteristics that favor one or the other AF, which is a stark difference between this paper and previous works on dynamic AF selection. We thus observe that the choice of schedule is largely problem-dependent and that further investigation in this direction is of utmost importance to fully understand and exploit the untapped potential of adaptive AF selection.

Background Since our approach is focused on PI and EI, we briefly recall their formulations:

PI(x) = Φ ∆(x) σ(x) and EI(x) = ∆(x) Φ ∆(x) σ(x) + σ(x) φ ∆(x) σ(x) if σ(x) > 0 0 if σ(x) = 0 with ∆(x) = y min -µ(x)
, where y min is the best observed value so far, and the mean µ(x) and the variance σ 2 (x) are queried from the posterior of the surrogate model -in our work a Gaussian Process (GP). Φ and φ are the Gaussian cumulative distribution function and probability density function, respectively. By definition, in a minimization problem, the maximization of PI drives the selection of new candidates for evaluation towards areas of the domain where the mean value of the GP model is low. On the contrary, EI balances exploitation (thanks to the first summation term) and exploration (second summation term) by alternating sampling in promising areas of the domain according to the mean value and unexplored ones, where the variance of the GP model is high.

Comparing both, EI is more explorative and PI more exploitative, see Appendix A for an example.

Related Work

We first present existing work on AFs and focus on different approaches considering AFs. Important findings are reported when the main target is to improve and develop better AFs, from enhancing EI [START_REF] Chao Qin | Improving the expected improvement algorithm[END_REF][START_REF] Balandat | Botorch: A framework for efficient monte-carlo Bayesian optimization[END_REF] to meta-learning neural AFs [START_REF] Volpp | Meta-learning acquisition functions for transfer learning in bayesian optimization[END_REF]. A different line of work is concerned with combining different AFs, e.g. by building a portfolio of AFs (EI, PI ¸, Upper Confidence Bound (UCB) with different hyperparameter settings) and then using an online multi-armed bandit strategy to assign probabilities of which AF to use at which step, called GP-Hedge [START_REF] Hoffman | Portfolio allocation for Bayesian optimization[END_REF]. Their work indicates that the performance of GP-Hedge highly varies with the number of arms and their respective hyperparameter settings. Similarly to GP-Hedge, [START_REF] Kandasamy | Tuning hyperparameters without grad students: Scalable and robust Bayesian optimisation with dragonfly[END_REF] update weights of their portfolio (UCB, EI, Thompson Sampling (TS) [START_REF] Thompson | On the likelihood that one unknown probability exceeds another in view of the evidence of two samples[END_REF], Top-Two Expected Improvement (TTEI) [START_REF] Chao Qin | Improving the expected improvement algorithm[END_REF]) in an online manner. They do not include PI as they observe it exhibits inferior performance compared to other single static AFs. In addition, robust versions of EI, PI and UCB can be combined to a multi-objective AF combining the strengths of the individual ones [START_REF] Cowen-Rivers | An empirical study of assumptions in Bayesian optimisation[END_REF]. In contrast to the existing literature, we here take a step back and ask ourselves what we could achieve by employing a simplistic approach of switching between AFs through a dynamic schedule.

It has also been shown in other optimization-related areas that dynamic choices are beneficial in terms of performance, e.g. in evolutionary computation [START_REF] Karafotias | Parameter control in evolutionary algorithms: Trends and challenges[END_REF]; [START_REF] Doerr | Theory of parameter control mechanisms for discrete blackbox optimization: Provable performance gains through dynamic parameter choices[END_REF], planning [START_REF] Speck | Learning heuristic selection with dynamic algorithm configuration[END_REF] and deep learning [START_REF] Adriaensen | Automated dynamic algorithm configuration[END_REF]. Recently, the introduction of Dynamic Algorithm Configuration (DAC) [START_REF] Biedenkapp | Dynamic algorithm configuration: Foundation of a new meta-algorithmic framework[END_REF] underlines the potential of training dynamic schedules (as opposed to selecting algorithm components on the fly, as is usually done in evolutionary computation [START_REF] Hansen | Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)[END_REF]).

Proposed Method

To evaluate the potential of dynamic AF schedules, we consider several different variants, summarized in Table 1. As baselines, we use static schedules, namely EI and PI. Two types of dynamic schedules are considered: alternating and switching. The former go back and forth between EI and PI, whereas the latter make a single switch from EI to PI after a pre-defined percentage of the budget has been spent. We introduce two alternating schedules as baselines for a dynamic approach: random and round robin. For the random schedule the AF is chosen uniformly at random before a new solution is sampled, while round robin simply alternates the AFs for each sample. Finally, three different switching schedules are considered, making a switch after 25 %, 50 % and 75 % of the budget remaining after the initial design (i.e., surrogate-based function evaluations), respectively. This makes it possible to investigate whether the conceptual idea of switching is beneficial and additionally gives insight into the effects of switching at different stages of the optimization process.

Experiments

We evaluate our schedules on the 24 single-objective noiseless BBOB functions of the COCO benchmark [START_REF] Hansen | COCO: A platform for comparing continuous optimizers in a black-box setting[END_REF] in dimensions 5 with 60 seeds (i.e., we perform 60 independent runs on each problem and for each AF schedule). We optimize our functions with an initial design (or: Design of Experiment (DoE)) of 3d = 15 data points and BO length of 20d = 100 function evaluations. Per seed, the initial design is the same for every schedule. To create the dynamic BO, SMAC3 [START_REF] Lindauer | SMAC3: A versatile bayesian optimization package for Hyperparameter Optimization[END_REF] is adapted accordingly and we use a GP as surrogate model. All experiments were conducted on a Slurm CPU cluster with 1592 CPUs available across nodes. For visualization, we consider the log-regret of the incumbent (best evaluated search point) and only visualize the evaluations proposed by BO, omitting the initial design. In the violin plots the log-regrets of the final incumbents of all 60 seeds are normalized to [0, 1] per BBOB function. In the convergence plots the log-regret of the incumbents of all 60 seeds over all evaluations are normalized to [0, 1] per BBOB function and the means with the 95 % confidence intervals are shown. The plots for each function can be found in Appendix B. The code can be found here: https://github.com/automl/pi_is_back.

Observations A first observation is that switching from EI to PI is in general beneficial when the function landscape has an adequate global structure (F1-F19). Here, the only exceptions are F5 and F19. F5 is a purely linear function, where EI performs best as it explores fast in the beginning and exploits sufficiently fast later on because of the simplicity of the landscape. For F19, we hypothesize that only PI is able to exploit a lucky initial design landing close to optima as indicated by the individual runs in Figure 22c.

In addition, we can see that PI works well for uni-modal and quite smooth functions (F1-F14), and even better when used after the switch. This observation is in line with PI's exploitative behavior. In these cases exploiting does not miss any other optima further away in the landscape. In contrast, PI is in general worse than EI and also not beneficial after the switch for multi-modal functions with weak global structure (F20-F24). Again, this is in line with our intuition because for these functions we have many important basins of attraction that have to be discovered before starting exploitation. The only exception is F23 (Figure 26), a rugged function with a high number of global optima, where the probability of starting in a basin of attraction is high and thus exploitation is a viable strategy. Also, the flatter the landscape the worse PI performs which can be also seen in F7 (slope with step, Figure 10).

Besides the static and the switching schedules, round robin switches from EI to PI and vice versa for each new function evaluation. The round robin schedule creates a step-wise progress, but is never the best strategy. Apparently, less frequent switching is preferable in order to take advantage of the strengths of each AF. On average, the random schedule performs similarly to round robin, but with a smoother progression. Most likely, the random schedule still switches too frequently for the AFs to work effectively.

On F16 we observe that a late switch from EI to PI performs best, see Figure 2b. We can also nicely spot the general boosts after switching after 25 %, 50 % and 75 %. F16 has a highly rugged and moderately repetitive landscape with many local optima with evident quality difference. Therefore, PI can be trapped in a bad local optimum if applied too early.

Key Insights To summarize our observations, the general optimal choice of static or dynamic AF schedules depends on the problem at hand. Plus, switching from EI to PI after a certain percentage of the budget of total number of function evaluations is beneficial and can boost performance. However, the optimal switching point depends on the problem. On average, switching after 25 % seems to be best and is a viable schedule if the problem's properties are unknown.

Conclusions and Future Work

In this work we acknowledge the modular structure of Bayesian Optimization (BO) and explore different acquisition function schedules composed of Expected Improvement (EI) and Probability of Improvement (PI), with the premise that a dynamic choice would boost BO performance. We evaluate our approach on a standard benchmark for black-box optimization, and observe that dynamic acquisition function (AF) schedules often outperform single static ones, although their performance depends on the problem at hand. As a general recommendation, if the properties of the problem to optimize are unknown, switching from EI to PI after 25 % of the budget of surrogate-based evaluations should be favored. We also highlight some key insights into the interplay of the nature of the problem and the performance of the proposed approach.

These preliminary findings suggest the potential of the dynamic choices in a BO pipeline merit further investigation. As future steps in this direction, finding the most appropriate schedule of different AFs is an open problem we could tackle with meta-learning. One way would be to train models recommending which AF to use at which step based on problem representation, i.e., typically Exploratory Landscape Analysis (ELA) features [START_REF] Mersmann | Exploratory landscape analysis[END_REF]. In addition, finding meaningful state features could enable an online selection of the AF via heuristics or Reinforcement Learning [START_REF] Adriaensen | Automated dynamic algorithm configuration[END_REF]. To this end, besides elapsed time and progress being default choices as state features, we could incorporate properties of the surrogate model to serve as a guide to select the AF. Furthermore, including more AFs and their hyperparameters in the portfolio, as well as dynamically changing other components of BO, like the surrogate model, the GP kernel, or the optimizer of the AF, might increase overall performance and sample-efficiency of BO.

A Exploration vs. Exploitation

We choose EI and PI to build our AF schedules because of their different search behavior. In Figure 3 we exemplarily show EI's explorative character and PI's exploitative character. Both behaviors can be beneficial depending on the problem to optimize.

Figure 2 :

 2 Figure 2: BBOB Function 16: Highly rugged and moderately repetitive landscape with many local optima. Dynamic schedules outperform static ones and we can clearly observe the boost gained after switching from EI to PI.

Figure 3 :

 3 Figure 3: EI vs. PI on BBOB function 12 (upper) and 21 (lower). The dots indicate the visited configuration at a certain step. We visualize all 20 seeds with an initial design size of 10 (not plotted) and 40 BO evaluations.

Acknowledgments

Our work has been financially supported by the the ANR T-ERC project VARIATION (ANR-22-ERCS-0003-01), by the CNRS INS2I project RandSearch, and by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF), and by RFBR and CNRS, project number 20-51-15009. Carolin Benjamins and Marius Lindauer acknowledge funding by the German Research Foundation (DFG) under LI 2801/4-1.


Carolin Benjamins

Anja Jankovic

Elena Raponi

Koen Van Der Blom

Marius Lindauer

Carola Doerr

Towards Automated Design of Bayesian Optimization via Exploratory Landscape Analysis

Bayesian optimization (BO) algorithms form a class of surrogate-based heuristics, aimed at efficiently computing high-quality solutions for numerical black-box optimization problems. The BO pipeline is highly modular, with different design choices for the initial sampling strategy, the surrogate model, the acquisition function (AF), the solver used to optimize the AF, etc. We demonstrate in this work that a dynamic selection of the AF can benefit the BO design. More precisely, we show that already a naïve random forest regression model, built on top of exploratory landscape analysis features that are computed from the initial design points, suffices to recommend AFs that outperform any static choice, when considering performance over the classic BBOB benchmark suite for derivative-free numerical optimization methods on the COCO platform. Our work hence paves a way towards AutoML-assisted, on-the-fly BO designs that adjust their behavior on a run-by-run basis.

Introduction

In optimization we often encounter black-box problems having no explicit formulation of the underlying function or its derivatives. To identify high-quality solutions for such functions, black-box algorithms rely on an iterative process of querying the quality of the solution candidates it suggests, and adjusting their strategy to obtain the next set of candidate solutions. This process is particularly challenging when the maximal number of evaluations that can be made before a final recommendation is expected is very small compared to the size of the design space. A standard approach for such settings is Bayesian optimization (BO) [START_REF] Mockus | Bayesian Approach to Global Optimization: Theory and Applications[END_REF], a family of surrogate-based optimization algorithms that obtains its candidate solutions via the following process: a first set of candidates, known as the initial design or design of experiment (DoE), is obtained from a classic sampling strategy, e.g., (quasi-)random distributions or space filling designs. After evaluation, these points are used to build a surrogate model, an approximation of the unknown objective function capturing the uncertainty. An acquisition function (AF) (also: infill criterion) determines which point to sample next based on the predicted mean and variance of the surrogate model, requiring no evaluations of the true problem. As long as the budget is not exhausted, the surrogate model is updated and the algorithm repeats the last steps. All steps of the BO pipeline are subject to different design choices, impacting its performance [START_REF] Lindauer | Towards assessing the impact of bayesian optimization's own hyperparameters[END_REF][START_REF] Cowen-Rivers | An empirical study of assumptions in Bayesian optimisation[END_REF][START_REF] Bossek | Initial design strategies and their effects on sequential model-based optimization: an exploratory case study based on BBOB[END_REF]. Of particular importance for the overall BO performance is the balance between exploration and exploitation to efficiently find the optimal solution, which is strongly influenced by the choice of the AF. Although many different AFs have been introduced (e.g., Probability of Improvement (PI), Expected Improvement (EI), Upper Confidence Bound (UCB) [START_REF] Alexander | Engineering Design via Surrogate Modelling -A Practical Guide[END_REF] and Thompson Sampling (TS) [START_REF] Thompson | On the likelihood that one unknown probability exceeds another in view of the evidence of two samples[END_REF] there are no guidelines on how to select the most appropriate AF given the characteristics of the problem at hand. Furthermore, AF design decisions have predominantly considered a static choice over the entire BO procedure. Only few works have considered dynamic choices when selecting AFs [START_REF] Hoffman | Portfolio allocation for Bayesian optimization[END_REF][START_REF] Kandasamy | Tuning hyperparameters without grad students: Scalable and robust Bayesian optimisation with dragonfly[END_REF][START_REF] Cowen-Rivers | An empirical study of assumptions in Bayesian optimisation[END_REF], and similarly to insights obtained in other domains of optimization, have shown the potential of opting for dynamic combinations AF rather than a static choice. Orthogonal to this, performance gains may also be possible through the application of algorithm selection (AS) techniques [START_REF] Rice | The algorithm selection problem[END_REF]. Given sufficiently heterogeneous problem instances, and based exclusively on their characteristics, choosing different AFs or AF schedules might result in better performance than applying the same AF (schedule) to all problems, which is known as per-instance algorithm selection (PIAS) [Kerschke et al., 2019]. More importantly, we observe that the properties of the initial design itself further influence the choice of the AF, so we may want to select different AFs per run, even for the same problem instance. This is in line with per-run algorithm selection, as suggested in [START_REF] Jankovic | Trajectory-based algorithm selection with warm-starting[END_REF], Kostovska et al., 2022] for a portfolio of iterative optimization heuristics.

Motivated by recent insights from the AS domain, we aim to understand the interplay between the problem (characterized through the initial design), the choice of the AF (schedule), and the BO performance. To this end, we benchmark our approach on a test suite of well-established black-box numerical optimization problems, the BBOB functions of the COCO framework [START_REF] Hansen | COCO: A platform for comparing continuous optimizers in a black-box setting[END_REF]. We build and train a model, called the Acquisition Function Selector (AFS) (Figure 1), which automatically selects the best suited AF (schedule) for a certain problem. To represent problems, we employ exploratory landscape analysis (ELA) [START_REF] Mersmann | Exploratory landscape analysis[END_REF]. ELA is a framework for the extraction and computation of low-level numerical landscape features of black-box problems (related to e.g., multimodality, separability, conditioning, presence of plateaus, etc.) from sampled points and their evaluations. Here, we use the evaluated initial design for computing the ELA features. Since BO anyway evaluates the initial design before starting the surrogate-based optimization process, the use of ELA features comes at no evaluation cost. Our AF schedules are composed of either PI (more exploitative), EI (more explorative) or combinations of both.

Our key insights are that (i) already a selector built on top of a naïve random forest regression model outperforms any static AF schedule, (ii) the choice of the AF model should depend on the problem instance, but also on the characteristics (and in particular the quality) of the initial design, and (iii) the schedules that switch from EI to PI improve over static EI and PI.

Experiments

Building our Acquisition Function Selector (AFS) (see Figure 1) consists of two steps. First, we gather performance data of our seven AF schedules (static EI, static PI, and five dynamic ones: random, round robin, as well as three variants of explore-exploit schedules that switch from EI to PI after 25 %, 50 % and 75 % of the budget). Second, we train our AFS in the following way. We build a multi-target regression model that maps the ELA features of the initial design to the performance of each AF schedule. Our AFS then selects the schedule with the best predicted performance. We evaluate our schedules on the 24 single-objective noiseless BBOB functions of the COCO benchmark [START_REF] Hansen | COCO: A platform for comparing continuous optimizers in a black-box setting[END_REF] in dimension 5 and 5 instances with 60 seeds). The size of the initial design is set to 10d = 50 data points and a budget of 40d = 100 function evaluations is used for the surrogate-based optimization part. The initial design is the same for every seed, so every schedule sees Figure 2: AFS selecting between static PI and EI adopts the shape of the better performing schedule and ranks first.

the same initial design. We adapt SMAC3 [START_REF] Lindauer | SMAC3: A versatile bayesian optimization package for Hyperparameter Optimization[END_REF] to enable dynamic AF schedules and we use a Gaussian Process (GP) as surrogate model. We extract ELA features based on samples of the initial design and their function evaluations using the flacco library [Kerschke and Trautmann, 2019]. Among the 300 features (grouped in sets) in the library, we choose 38 cheap-to-compute features not requiring further function evaluations, namely those from y-Distribution, Meta-Model, Dispersion, Information Content and Nearest-Better Clustering feature sets, following practices in the literature [START_REF] Belkhir | Surrogate assisted feature computation for continuous problems[END_REF][START_REF] Jankovic | Towards feature-based performance regression using trajectory data[END_REF]. In our setup, ELA feature computation for an initial design of 50 points takes about 0.4 s, which is negligible compared to an expensive function evaluation. In this work, we restrict our attention to continuous problems, but ELA may also work well for discrete (and possibly mixed-integer) search spaces [START_REF] Pikalov | Parameter tuning for the (1 + (λ , λ )) genetic algorithm using landscape analysis and machine learning[END_REF]. We then train a standard random forest (RF) [START_REF] Tin | Random decision forests[END_REF][START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] Results Before we learn to select from our portfolio containing static and dynamic schedules, we first consider selecting either static PI or static EI. We can observe that AFS adopts the shape of the better performing acquisition function (Figures 2a and2b) and aggregated over all functions ranks first (Figure 2c), demonstrating the potential to select between AFs based on the initial design. We then extend the AFS to select between all seven proposed schedules. We introduce the virtual best solver (VBS), which is our oracle, always selecting the best schedule per run. The general optimal choice of static or dynamic AF schedules depends on the problem at hand (see figures in Appendix B).

As a rule of thumb, switching from EI to PI after a certain percentage of the budget of total number of function evaluations is beneficial and can boost performance, see Figure 3. However, the optimal switching point is highly problem-dependent and likely run-dependent. On average, switching after 25 % seems to be a viable general rule if the problem's properties are unknown. Moreover, due to the initial design, we gain information about the problem landscape based on ELA features for that specific run. Already with a simple, non-tuned RF, we are able to select well-performing schedules with the AFS. In this case, the AFS (apart from the VBS) also shows the best anytime-performance which is not necessarily guaranteed as the AFS is trained on the final performance of the other schedules. Again, we rank across all functions and instances and observe that the mass of ranks for AFS is centered on the first, in contrast to all other schedules, see Figure 4. In summary, our AFS is able to propose appropriate schedules from a portfolio based on the initial design per run. selection of the acquisition function can benefit the performance of Bayesian optimization (BO) approaches. More precisely, we have seen that a landscape-aware selection of the AF, choosing from a portfolio of seven different AF schedules, outperforms any static choice. This advantage was realized with a naïve random forest regression approach, and it seems likely that further improvements can be expected from a proper model selection, e.g., using auto-sklearn [START_REF] Feurer | Auto-sklearn 2.0: The next generation[END_REF].

Conclusion and Future Work

Our long-term objective is the development of modular BO algorithms that are trained to select their modules, including the AF, on the fly, i.e., during the optimization process. Such dynamic algorithm configuration (DAC) [START_REF] Biedenkapp | Dynamic algorithm configuration: Foundation of a new meta-algorithmic framework[END_REF][START_REF] Biedenkapp | Theory-inspired parameter control benchmarks for dynamic algorithm configuration[END_REF] approaches were recently shown capable of outperforming classic (static) hyperparameter optimization approaches. A key issue in the design of DAC is the identification of features that the trained model can use to guide its selection. Our results provide strong motivation to consider exploratory landscape analysis for this purpose. Blending the ELA features with information obtained from the surrogate model clearly is a natural next step for our work.

In our study, we have used a relatively large initial design size (10d points or 1/3 of the total budget).

Considering [START_REF] Bossek | Initial design strategies and their effects on sequential model-based optimization: an exploratory case study based on BBOB[END_REF], an intermediate decision whether to continue with the initial design or to start the surrogate-based optimization should bring additional performance gains. To realize this potential, a thorough investigation of suitable ELA features is needed, since some features cannot be considered reliable when based on such comparatively small sample sizes [START_REF] Belkhir | Surrogate assisted feature computation for continuous problems[END_REF]. We see our work as a continuation of the per-run algorithm selection approach suggested in [START_REF] Jankovic | Trajectory-based algorithm selection with warm-starting[END_REF], Kostovska et al., 2022]. In particular, we believe that it is not only the instance per se that should guide the selection of algorithm components and configurations, but also the particular trajectory that the algorithm follows in a particular run. In the future, we aim to develop a deeper understanding for when to switch from initial design sampling to using explorative AFs or exploitative ones. While in this first proof-of-concept study we have only considered EI and PI as AFs, in future studies we intend to include other AFs such as Upper Confidence Bound (UCB) [START_REF] Alexander | Engineering Design via Surrogate Modelling -A Practical Guide[END_REF], Top-Two Expected Improvement (TTEI) [Qin et al., 2017], and Thompson Sampling (TS) [START_REF] Thompson | On the likelihood that one unknown probability exceeds another in view of the evidence of two samples[END_REF]. Table 1: Schedule portfolio of PI, EI and their combinations. AFS is a selector that aims to predict which of the first seven schedules is the best. VBS is a perfect selector and forms the ground truth for AFS.

B Additional Results

In this section we provide all boxplot and convergence figures for each BBOB function. Please note that because the VBS is selected based on final performance, it does not always show the best anytime-performance.

We first present a more in-depth discussion of the results on specific BBOB functions, focusing on the manually defined schedules. A first observation is that switching from EI to PI is in general beneficial when the function landscape has an adequate global structure (F1-F19). Here, the only exceptions are F5 and F19. F5 is a purely linear function, where EI performs best as it explores fast in the beginning and exploits sufficiently fast later on because of the simplicity of the landscape. For F19, we hypothesize that only PI is able to exploit a lucky initial design landing close to optima.

In addition, we can see that PI works well for uni-modal and quite smooth functions (F1-F14), while performing even better when used after the switch. This observation is in line with PI's exploitative behavior. In these cases exploiting does not miss any other optima further away in the landscape.

In contrast, PI is in general worse than EI and also not beneficial after the switch for multi-modal functions with weak global structure (F20-F24). Again, this is in line with our intuition because for these functions we have many important basins of attraction that have to be discovered before starting exploitation. The only exception is F23 (Figure 27), a rugged function with a high number of global optima, where the probability of starting in a basin of attraction is high and thus exploitation is a viable strategy. Also, the flatter the landscape the worse PI performs which can be also seen in F7 (slope with step, Figure 11).

Besides the static and the switching schedules, round robin switches from EI to PI and vice versa for each new function evaluation. The round robin schedule creates a step-wise progress, but is never the best strategy. Apparently, less frequent switching is preferable in order to take advantage of the strengths of each AF. On average, the random schedule performs similarly to round robin, but with a smoother progression. Most likely, the random schedule still switches too frequently for the AFs to work effectively.

On F16 we observe that a late switch from EI to PI performs best, see Figure 20b. We can also nicely spot the general boosts after switching after 25 %, 50 % and 75 %. F16 has a highly rugged and moderately repetitive landscape with many local optima with evident quality difference. Therefore, PI can be trapped in a bad local optimum if applied too early.

Figure 1 :

 1 Figure 1: How to select acquisition function (AF) schedules Based on the evaluated initial design of a BO run, we compute the ELA features. With those we query a trained RF and observe the predicted performance of each AF schedule. The Acquisition Function Selector returns the AF schedule with the best predicted performance to use for surrogate-based evaluations.

  Ranks aggregated over all BBOB functions.

Figure 4 :

 4 Figure 4: Ranks of schedules, aggregated over all BBOB functions, instances and seeds. AFS has the most favorable rank distribution.

Checklist 1 .

 1 For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] (c) Did you discuss any potential negative societal impacts of your work? [No] We see no potential negative societal impact because our method is about speeding up an existing black-box optimization algorithm, therefore reducing required resources. (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] 3. If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] https: //github.com/automl/BO-AFS (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes] (b) Did you mention the license of the assets? [N/A] (c) Did you include any new assets either in the supplemental material or as a URL? [No] (d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] We create our own data. (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] 5. If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] (0.25) explore-exploit (0.5) explore-exploit (0.75) VBS (Virtual Best Solver) AFS (AF Selector)

Figure 28 :

 28 Figure 5: BBOB Function 1

  :2211.09678v1 [cs.LG] 17 Nov 2022

	Initial Design	ELA Features	Performance Regression	AF Selector	AF Schedule

), 6th Workshop on Meta-Learning at NeurIPS 2022, New Orleans.

arXiv

  Switching AFs and especially selecting AF schedules (AFS) is beneficial. Scaled logregret averaged over 5 instances of all 24 BBOB functions for 5 dimensions with the 95 % confidence interval for the schedules from Table1.

		0.8
	log regret (scaled)	0.5 0.6 0.7	schedule explore-exploit (0.25) explore-exploit (0.5) explore-exploit (0.75) random
			round robin
		0.4	static (EI) static (PI)
			VBS
			AFS
		0	20	40 BO evaluations 60	80	100
	Figure 3:

Using a standard test bed for benchmarking single-objective numerical black-box optimization techniques, the BBOB suite of the COCO environment, we have shown in this work that a per-run

Acknowledgments

Our work has been financially supported by the the ANR T-ERC project VARIATION (ANR-22-ERCS-0003-01), by the CNRS INS2I project RandSearch, and by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF), and by RFBR and CNRS, project number 20-51-15009. Carolin Benjamins and Marius Lindauer acknowledge funding by the German Research Foundation (DFG) under LI 2801/4-1.