Jérôme Bolte
email: jerome.bolte@tse-fr.eu

Laurent Miclo
email: laurent.miclo@math.cnrs.fr

Stéphane Villeneuve
email: stephane.villeneuve@tse-fr.eu

Swarm gradient dynamics for global optimization: the mean-field limit case

Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge-Kantorovich gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide class of global optimization methods. Due to an inbuilt combination of a gradient-like strategy and particles interactions, we call them swarm gradient dynamics. As in the original paper of Holley-Kusuoka-Stroock, the key to the existence of a schedule ensuring convergence to a global minimizer is a functional inequality. One of our central theoretical contributions is the proof of such an inequality for one-dimensional compact manifolds. We conjecture the inequality to be true in a much wider setting. We also describe a general method allowing for global optimization and evidencing the crucial role of functional inequalities à la Łojasiewicz.

Introduction

The global minimization of a nonconvex function is one of the most challenging problems in modern optimization. There are few global optimization methods that provide reasonable convergence guarantees, the most famous is probably the simulated annealing, whose premises are found in [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF], or the moment method [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], and their many variants. On the other hand, metaheuristics methods are numerous and have some notable empirical success: they orchestrate interactions between local and global strategies, combine random and deterministic procedures, and often end up with methods using optimizing agents. Some examples of metaheuristics are inspired by analogies with biology, as evolutionary algorithms [START_REF] Fogel | Evolutionary computation towards a new philosophy of machine intelligence[END_REF], ethology (e.g., ant colonies [START_REF] Dorigo | Ant Colony optimization theory: a survey[END_REF]), or particle swarms, see e.g., [START_REF] Kennedy | Particle Swarm optimization[END_REF]. The goal of this paper is to introduce a new family of swarm methods through gradient descent in the Monge-Kantorovich space and give general guarantees for their convergence to global minimizers.

Let us be more specific and consider the problem of solving min M U, (P)

where U : M → R is a differentiable function defined on a compact Riemannian manifold M . In order to introduce our swarm methods, we need first some considerations on simulated annealing.

Three views on simulated annealing Our starting point is indeed the famous simulated annealing method. In its time-continuous form, and when the state space is flat M (e.g., a flat torus), it is a solution X (X t ) t≥0 of the time-inhomogeneous Langevin-like stochastic differential equation

dX t = -β t ∇U (X t ) dt + √ 2dW t , (a)
where (W t ) t≥0 is a Brownian motion and β t → +∞1 is a time-dependent parameter tuned so that the expectation of U (X t ) tends to min M U . The formulation (a) can be extended to any compact Riemannian manifold M , but this requires more involved notations, see e.g., Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] or Emery [START_REF] Émery | Stochastic calculus in manifolds[END_REF]. The intuitive interpretation is quite natural, the method combines local gradient search with a vanishing Brownian exploration of the feasible set M . Although the method is often used as a heuristic, its proof has been made rigorous in various frameworks via different approaches, the two main ones being based on large deviations, see e.g., Azencott et al. [START_REF] Azencott | Simulated annealing. Parallelization techniques[END_REF], and on functional inequalities, cf. Holley, Kusuoka and Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. Key to the foundational approach of [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], is the establishment of a generalized log-Sobolev inequality followed by hypercontractivity arguments. This approach was then simplified by Miclo [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre [Simulated annealing on R n . Study of the free energy evolution[END_REF] via the identification of the relative entropy as a convenient Lyapunov function. In order to explain the role of the log-Sobolev inequality, the relative entropy, the mechanisms behind the convergence properties, and understand the scope of the method, we view simulated annealing along three complementary angles:

(a) the SDE form of the algorithm: the overdamped Langevin dynamics (a) above, (b) the PDE counterpart of (a), which describes the time evolution of the density t → ρ(t) of X t . It assumes the form of a Fokker-Planck equation,

d dt ρ = β t div(ρ∇U ) + ∆ρ, t ≥ 0. (b)
(c) Otto's formalism [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], which we recall briefly in the Appendix, allows one to interpret the latter as a gradient-like system in the space of probabilities on M endowed with Monge-Kantorovich metric:

d dt ρ(t) = -grad W U βt [ρ(t)], (c)
with

U β [ρ] = β M U ρ d + M ρ log ρ d
, where is the Riemannian measure of M . This quantity is also known as the relative entropy of ρ with respect to the Gibbs measure whose density is proportional to exp(-βU ).

This triple perspective, mainly due to [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], is not new and has known a recent success in sampling [START_REF] Dalalyan | Theoretical guarantees for approximate sampling from smooth and logconcave densities[END_REF][START_REF] Durmus | Non-asymptotic convergence analysis for the unadjusted Langevin algorithm[END_REF], optimization [START_REF] Ma | Sampling can be faster than optimization[END_REF][START_REF] Ma | Is there an analog of Nesterov acceleration for MCMC?[END_REF] and machine learning [START_REF] Raginsky | Non-convex learning via stochastic gradient Langevin dynamics: a non-asymptotic analysis[END_REF][START_REF] Cheng | Convergence of Langevin MCMC in KL-divergence[END_REF].

Depending on the form we adopt to study the dynamics, subsequent results or developments may be considerably easier to understand. Indeed, while (a) classically provides operational algorithms through discretization, (b) offers a tractable version amenable to classical PDE analysis methods as Lyapunov methods. As for the last angle, (c), it confers a sharp geometrical content to the method and allows to interpret the essential tools of convergence through classical intuitive geometric ideas. An essential fact about (c) is that functional inequalities, as the log-Sobolev inequality, may be seen as Łojasiewicz gradient inequalities. In our case it means that there exists an exponent γ ∈ (0, 1) such that the slope of (U β -min U β ) γ is bounded away from zero (save at the stationary measure). This reparametrization sharpens the energy landscape while leaving unchanged level sets: this allows for a direct convergence analysis of the gradient method (c), see [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF] and references therein. In the simulated annealing case the log-Sobolev inequality of Holley, Kusuoka and Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] turns out to be an instance of such an inequality, see [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF].

Swarm gradient dynamics The triple-perspective (a)-(b)-(c) we used to describe the strategy of simulated annealing can be generalized to a much larger framework. For this, we adopt the angle (c) under which we observe that it is natural to consider more general convex functions ϕ than R + r → r ln(r) in the Boltzmann entropy,

B[ρ] = M ρ log ρ d .
Referring to the results in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], we may indeed use a whole family of convex functionals

H[ρ] = M ϕ(ρ) d , leading to a penalized cost U β [ρ] = β M U ρ d + H[ρ]
and to the triplet of "equivalent" minimizing dynamics modeled on (c), (b), (a),

d dt ρ(t) = -grad W U βt [ρ(t)] (1)
d dt ρ = β t div(ρ∇U ) + divρ∇ϕ (ρ) (2)
dX t = -β t ∇U (X t ) dt + √ 2α(ρ)dW t , (3)
with lim t→∞ β t = +∞, ϕ, α are some positive functions to be specified, and ρ t is the probability density of X t for each t. The fact that a particle interacts with its law may be considered as a swarm effect, this is why we call these dynamics swarm gradient dynamics The key to convergence: functional inequalities The central question is that of the convergence properties to a global minimizer. In particular, an essential question is: what are the assumptions ensuring that the global minimization min M U problem is solved by the above?

In simulated annealing, the essential tool for convergence is the log-Sobolev inequality of Holley, Kusuoka and Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. We also recalled that this inequality can be advantageously thought as a Łojasiewicz inequality when considering the problem along the gradient system angle (c). We follow, therefore, the same protocol but in a reverse way: we formally write Łojasiewicz inequalities using the Monge-Kantorovich formalism, which reveals, in turn, the functional inequalities we would like to have at our disposal. This leads us to consider:

M |∇ϕ (ρ) -∇ϕ (µ β )| 2 ρ d ≥ c(β) Ω M ϕ(ρ) -ϕ(µ β ) -ϕ (µ)(ρ -µ β ) d (4)
where c, Ω : (0, +∞) → R + are positive functions having specific properties and where µ β is the unique stationary measure of U β . We are at the heart of this paper and our central result: proving such a functional inequality under adequate assumptions. Our result holds in compact one-dimensional manifolds for power-like potential functions ϕ. As a consequence, we obtain a full convergence result of our global methods on compact one-dimensional manifolds. We also evidence the general mechanisms of global convergence, and for completeness, we sketch the form that operational algorithms could take (but we postpone their study to another paper).

Apart from the interest of our work for optimization, we believe that it raises important questions and hopes on the validity domain of the family of inequalities in [START_REF] Belaribi | Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation[END_REF]. Positive outcomes would lead to new results in optimization and in other fields.

Related works

The quantitative comparison between entropy-type functional and its time derivative, often called the entropy production or dissipation, dates back to [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre [Simulated annealing on R n . Study of the free energy evolution[END_REF], where it was exploited through the logarithmic Sobolev inequality of [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. Using Otto's formalism, this can be, in turn, reinterpreted as an approach à la Łojasiewicz [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF]. Equation (2) may be seen as a formal generalization of porous media equation and fast diffusion equations -which corresponds to the case when ϕ is a potential. These have been studied by several authors using the Monge-Kantorovich framework, see, e.g., Otto [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF], and Carrillo, McCann and Villani [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF][START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF]. Their asymptotic analysis is through the Bakry-Emery method [START_REF] Bakry | Diffusions hypercontractives[END_REF]: it consists in the second-order time differentiation of the entropy. Contrary to [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF][START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre [Simulated annealing on R n . Study of the free energy evolution[END_REF] and our current approach, this approach requires convexity, which makes it unsuitable for general global minimization.

The article of Iacobelli, Patacchini, and Santambrogio [START_REF] Iacobelli | Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour[END_REF] is also connected to our approach since they consider ultrafast diffusion equations, which corresponds to a negative exponent m in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. However, the hypocoercive bounds they obtain do not seem well-suited to extensions to time-inhomogeneous situations since they do not lead to a differential inequality satisfied by the entropy-like functionals.

The uniqueness of the stationary measure, i.e., of the minimizer of U β , is not a new result, it can be found in, e.g., Carrillo, Jüngel, Markowich, Toscani and Unterreiter [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF]. For the sake of completeness, we provide a proof in the next section. Let us conclude by mentioning a few works using nonlinear diffusion where nonlinearities generally affect the drift coefficient but not the diffusion term as here. In Eberle, Guillin, and Zimmer [START_REF] Eberle | Quantitative Harris type theorems for diffusions and McKean-Vlasov processes[END_REF], the authors use coupling techniques, Carillo, Gvalani, Pavliotis and Schlichting [START_REF] Carrillo | Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus[END_REF] treat the case of interaction potentials while Delarue and Tse [START_REF] Delarue | Uniform in time weak propagation of chaos on the torus[END_REF] consider chaos propagation. Denote by d the distance on M , and the natural Riemannian measure. Up to a normalization factor, assume (M ) = 1. Let P(M ) be the space of probability measures on M equipped with the Monge-Kantorovich distance defined through

Presentation of the problem

W 2 2 (µ, ν) = inf M d 2 (x, y) p(dx, dy) : p is a coupling of µ and ν on M 2 ,
for any µ, ν in P(M ). The extreme values of U play a special role in our approach, one defines

osc(U ) := max M U -min M U, (5)
which we may assume positive -since otherwise, the problem would be trivial. We make the following regularity assumptions: Assumption (A). The manifold M and the function

U : M → [0, +∞[ are of class C 2 .
The C 2 regularity assumptions are simple means to obtain existence results for the gradient evolution in the Monge-Kantorovich space as in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Ferreira | Gradient flows of time-dependent functionals in metric spaces and applications to PDEs[END_REF] We embed our initial problem min M U in P(M ) and consider the relaxed minimization problem

U[ρ] := M U ρ.
Using the monotonicity of the integral, one easily sees that min

P(M ) U = min M U.
Let β > 0, we introduce a penalized relaxation of U in the metric space (P(M ), W 2 ) through

U β [ρ] = βU[ρ] + H[ρ] = β M U ρ d + H[ρ] (6)
where

H[ρ] =    M ϕ(ρ) d if ρ is absolutely continuous w.r.t the Riemannian measure + ∞ otherwise,
with ϕ : [0, +∞) → R + is strictly convex and C 2 on (0, +∞). Up to a multiplicative factor, the first term in ( 6) is the classical relaxation of U within the probability space over M . On the other hand, as in simulated annealing, the second term acts as a penalization forcing the minimizer to be unique and to have a density with respect to (see Lemma 1). As we shall see below, letting β go to infinity allows one to recover the initial problem.

Remark 1. (a) (Power-like penalizations) A strong focus will be put on the class of power-like functions. For any m ∈ (0, +∞) \ {1}, define the convex function ϕ m : R

+ → R + via ∀ r ≥ 0, ϕ m (r) r m -1 -m(r -1) m(m -1) (7)
Let us observe that ϕ m is a strictly convex function, C 2 on (0, +∞) and such that ϕ m (1) = 0, ϕ m (1) = 0 and ϕ m (1) = 1 for every admissible m. By the Taylor-Lagrange formula, we deduce that ϕ m is always positive, except at 1. The convex function ϕ 1 : R + → R + defined by ϕ 1 (r) := r ln(r) -(r -1) is recovered as the limit of ϕ m when m goes to 1 and corresponds to the Boltzmann entropy. Hereafter, we will in particular consider functions ϕ that are constructed by gluing together two different functions ϕ m at 1. (b) (Regularity of the penalization) Observe as well, from [1, Theorem 9.3.9, p.212] and [1, Proposition 9.3.2, p.210], that the function H is lower for the Monge-Kantorovich distance, and geodesically convex in P(M ).

The approach we adopt is through the one-parameter family of problems val(P β ) := inf

P(M ) U β (P β )
where the parameter β > 0 is the inverse of a penalization parameter or the inverse of "the temperature" according to the simulated annealing literature. It ultimately tends to ∞, and one has an elementary but important fact:

Proposition 1 (A global optimization principle). Assume (A) and that ϕ : [0, +∞) → R + is strictly convex and C 2 on (0, +∞).

Then (i) lim β→+∞ 1 β val(P β ) = min M U,
(ii) if µ β is a sequence of solutions to (P β ), the weak* limit points of µ β have a support concentrated on the set of minimizers of U Proof. As a first observation, it is clear that min

P(M ) M U ρ = min M U.
Fix > 0 and choose ρ to be an -minimizer of U[ρ] = M U ρ. The above observation yields U[ρ ] ≤ min M U + . Since dom H contains smooth densities, one can also assume that H(ρ ) is finite. Take β > 0 and let µ β, be an -solution to (P β ), that is

1 β U β [µ β, ] = U[µ β, ] + 1/βH[µ β, ] ≤ U(ρ) + 1/βH[ρ] +
for all ρ. Thus choosing ρ = ρ yields

1 β U β [µ β, ] ≤ min M U + 1/βH[ρ ] + 2 .
Letting β goes to infinity yields

lim sup β→∞ 1 β U β [µ β, ] ≤ min M U + 2 .
Whence lim sup β→∞, →0

1 β U β [µ β, ] ≤ min M U . Since U β ≥ min M U
P(M ) U β -min M U (8)
tends to zero when β tends to +∞ by the previous result. The quantity gap(β) measures the approximation abilities of the problem (P β ) with respect to the initial problem min M U .

Variational considerations and stationary measures

Let us denote •, • x the Riemannian metric on the tangent at x to M and |•| x the corresponding norm. We generally omit the dependence in x.

Let us analyze the first-order conditions for the above problem (P β ) through the lenses of the Monge-Kantorovich metric. We shall freely use the definition of Monge-Kantorovich subgradients and related objects. As they are only central to our understanding but not to our proofs, we refer to [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for details and to the Appendix for some elementary considerations. The subgradient of U β with respect to the Monge-Kantorovich metric has a domain contained in L 1 (M ) and is formally given by

grad W U β [ρ] = -div(ρ(β∇U + ∇ϕ (ρ)))
for any admissible ρ in L 1 (M ). Stationary solutions of (2), with β t ≡ β, are thus probability densities µ solution to

div(µ(β∇U + ∇ϕ (µ))) = 0, (9)
which is to be understood in the standard weak sense. By integration by parts, we have for

f ∈ C 2 (M ), M div(µ∇ϕ (µ))f d = - M µ∇ϕ (µ), ∇f d = - M µϕ (µ) ∇µ, ∇f d = - M ∇(µα(µ)), ∇f d = M µα(µ) f d = {µ>0} α(µ) f µd
where the function α : (0, +∞) → R + is given by

∀ r > 0, α(r) 1 r r 0 sϕ (s) ds.
On the other hand, we have

M div(µβ∇U )f d = - M β∇U, ∇f µ d .
Finally, µ is a stationary solution to [START_REF] Carrillo | Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus[END_REF] if

∀ f ∈ C 2 (M ), {µ>0} L µ [f ] µd = 0 with L µ [f ] = α(µ) f -β∇U, ∇f . Remark 2 (Infinitesimal generator). Observe that the choice ϕ = ϕ 1 leads to α(r) = 1, that is to L µ [f ] = f -β∇U, ∇f .
This is the infinitesimal generator of the classical (overdamped) Langevin SDE. Up to a multiplicative constant, this is the only choice for the operator L µ to be independent of µ.

Let us apply formally the above relationship to the function f = βU + ϕ (µ), assuming here that the stationary density µ is smooth enough. We obtain

3 {µ>0} |∇(βU + ϕ (µ))| 2 µ d = - {µ>0} L µ [βU + ϕ (µ)] µd (10) = 0,
where the last equality comes from the stationarity of µ. Therefore, βU + ϕ (µ) is constant on every connected component of the set {µ > 0}. We analyze this condition in the next paragraph.

Uniqueness of the stationary density

The main ingredient to study the uniqueness of the stationary density is the relation

βU + ϕ (µ) = c, (11)
where the constant c depends on the considered connected component of the support {µ > 0}.

Define I ϕ ((0, +∞)) and denote by ψ : I → (0, +∞) the inverse of ϕ .

Lemma 1 (Existence and uniqueness of the minimizer of (P β )). Assume ϕ (0) = -∞, then there exists an unique stationary density µ β solution to [START_REF] Carrillo | Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus[END_REF]. Moreover, (i) µ β is positive everywhere on M and is characterized by the relation

µ β = ψ(c * -βU ), (12)
where c * is a normalization parameter characterized by the condition

M ψ(c * -βU ) d = 1. (ii) µ β is the global minimizer of U β .
Proof. Let us start with (i) and by showing that a stationary density µ is everywhere positive. Towards a contradiction, assume that the set {µ = 0} is nonempty and let M 1 be a connected component of the open set {µ > 0} with ∂M 1 = ∅. Let (x n ) n∈N be a sequence of elements of M 1 converging to a point of the boundary ∂M 1 . According to [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF], we have for every n ∈ N

βU (x n ) + ϕ (µ(x n )) = c M 1
where the left-hand side term converges to -∞ which is absurd. Therefore, M 1 = M and equation ( 11) is valid everywhere on M . Set c c M 1 . Being strictly convex, the function ϕ is one-to-one and onto between (0, +∞), and by definition of ψ, equation ( 11) rewrites:

µ = ψ(c * -βU ) (13)
As µ is a density function, we must have

M ψ(c * -βU ) d = 1. ( 14
)
Since ψ is strictly increasing and satisfies lim inf I ψ = lim -∞ ψ = 0 and lim sup I ψ = +∞, there is a unique value c * ∈ R that satisfies equation ( 14). We have

µ = ψ(c * -βU )
which ends not only the proof of the uniqueness of the stationary density µ but also gives its existence and its explicit form. To see (ii) and that µ is the global minimizer, we observe that

U β [ρ] -U β [µ] = M (ϕ(ρ) -ϕ(µ)) d + M βU (ρ -µ) d = M (ϕ(ρ) -ϕ(µ)) d + M (c * -ϕ (µ)) (ρ -µ) d = M ϕ(ρ) -ϕ(µ) -ϕ (µ) (ρ -µ) d ,
which is positive whenever ρ = µ by strict convexity of ϕ.

When no confusion can occur we simply write µ for µ β . We gather the assumptions we need regarding ϕ within Assumption (B). ϕ : [0, +∞) → R is convex, twice differentiable on (0, +∞) with ϕ > 0, and satisfies ϕ (0) = -∞.

Global minimization dynamics

Minimizing dynamics We are now in a position to provide dynamical systems meant to solve the problem (P). Inspired by Holley, Kusuoka, and Stroock's approach to simulated annealing [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], as it was simplified in [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre [Simulated annealing on R n . Study of the free energy evolution[END_REF], and using as well the gradient view provided by Otto's formalism (see Appendix), we consider, formally, the gradient system

d dt ρ(t) = -grad W U βt [ρ(t)] a.e. on R + , ρ(0) = ρ 0 , (15)
where the term β t is a C 1 positive time-varying parameter and where we use Newton's notation, d dt ρ here, for time derivatives. The initial distribution ρ 0 is chosen in the domain of U β , that is in the domain of H. The time-dependent density ρ turns out to satisfy the following partial differential equation

d dt ρ = div(ρ (β t ∇U )) + div(ρ(∇ϕ (ρ))), ρ(0) = ρ 0 . (16)
The time-varying parameter β t is traditionally interpreted as an inverse of a temperature which typically cools down, i.e., lim

t→∞ β t = +∞ (17)
Here we also interpret this parameter as the inverse of a penalty term echoing the static formula [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF].

Remark 3. (a) (Simulated annealing) When ϕ = ϕ 1 , ( 16) boils down to the famous simulated annealing dynamics

d dt ρ = β t div(ρ∇U ) + ∆ρ (18)
which, by a famous "nonconvex" extension of the log-Sobolev inequality, due to Holley, Kusuoka and Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], is known to generate measures concentrating on the set of global minimizers of U whenever the temperature schedule is finely tuned. (b) (Porous media) Taking U constant and ϕ = ϕ m in ( 16) with m > 0, the dynamic corresponds to the porous media equation

d dt ρ = m∆ (ρ m ) .
The case m > 1 refers to the slow diffusion case while the case m < 1, for which ϕ (0) = -∞ refers to the fast diffusion situation, (see Vazquez [START_REF] Vazquez | The Porous Medium Equation. Mathematical Theory[END_REF] and Otto [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]).

Existence results and evolution equations Following the pioneering work of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], the non-autonomous theory for Monge-Kantorovich gradient flows has recently been developed in [START_REF] Ferreira | Gradient flows of time-dependent functionals in metric spaces and applications to PDEs[END_REF]. In the line of [22, Theorem 4.4, Theorem 5.4] and the existence results of [START_REF] Iacobelli | Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour[END_REF], we assume that ( 15) and ( 16), have a common unique solution curve t → ρ(t) in (P(M ), W 2 ), which satisfies in addition

t → U βt [ρ(t)] and t → ρ(t) are absolutely continuous, ( 19
) d dt U βt [ρ(t)] = M (ϕ (ρ) + β t U ) d dt ρ d + d dt β t M U ρ d , (20)
where the time derivatives are taken for almost all times.

Functional inequalities Under hypothesis (A), (B) and some extra-assumptions on ϕ related to the geometry of the penalized cost, we intend to prove that the dynamics ( 15)-( 16) has global optimizing properties, in the sense that the global cost

U βt [ρ] = M (ϕ(ρ) + β t U ρ) d (21)
evaluated along the trajectory t → ρ(t) given by ( 15)-( 16) should converge to the value of (P), i.e., lim

t→+∞ U βt [ρ] = val(P) = min M U.
As it is customary in the analysis of PDEs the key to convergence is given by "entropyenergy" or "entropy-production" functional inequalities. In the "gradient or in the optimization world" , these can often be seen as Łojasiewicz type inequalities, see [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF] and references therein. They connect the cost U β to the norm of its gradient grad W U β and to the constant β:

grad W U β 2 ≥ c(β) Ω U β (ρ) -min U β , (22)
where c, Ω : (0, +∞) → R + are positive functions, with Ω being increasing and null at zero. Reexpressing U β by means of its stationary density [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] gives

U β (ρ) -U β (µ) = M ϕ(ρ) -ϕ(µ) d + M βU (ρ -µ) d = M ϕ(ρ) -ϕ(µ) d + M (c * -ϕ (µ)) (ρ -µ) d = M ϕ(ρ) -ϕ(µ) -ϕ (µ) (ρ -µ) d . Because ϕ is convex, we obtain U β (ρ) ≥ U β (µ) thus U β (µ) = min U β .
As a consequence, inequality [START_REF] Ferreira | Gradient flows of time-dependent functionals in metric spaces and applications to PDEs[END_REF] writes

M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d ≥ c(β) Ω M ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) d (23)
where c, Ω : (0, +∞) → R + are positive functions. A typical example is given by the log-Sobolev inequality of Holley, Kusuoka, and Stroock which can be written as

M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d ≥ C HKS (β) M ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) d ( 24
)
where ϕ(r) = ϕ 1 (r) = r ln(r) -r + 1 and

lim β→+∞ 1 β ln(C HKS (β)) ≥ -osc(U )
(see [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] for the precise description of the l.h.s. in terms of the landscape of U ). When U is convex and ϕ is power-like, one can also recover Gagliardo-Nirenberg inequalities of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], see [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF] for connections with Łojasiewicz inequalities.

Convergence mechanisms for a fixed penalization parameter As previously mentioned, we adapt the approach of [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre [Simulated annealing on R n . Study of the free energy evolution[END_REF] developed in the Boltzmann entropy case (ϕ = ϕ 1 ) to our generalized class of relaxations.

Let us provide a first account of the general method through the constant parameter case. For ρ ∈ P(M ) having a density with respect to , set

I[ρ] = U β (ρ) -min P(M ) U β (25) J [ρ] = T |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d (26)
where the quantities may take infinite values and where µ is the unique stationary density (see Lemma 1), so that

I[µ] = 0.
At this stage, we do not assume that β > 0 depends on time.

By time differentiation, using ( 19)-( 20) and the evolution equation, we obtain

d dt I[ρ(t)] = M ϕ (ρ) d dt ρ d + βU d dt ρ d = M (ϕ (ρ) + βU ) d dt ρ d = M (ϕ (ρ) + βU )div(ρ(β∇U + ∇ϕ (ρ))) d = - M ∇(ϕ (ρ) + βU )(β∇U + ∇ϕ (ρ))) ρd = - M |∇ϕ (ρ) + β∇U | 2 dρ.
Whence, if we have some inequality à la Łojasiewicz like (23) (as for instance the log-Sobolev inequality of Holley, Kusuoka, and Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] when ϕ = ϕ 1 ), we derive a differential inequality for the one-variable function I[ρ],

d dt I[ρ(t)] ≤ -c(β) Ω(I[ρ(t)]) (27)
This implies in turn that I[ρ(t)] converges. If this limit was not zero, the fact that Ω is positive out of 0 would imply, through [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], that the decrease-rate would be perpetually lower than a negative constant, which is absurd. This allows proving that I[ρ(t)] tends to zero as t → ∞.

We thus have proved the first part of:

Theorem 1 (Convergence with a non-vanishing penalty parameter). Assume that (A), (B) are satisfied, and that there exist c > 0, Ω : R + → R + increasing, such that an inequality of the type

M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d ≥ c Ω M ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) d , (28)
holds true whenever ρ is measurable and the left hand side is finite. Then

(i) U β [ρ(t)] → min U β .
(ii) If moreover Ω(s) = Θ(s 2θ ) at 0, with θ ∈ (0, 1), then ρ(t) tends to µ β for the Monge-Kantorovich metric, i.e., for the weak* topology.

Proof. Item (i) is already proved. For (ii), from Ω(s) = Θ(s 2θ ) holds, we deduce that the lower semicontinuous function U β satisfies a Łojasiewicz inequality as in [6, Theorem 2], so that one may assert that the curve t → ρ(t) has a finite Monge-Kantorovich length; convergence rates depending on θ are also available.

What are the conditions for the above inequality (28) to be valid, is a delicate open question. The subject of the next section, and the central result of this paper, is to establish such inequalities for one-dimensional compact manifolds and a family of power-like potentials ϕ.

A functional inequality on the circle

Main theorem

For m ∈ (0, 1/2), set

∀ r ≥ 0, ϕ m,2 (r)    ϕ m (r) if r ∈ (0, 1], ϕ 2 (r) = (r-1) 2 2 if r ∈ (1, +∞). (29)
The function ϕ m,2 is convex on [0, +∞) and C 2 on (0, +∞). The latter property is a consequence of the fact that

ϕ m (1) = 0 ϕ m (1) = 0 ϕ m (1) = 1.
Observe also that ϕ m,2 is concave on (0, +∞) with ϕ m,2 (0) = -∞, so that Lemma 1 applies. Let us recall that the unique solution of (P β ) is denoted by µ = µ β and that

ψ m,2 = [ϕ m,2 ] -1 .
This section is devoted to the proof of:

Theorem 2 (A new functional inequality on the circle). Assume that M is the circle T R/(LZ) of perimeter L > 0 endowed with its usual Riemannian structure. Then there exists a constant c(β), depending on µ min and L, such that for any measurable density ρ on

T T |∇ϕ m,2 (ρ) -∇ϕ m,2 (µ)| 2 ρ d ≥ c(β) Ω T ϕ m,2 (ρ) -ϕ m,2 (µ) -ϕ m,2 (µ)(ρ -µ) d
where, for β large,

c(β) = O β -3(2-m) 1-2m Ω(r) =        r 3 2 if r ∈ [0, 1) r 1-2m 2(1-m) if r ≥ 1.
Corollary 1 (An inequality à la Talagrand). Under the assumptions of the previous theorem, for any measurable density ρ on T,

T ϕ m,2 (ρ) -ϕ m,2 (µ) -ϕ m,2 (µ)(ρ -µ) d ≥ d(β) ω W 2 (ρ, µ)
where

d(β) = O β -3(2-m) 1-2m
, and

ω(r) =        8/5 r 5 8 if r ∈ [0, 1) 4(1 -m)/(3 -2m) r 3-2m 4(1-m) if r ≥ 1. (30)
The rest of this section is devoted to the proof of this theorem (the corollary will follow easily using a generalization of Otto-Villani theorem [START_REF] Blanchet | A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions[END_REF]). Most of the intermediary results we provide are valid for a general compact C 2 manifold, thus, unless otherwise stated, we assume for the moment that M is arbitrary. In the remaining subsections of the present section, for the sake of simplicity, we shall often write ϕ = ϕ m,2 and ψ m,2 = ψ.

Some estimates

Let us define the positive quantities

µ min min M µ and µ max max M µ. ( 31
)
Proposition 2 (Bounds for the stationary measure). We have, for any 0 < m < 1,

(1 + (1 -m)βosc(U )) 1 m-1 ≤ µ min ≤ µ max ≤ βosc(U ) + 1.
where we recall that osc(U ) = max M U -min M U .

Proof. The stationary measure µ satisfies for every x ∈ M , µ(x) = ψ(c * -βU (x)) for some real constant c * and with β > 0 (recall (13)). Because ψ is nondecreasing , we have

∀x ∈ M, ψ(c * -β max M U ) ≤ µ(x) ≤ ψ(c * -β min M U ).
Integrating with respect to the probability measure , we obtain

ψ(c * -β max M U ) ≤ 1 ≤ ψ(c * -β min M U ).
Because ϕ is nondecreasing and ϕ (1) = 0, we obtain the following bounds for the constant c *

β min M U ≤ c * ≤ β max M U.
Finally, for every x ∈ M , ψ(-βosc(U )) ≤ µ(x) ≤ ψ(βosc(U )). Because βosc(U ) ≥ 0, we have for any m ∈ (0, 1),

(

-(m -1)βosc(U )) 1 m-1 ≤ µ min ≤ µ max ≤ 1 + βosc(U ), 1
which ends the proof.

The following formal observation is essential. When the potential function ϕ is the entropy function ϕ 1 , the density of ρ with respect to µ plays a pivotal role in the establishment of the log-Sobolev inequality [START_REF] Hastie | The Elements of Statistical Learning Data Mining, Inference, and Prediction, Second Edition[END_REF], see [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. In our case, the counterpart is the function

ψ(ϕ (ρ) -ϕ (µ)).
It is also convenient to use the quantity

g := ϕ (ρ) -ϕ (µ) (32)
so that

ρ = ψ(g + ϕ (µ)). ( 33
)
An upper bound for the reduced cost I[ρ] This necessitates three steps.

Lemma 3. For any measurable density ρ, we have

ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) ≤ ϕ(ψ(g + ϕ (µ max ))) -ϕ(µ max ) -ϕ (µ max )(ψ(g + ϕ (µ max )) -µ max )
Proof. By definition of g, we have

ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) = ϕ(ψ(g + ϕ (µ))) -ϕ(µ) -ϕ (µ)(ψ(g + ϕ (µ)) -µ).
Fix x ∈ M and consider the function F defined on (0, +∞) by

∀ r > 0, F (r) ϕ(ψ(g(x) + ϕ (r))) -ϕ(r) -ϕ (r)(ψ(g(x) + ϕ (r)) -r)
To prove the result, it is sufficient to show that F is nondecreasing . For r > 0, we compute

F (r) = ϕ (ψ(g(x) + ϕ (r)))ψ (g(x) + ϕ (r)))ϕ (r) -ϕ (r) -ϕ (r)(ψ(g(x) + ϕ (r)) -r) -ϕ (r)[ψ (g(x) + ϕ (r)))ϕ (r) -1] = ϕ (ψ(g(x) + ϕ (r)))ψ (g(x) + ϕ (r)) -(ψ(g(x) + ϕ (r)) -r) -ϕ (r)ψ (g(x) + ϕ (r)) ϕ (r) = (g(x) + ϕ (r))ψ (g(x) + ϕ (r)) -(ψ(g(x) + ϕ (r)) -r) -ϕ (r)ψ (g(x) + ϕ (r)) ϕ (r) = g(x)ψ (g(x) + ϕ (r)) -(ψ(g(x) + ϕ (r)) -r) ϕ (r) = g(x)ψ (ϕ (s)) -(s -r) ϕ (r)
where we set s ψ(g(x) + ϕ (r)) in the last equality. Because ψ = (ϕ ) -1 , we have

ψ (ϕ (s)) = 1 ϕ (s)
and we get for r > 0,

F (r) = [g(x) -ϕ (s)(s -r)] ϕ (r) ϕ (s) = [ϕ (s) -ϕ (r) -ϕ (s)(s -r)] ϕ (r) ϕ (s) .
Because, the function ϕ is convex and ϕ is concave, we have ϕ (r)/ϕ (s) is positive and the quantity ϕ (s) -ϕ (r) -ϕ (s)(s -r) is nonnegative, so that F ≥ 0 on (0, +∞) and F is thus nondecreasing .

We define further ξ max (s) ψ(s+ϕ (µ max )) for any real number s and set ρ max ξ max (g). Therefore, Lemma 3 can be rewritten

ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) ≤ ϕ(ρ max ) -ϕ(µ max ) -ϕ (µ max )(ρ max -µ max ). (34)
We are in position to give an upper bound for I[ρ].

Lemma 4 (An upper bound for the reduced cost I). For every ρ ∈ P(M ) such that g ∈ L 2 ( ), we have

I[ρ] ≤ M g 2 (x) (dx).
Proof. Because ϕ is convex, we have

ϕ(µ) -ϕ(ρ) -ϕ (ρ)(µ -ρ) ≥ 0 and ϕ(µ max ) -ϕ(ρ max ) -ϕ (ρ max )(µ max -ρ max ) ≥ 0.
Adding the latter positive quantity to the right-hand-side of equation [START_REF] Ma | Sampling can be faster than optimization[END_REF] gives

0 ≤ ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) ≤ (ϕ (ρ max ) -ϕ (µ max ))(ρ max -µ max ) = g(ξ max (g) -ξ max (0)).
Recalling that ϕ has been constructed by gluing ϕ m and ϕ 2 at 1 (see equation ( 29)), we have that ψ = (ϕ ) -1 is convex and increasing with 0 ≤ ψ ≤ 1. Therefore, ξ max is convex and we have ψ (ϕ (µ max ))g ≤ ξ max (g) -ξ max (0) ≤ ψ (g + ϕ (µ max ))g.

Whence,

|ξ max (g) -ξ max (0)| ≤ |g|, which gives the desired result.

A lower bound for the squared Monge-Kantorovich gradient J [ρ] Once more several steps are necessary to obtain a bound. Let us define the function θ : R → R via

∀ r ∈ R, θ(r) = r 0 ψ(s + ϕ (µ min )) ds. (35)
We observe first that:

Lemma 5 (Lower bound for θ). Assume 0 < m < 1 2 . For any r ∈ R,

|θ(r)| ≥ 2 3 min 1 (C 0 + (1 -m)) 3 2 -η , ψ (ϕ (µ min )) min(|r| 3/2 , |r| η ),
where C 0 = 1 -(1 -m)ϕ (µ min ) and η := 1-2m 2(1-m) ∈ (0, 1/2). Proof. Assume first that r > 0. Because ψ is convex, we have for every s,

ψ(s + ϕ (µ min )) ≥ µ min + ψ (ϕ (µ min ))s. But µ min > 0 thus θ(r) ≥ ψ (ϕ (µ min )) r 0 √ s ds = 2 3 ψ (ϕ (µ min ))r 3/2 .
Now, assume that r < 0. By a change of variables, t = -s and τ = -r > 0, we get

-θ(r) = τ 0 ψ(ϕ (µ min ) -t) dt.
Remembering that ψ is both convex and positive, we get, for all t, ψ(ϕ (µ min ) -t) ≥ ψ (ϕ (µ min ) -τ )(τ -t).

We deduce that -θ(r) ≥

2 3 ψ (ϕ (µ min ) -τ )r 3/2 .
We shall now use the explicit form of the derivative of ψ given by ψ (τ )

   (1 + (m -1)τ ) 2-m m-1 if τ ∈ (-∞, 0) 1 if τ ∈ (0, +∞) (36)
By definition of µ min , we have µ min < 1 whenever U is not constant over M . Therefore ϕ (µ min ) ≤ 0 < τ , so that 1) .

ψ (ϕ (µ min ) -τ ) = 1 -(1 -m)ϕ (µ min ) + (1 -m)τ 2-m 2(m-
Set C 0 = 1 -(1 -m)ϕ (µ min ), α = 2 -m 2(1 -m) ∈ (1, 3/2) and η = 3/2 -α = (3(1 -m) -(2 -m))/(2(1 -m)) = (1 -2m)/2(1 -m).
Therefore, -θ(r) ≥ 2 3

τ 3/2 (C 0 + (1 -m)τ ) α = 2 3
τ 3/2 (C 0 + (1 -m)τ ) α 1 0<τ ≤1 + τ 3/2 (C 0 + (1 -m)τ ) α 1 τ >1 ≥ 2 3 τ 3/2 (C 0 + (1 -m)) α 1 0<τ ≤1 + τ 3/2 (C 0 τ + (1 -m)τ ) α 1 τ >1 ≥ 2 3 1 (C 0 + (1 -m)) α min(τ 3/2 , τ η ).
Remark 4 (On constants). For reasons that will appear later during the study of our global optimization method, it is useful to have a compact expression for the inverse of the constant 2/(3(C 0 + (1 -m)) α ) appearing in Lemma 5. Using the equality ψ (ϕ (µ min )) = 1/ϕ (µ min ), this inverse writes

C 1 (µ min ) := 3 2 max (1 + (1 -m)(1 -ϕ (µ min ))) 2-m 2(1-m) , ϕ (µ min ) > 1.
Observe that, as a function of µ min , C 1 is decreasing and therefore is bounded above by

3 2 max 1 + (1 -m) 1 -ϕ [1 + (1 -m)βosc(U )] 1/(m-1) 2-m 2(1-m) ; ϕ ((1 + (1 -m)βosc(U )) 1/(m-1) ) ,
according to Proposition 2. When β goes to +∞, this bound behaves as O β

2-m 2(1-m) . Finally note that C 1 (µ min )|θ(r)| ≥ min(|r| 3/2 , |r| η ) for all r. (37)
We now turn to the desired lower bound for the squared Monge-Kantorovich gradient J [ρ].

Lemma 6 (Lower bound for the squared Monge-Kantorovich gradient). We have

J [ρ] = M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d ≥ M |∇θ(g)| 2 d .
Proof. Taking into account that both ϕ and ψ are nondecreasing functions, we get

ρ = ψ(g + ϕ (µ)) ≥ ψ(g + ϕ (µ min ))
= (θ (g)) 2 .

It ensues

M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d = M |∇g| 2 ρ d ≥ M |∇g| 2 (θ (g)) 2 d = M |θ (g)∇g| 2 d = M |∇θ(g)| 2 d

Proof of Theorem 2 and its corollary

Assume for the moment that M is arbitrary.

In the previous section, we have proved two inequalities:

I[ρ] ≤ M g 2 d and J [ρ] ≥ M |∇θ(g)| 2 d .
To reach a conclusion, it suffices to relate the quantities When β is large enough C 1 (µ min ) ≥ 1, we may thus write

g 2 ≤ C 2 (µ min ) max(|h| 4 3 , |h| 2 η ),
where we have set h := θ(g) and

C 2 = C 2 η
1 (µ min ). Whence, taking the supremum and integrating yields

M g 2 d ≤ C 2 (µ min ) max(||h|| 4 3 ∞ , ||h|| 2 η ∞ ).
Recall that the function Ω : R + → R + is such that

Ω(r) r 3 2 if r ∈ [0, 1)
r η if r ≥ 1.
Let us prove that the increasing function Ω satisfies the inequality

∀x > 1, ∀y > 0, Ω(xy) ≤ x 3 2 Ω(y). ( 38
)
Let us consider two cases: First case: y ≥ 1. Because x > 1, this implies xy > 1. Thus,

Ω(xy) = (xy) η = x η Ω(y) ≤ x
Second case: y < 1. When xy > 1, the inequality follows as above. On the other hand, if xy < 1, we have Ω(xy) = (xy)

3 2 = x 3 2 Ω(y).
Therefore, by using the fact that Ω is increasing and satisfies [START_REF] Raginsky | Non-convex learning via stochastic gradient Langevin dynamics: a non-asymptotic analysis[END_REF], we get

Ω(I[ρ]) ≤ Ω M g 2 d ≤ (C 2 (µ min )) 3 2 ||h|| 2 ∞ .
To end the proof, it remains to compare

||h|| 2 ∞ and M |∇h| 2 d .
It is only at this point that we use the assumption about the dimension of M . Let us start first by an observation regarding regularity and prove that if J is finite then ρ must be continuous. Observe first that for γ > 0 and any measurable ρ, we have

J (ρ) = M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d (39) = M |∇ϕ (ρ)| 2 ρ d + M |∇ϕ (µ)| 2 d -2 M ∇ϕ (µ)∇ϕ (ρ)ρ d (40) ≥ 1 2 M |∇ϕ (ρ)| 2 ρ d - M |∇ϕ (µ)| 2 ρ d ( 41
)
where we have used 2|∇ϕ (µ)∇ϕ (µ

)| ≤ 2|∇ϕ (µ)| 2 + 1 2 |∇ϕ (ρ)| 2 . Setting v(r) = r 1 √ sϕ (s) ds, for r > 0 we see that M |∇ϕ (ρ)| 2 ρ d = M |∇v(ρ)| 2 d .
Thus the finiteness of J [ρ] implies that M |∇v(ρ)| 2 d is finite too. Assuming now that dim M = 1, standard results ensures that v(ρ) is absolutely continuous and thus so is ρ (so that we have furthermore (v(ρ)) 2 d < +∞ and v(ρ) belongs to the Sobolev space W 1,2 (M )).

Observe that we also obtain that ρ is positive everywhere. Since the function ρ -µ is continuous and satisfies M ρ -µ d = 0, there exists at least one point x 0 in M , such that ρ(x 0 ) -µ(x 0 ) = 0. It follows from (32) that g(x 0 ) = 0 and from (35) that h(x 0 ) = 0 (where h = θ(g)). For any x ∈ R/(LZ), denote by [x 0 , x] the shortest segment in R/(LZ) with boundary points {x 0 , x}. Since h is absolutely continuous:

h 2 (x) = [x 0 ,x] h (y) (dy) 2 ≤ ([x 0 , x]) [x 0 ,x] (h (y)) 2 (dy) ≤ L 2 M |∇h| 2 d .
Gathering the previous results gives

J [ρ] ≥ c(β)Ω(I[ρ]), with c(β) = 2 L C 2 (µ min ) -3 2 (42) Since 2/η = 4(1 -m)/(1 -2m) and for β large enough, C 1 = O β 2-m 2(1-m) , one has C 2 = O β 2(2-m) 1-2m and
c(β) = O β -3(2-m) 1-2m
.

Let us conclude with the proof of Corollary 1. The key is to observe that 1/ √ Ω is integrable at 0, so that the inequality is a Łojasiewicz gradient inequality. It suffices to use the generalization of Otto-Villani theorem provided in [6, Theorem 1(i)].

Time-dependent swarm gradient methods

Time dependence is key to obtain convergence to the actual global minimum: the penalty schedule β is tuned so that the exploratory forces embodied in H are sufficiently active at the beginning of the process while progressively loosing their influence on the dynamics as global goals have been achieved. In this second phase, as the diffusion process generated by H fades away, the gradient dynamics of U dominates and somehow terminates the process. As in the famous simulated annealing method, the presence of a functional inequality is fundamental for the dynamical system to converge.

In the remainder, unless otherwise stated we use a general potential ϕ.

Main convergence results

Convergence under a functional inequality The following general theorem shows the global optimization properties of ( 16), provided that a functional inequality is available (as in simulated annealing or as in Theorem 4 below).

Theorem 3 (Global optimization under a weak functional inequality). Assume the set of hypothesis A, B are met, and that β, U, satisfy a functional inequality of the type:

M |∇ϕ (ρ) -∇ϕ (µ)| 2 ρ d ≥ c(β) Ω M ϕ(ρ) -ϕ(µ) -ϕ (µ)(ρ -µ) d
where c, Ω : (0, +∞) → R are positive with Ω being nondecreasing. If the penalization schedule t → β(t) satisfies

lim t→+∞ β(t)/c(β(t)) = 0 (43) +∞ 1 c(β(t)) dt = +∞ (44)
To give an upper bound of the second term, we write

βt M U (ρ -ν t ) d = βt M U ρ d - M U ν t d ≤ osc(U ) βt ,
where the last inequality uses that ρ and ν t are probability densities on M . Finally, we obtain

v(t) ≤ -c(β t )Ω(v(t)) + osc(U ) βt ,
We are thus led to consider differential inequalities of the type

v ≤ -c(β)Ω(v) + δ β
where v : R + → R + is a nonnegative function, d > 0 is a positive constant and Ω is an nondecreasing function taking positive values on (0, +∞).

Our goal is now to give conditions on the inverse temperature scheme β : R + → (0, +∞) ensuring that v converges to zero for large times: Proposition 7 (Schedule conditions). Assume that for large times, (43) and (44) hold, then

lim t→+∞ v(t) = 0.
The proof of Proposition 7 is based on the two following observations. Lemma 8. Under the assumptions of Proposition 7, we have

lim inf t→+∞ v(t) = 0.
Proof. Towards a contradiction assume that there exist > 0 and T 0 ≥ 0 such that v(t) ≥ for all t ≥ T 0 . Then, Ω being nondecreasing,

∀ t ≥ T 0 , v(t) ≤ -c(β(t))Ω( ) + δ β(t) .
From the first condition (43), there exists T 1 ≥ T 0 such that

∀ t ≥ T 1 , β(t) ≤ 1 2 c(β(t))Ω( )
and thus

∀ t ≥ T 1 , v(t) ≤ - 1 2 c(β(t))Ω( ).
This implies

∀ t ≥ T 1 , v(t) ≤ v(T 1 ) - 1 2 Ω( ) t T 1 c(β(s)) ds,
so letting t go to infinity, due to the second condition (44), lim t→+∞ v(t) = -∞, in contradiction with the nonnegativity of v.

Our second ingredient is:

Lemma 9. Under the assumptions of Proposition 7, fix > 0. Then there exists T ≥ 0, such that

∀ t ≥ T, v(t) = ⇒ v(t) < 0 .
Proof. Indeed, consider

T inf τ ≥ 0 : ∀ t ≥ τ, - 1 2 c(β(t))Ω( ) + δ β(t) ≤ 0
which is finite by assumption (43). Proceeding as in Lemma 8, for any t ≥ T such that v(t) = , we may assert that

v(t) ≤ - 1 2 c(β(t))Ω( ) < 0.
The proof of Proposition 7 is as follows: by Lemma 8, for any arbitrary small level > 0, v will always go below at some point, but by Lemma 9 there exists a time T after which it will no longer be able to cross it upward. Whence v must be below for large times. This concludes the proof of Proposition 7.

Let us now come back to the proof of Theorem 3: since v(t) tends to zero, we have for all > 0 a T 0 such that

I[t, ρ] M ϕ(ρ) d + M β t U ρ d - M ϕ(ν t ) d + M β t U ν t d ≤ that is M U ρ d - M U ν t d ≤ β -1 t M ϕ(ν t ) d - M ϕ(ρ) d + which implies in turn M U ρ d -min M U ≤ β -1 t M ϕ(ν t ) d + + M U ν t d -min M U ≤ gap(β t ) + β -1 t .
The value of being arbitrary, the results follows.

One-dimensional global optimization

We may now combine Theorem 3 with the functional inequality we obtained previously in Theorem 2.

Theorem 4 (Global optimization by swarm gradient in dimension 1). Assume that M is the circle T R/(LZ) endowed with its usual Riemannian structure, that the function ϕ is as in (29), i.e., ϕ = ϕ m,2 and that the schedule is given by

β(t) = kt 1/γ , with k > 0 and γ = 3(2-m) 1-2m ∈ [6, +∞). Then lim t→∞ U[ρ(t)] = min M U (47)
Proof. Theorem 2 provides a functional inequality as required by Theorem 3. From this inequality we may assume that c(β) = κβ -γ for some κ > 0. Now, using Remark 5, we choose

β(t) = c -1 (t) = κ -1 t

A stochastic view on the dynamics: particles swarm optimization

In this last section, we sketch a possible discrete approach of the swarm gradient dynamics; it will be properly developed in future works. It relies on the diffusion process associated with the evolution equation

∀ t ≥ 0, d dt ρ = div(ρ(β t ∇U + ∇ϕ (ρ))). (48)
To evidence this link, let us use the formal integration by parts presented in Section 2.2 to obtain at any time t ≥ 0 and for every f

∈ C ∞ (M ), {ρt>0} L t,ρ [f ] ρ t d = M f div(ρ(β t ∇U + ∇ϕ (ρ))) d
where

L t,ρ [f ] = α(ρ) f -β t ∇U, ∇f (49)
with α : (0, +∞) → R + given by

∀ r > 0, α (r) 1 r r 0 sϕ (s) ds.
It is then natural to associate to equation (48) a Markov process (X t ) t≥0 whose infinitesimal generator is given by (49) and whose law has density ρ(t) for all t ≥ 0. Due to the dependence of the evolution on the density, such a Markov process is said to be nonlinear. When M is a flat torus of dimension d, consider the stochastic differential equation

X t = X 0 - t 0 β s ∇U (X s ) ds + t 0 α(ρ(X s )) dB s , (50)
where (B s ) s≥0 is a Brownian motion of dimension d. If (X, ρ) is a solution of the stochastic differential equation (50) then an application of the Itô formula shows that ρ is a solution of equation ( 48). This observation can be extended to any compact Riemannian manifold. Two particular choices for which the existence of (X, ρ) has been established in the literature are (i) ϕ(ρ) = ρ log(ρ), we have α ≡ 1, so L t,ρ does not depend on ρ and is the Langevin generator • -β t ∇U, ∇• . The existence and uniqueness of a strong solution to the extension of (50) on a compact Riemannian is well-known, as soon as ∇U is Lipschitz, which is true if U is smooth.

(ii) ϕ(ρ) = ρ m which corresponds to the equation of porous media. On R, the existence and uniqueness of a strong solution to (50) is proven by Benachour, Chassaing, Roynette and Vallois [START_REF] Benachour | Processus associés à l'équation des milieux poreux[END_REF] (see Belaribi and Russo [START_REF] Belaribi | Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation[END_REF] for more general functions ϕ).

The main difficulty in investigating the existence and uniqueness of (50) is the presence of the density ρ s . It can be relaxed if it could be replaced by integrals of smooth functions with respect to ρ s (x) (dx). It leads us to consider convolutions of ρ s with respect to some smooth kernels, as those traditionally used in statistics for density estimation, cf. e.g. Silverman [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF]. In our geometric setting, it seems natural to resort to the heat kernel (associated to the Laplace-Beltrami operator). To avoid the introduction of further notation, let us just consider the case of a flat torus M = (R/Z) d , endowed with its usual Riemannian structure. Let K : (R/Z) d → R + be a smooth function with support in [-1/4, 1/4] d (seen as a subset of (R/Z) d ) and such that M K d = 1. For any probability density ρ on M and h ∈ (0, 1), which is a bandwidth parameter, set

∀ x ∈ M, ρ h (x) h -d M K((x -y)/h) ρ(y) (dy). ( 51
)
When h is small, ρ h is an approximation of ρ. Replacing in (50), ρ s by ρ h,s , we end up with the stochastic differential equation

X t = X 0 - t 0 β s ∇U (X s ) ds + t 0 α(ρ h (X s )) dB s , (52)
which is simpler to investigate, taking into account the usual mean field theory (see for instance Del Moral [START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF]). Furthermore, (50) admits natural particles approximations. Indeed, (51) can be extended to any probability measure π on M , by replacing ρ(y) (dy) by π(dy). In particular, it makes sense for the empirical measure of N particles, which is a crucial feature for particle approximations. More precisely, consider a system of N particles, X 1 , X 2 , ..., X N whose joint evolution is described by the stochastic differential equations,

∀ n ∈ N , dX n (t) = -β t ∇U (X n (t)) + α(ρ N,h (X n (t))) dB n (t) (53)
where the (B n (t)) t≥0 , for n ∈ N , are independent Brownian motions of dimension d, and where

∀ x ∈ M, ρ N,h (x) 1 N N n=1 1 h K x -X n (t) h
namely ρ N,h is given by (51) when ρ is replaced by the empirical measure

ρ N (t) 1 N N n=1 δ Xn(t)
Resorting to mean field theory and chaos propagation, when N tends to infinity, the random evolution (ρ N (t)) t≥0 converges in probability toward the dynamical system (ρ(t)) t≥0 (for related initial conditions), the law of (X 1 (t)) t≥0 converges toward that of (X t ) t≥0 , solution of (52), and the trajectories of any fixed finite number of particles become asymptotically independent.

Putting together these observations, we believe that in addition to (β t ) t≥0 , schemes (h t ) t≥0 and (N t ) t≥0 can be found, with lim t→+∞ h t = 0 and lim t→+∞ N t = ∞, so that for large times t, the corresponding empirical measures ρ Nt concentrate around the set of global minima of U . There are several ways to increase the number of particles, the most natural one might be to duplicate some of the current particles -but it is also possible to make them appear in independent random positions.

This procedure would provide a new stochastic algorithm for global minimization. At least up to the simulation of particle systems such as (53), but this can be done through the traditional Euler-Maruyama scheme.

At the current stage of our understanding, we are not able to properly compare the theoretical or practical performances of both approaches. We merely provide some thoughts in that direction. One of the motivations of our study was to use approximating particles in order to improve the global knowledge of the energy landscape, with the hope that a swarm of interacting particles might do better than a swarm of independent ones (as in simulated annealing where particles follow Langevin-like dynamics with independent Brownian motions).

What follows is, therefore, informal; it is only meant to illustrate through rough approximations and simple dynamics that interacting swarm methods may be more advantageous.

Exit times of swarm methods: interacting particles vs. independent particles We examine exit times from wells for Langevin dynamics and our swarm method when β is large enough. For simplicity, we assume the schedule β t to be constant.

Consider the d-dimensional torus T d and the stochastic differential equation:

dZ t = √ 2dB t -β∇U (Z t ) dt. (54)
where For the "swarm-like" Markov process X (X t ) t≥0 whose evolution is described by (3), we proceed similarly and consider the exit time

(B t ) t≥0 is a d-dimensional
τ swarm t inf{s ≥ 0 : X t+s ∈ W }.
As it will be pleaded below through heuristic considerations, when β is large enough, we expect:

-when b is a non-global minimizer:

τ swarm t << τ Lang t , (55)
-when b is a global minimizer:

τ swarm t >> τ Lang t . (56)
In other words, in the presence of interaction forces as in (3), particles escape faster from non-global minimizers and stay longer in the vicinity of global ones.

Heuristic elements for the comparison For large β > 0, the time t and the position Z t = z being fixed, τ Lang t is of order exp(hβ) in the following sense:

lim β→+∞ 1 β ln(E z [τ Lang t ]) = h (57)
Actually, finer results are available, Eyring-Kramers formula provides the pre-exponential factors in terms of the Hessian of U at b and on the boundary of W , see, e.g., Chapter 11 of Bovier and den Hollander [START_REF] Bovier | Grundlehren der Mathematischen Wissenschaften[END_REF].

Let us come back to our swarm dynamics and the Markov process X (X t ) t≥0 whose evolution is described by [START_REF] Bakry | Diffusions hypercontractives[END_REF]. Suppose that at some time t ≥ 0, X t is close to b and consider the exit time

τ t inf{s ≥ 0 : X t+s ∈ W }
This random time is more complex to apprehend than the exit time of the Langevin dynamic -since the evolution of (X s ) s≥0 uses its marginal distributions. Yet, as we will see, several natural approximations heuristically suggest that it is generally shorter than τ Lang t .

Let us investigate the approximations we mentioned. The diffusive coefficient of X s at time s ≥ 0 within (3) is α(ρ s (X s )). According to the previous sections, for large s ≥ 0, the law ρ s of X s is close to the invariant probability measure µ β . In the large time regime, it is thus natural to approximate the swarm dynamics by the process ( X s ) s≥0 defined by

∀ s ≥ 0, d X s = 2α(µ β ( X s ))dB s -β∇U ( X s ) ds
and denote τ t inf{s ≥ 0 : X t+s ∈ W } assuming once more that X t is close to b.

We now provide two lemmas suggesting an even simpler approximation together with some estimation of the exit time.

To simplify the notation, we assume, with no loss of generality, that min T d U = 0.

Lemma 10 (An approximation of the diffusive coefficient). There exists a constant C > 0 such that for β ≥ 1 large enough,

c β ≤ Cβ d d+2 .
Furthermore, for any x ∈ T d such that U (x) > 0,

lim β→+∞ 1 β α(µ β (x)) = U (x).
Proof. Denote x 0 a global minimum of U , namely satisfying U (x 0 ) = 0. Due to the regularity of U , there exists r > 0 small enough and C > 0 big enough so that

∀ x ∈ B(x 0 , r), U (x) ≤ C x -x 0
where B(x 0 , r) is the ball centered at x 0 and of radius r in T d . We deduce:

1 ≥ B(x 0 ,r) ψ(c β -βU (x)) (dx) ≥ B(x 0 ,r) c β -βU (x) (dx) ≥ B(x 0 ,r) c β -Cβ x -x 0 2 (dx) = (B(x 0 , r))c β -Cβ B(x 0 ,r) x -x 0 2 (dx)
where in the second bound, we used that x -x 0 2 (dx)

∀ s ∈ R, ψ(s) ≥ s due to the fact that ∀ s ∈ R, ϕ (s)
Note that for small r > 0, (B(x 0 , r)) is of order r d and B(x 0 ,r) x -x 0 2 (dx) of order r d+2 .

Thus taking r = β -1 d+2 for β ≥ 1 large enough, we end up with the first announced result. It follows that for any x ∈ T d with U (x) > 0, we have for large β > 0, c β -βU (x) ∼ -βU (x) so taking into account the behavior of α at 0 and ψ at -∞, we deduce the second announced result.

The above lemma suggests that the swarm process may be seen, for large times, close to the process X app ( Xs ) s≥0 given by

∀ s ≥ 0, dX app t = 2βU (X app s )dB s -β∇U (X app s ) ds. (58)
The corresponding exit time is:

τ app t inf{s ≥ 0 : X app t+s ∈ W } assuming that X app t is close to b.
In order to estimate this quantity, we shall use the following:

Lemma 11 (On exit times and parametrization). Consider the diffusion Z ( Z t ) t≥0 satisfying

d Z t = 2βS( Z t )dB t -β∇U ( Z t ) dt ( 59
)
where S is positive and smooth function on T d . Let

τ t inf{s ≥ 0 : Z t+s ∈ W }.
Then, assuming Z t = z for some time t,

∀ β, E z [ τ t ] = C β (60)
where C > 0.

Proof. Let us denote the dependence of the exit time through β by using the τ t,β instead of τ t . The scaling property of the Brownian motion shows that the law of the corresponding exit time τ t,β coincides with the law of τ t,1 /β, so that τ β is now of order 1/β in the sense that

∀ β, E z [ τ t,β ] = E z [ τ t,1 ] β ( 61
)
where z is the common starting point of the underlying diffusions Z.

Let us now proceed to a discussion on the exit times of interacting and independent particles process from a well with bottom b.

-The local minimum b is not global. In this case, we may "infer" from (61) that τ app t is of order 1/β 1, implying that X app exits very rapidly from W and in the interval [t, t + τ app t ]. Due to our approximations, we hope for the same behavior for τ swarm t : X should indeed escapes from W in a time of order 1/β. On the other hand, when at a large time t ≥ 0, the position of the simulated annealing algorithm X t happens to be close to b, then the exit time τ swarm t is of order exp(hβ). This is the reason behind the exit time comparison (55).

-The local minimum b is global. In that case, Lemma 11 does not apply since U approaches 0 (recall that min M U = 0). But this also means that the diffusive coefficient vanishes as one is closer to b, so that gradient forces dominate, yielding a trapping effect near b. This trapping effect is very likely to be superior to the case when the diffusion is constant as for the Langevin diffusion.

This suggests the exit time comparison (56).

Gathering these observations, and assuming our heuristic considerations have some validity, we expect the swarm algorithm to converge faster (and, hopefully, in a stronger sense) towards the global minima than the classical simulated annealing method.

Appendix

6.1 On the differential calculus in the probability space P(M )

In this section, we provide some ingredients for an understanding of the equivalence between (a), (b), and (c).

We recall beforehand that the pushforward of a measure µ through a Borel map T : M → M , denoted T #µ, is defined by (T #µ)(A) = µ(T -1 (A)) for all Borel sets A ⊂ M . We recall that for f : M → R bounded, we have Let us sketch some essential ideas to view (P(M ), W 2 ) through a "Riemannian space" perspective. To do this, we use, sometimes very informally, some of the ideas of the book [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], Chapter 8.

First, we need to have a notion of an absolutely continuous path on P(M ). Given a family of sufficiently regular vector fields4 on M denoted by b t , t ∈ (-η, η) for η > 0, we consider the ordinary differential equation

∂ t φ t (x) = b t (φ t (x)), φ 0 (x) = x ∈ M. ( 62
)
This defines a family of diffeomorphisms φ t on M ; for any µ ∈ P(M ), we define µ t = φ t #µ.

If we consider vector fields on M such that η -η M |b t (x)| 2 dµ t dt < +∞, one may derive, using integration by parts, that µ t satisfies the continuity equation

∂ t µ t + div M (µ t b t ) = 0
in the sense of distribution. Roughly speaking, curves (µ t ) t∈(-η,η) defined by the above equation are the absolutely continuous curves on P(M ) (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Theorem 8.3.1.]). When one uses the derivative ∂ t µ t | t=0 to define a "tangent space" to P(M ) at µ, we might proceed to the following identification: This provides a "differential structure" to P(M ), so we can now turn to the "Riemannian" interpretation of the Monge-Kantorovich distance by Otto to understand the equivalence between the views (b) and (c). For this, we may use the two following "metrics": where φ b t is the flow associated to the ODE (62) with vector field b. Then by using Riesz representation idea, we may formally associate two "concepts" of gradients to this differential mapping, grad W J [µ] ∈ Tµ P(M ) and ∇ W J [µ] ∈ Tµ P(M )

dJ [µ](b) = grad W J [µ], div M (µb ) = - M ∇ W J [µ], b(x) dµ,
where b is an arbitrary L 2 vector field. Once gradients are defined, one can consider gradient curves for J through:

∂ρ t = div M (ρ t ∇ W J [ρ t ]) = -grad W J [ρ].
The interested reader may consult [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for a proper derivation of these concepts and the use of subgradient curves which are actually necessary to define properly the gradient systems we use in this paper. Even though J = U β is not differentiable (it is not actually defined on the whole of P(M )), we proceed formally and we obtain

∇ W U β [µ] = β∇U + ∇ρ ρ .
The corresponding Monge-Kantorovich gradient equation therefore writes

d dt ρ(t) = -grad W U βt [ρ(t)],
that is d dt ρ = β t div M (ρ∇U ) + ∆ρ, t ≥ 0.

which is exactly the Fokker-Planck equation (b). This shows the equivalence between (b) and (c). Similar considerations apply to the general swarm methods considered in Section 2.

We now establish the connection between the points of view (a) and (b). We consider the solution (X t ) t to the time-inhomogenous Langevin-like stochastic differential equation where β t is assumed to be continuous. For every smooth function φ on M , Itô's formula gives for t ≥ 0 and h > 0, E[φ(X t+h )] = E[φ(X t )] + E t+h t (-β s ∇U (X s ), ∇φ(X s ) + ∆φ(X s )) ds .

Denoting by p(t, x) the density function of the probability distribution of X t , we deduce

2. 1 A

 1 family of relaxations in the probability space P(M ) Consider the nonconvex minimization problem: Find a global minimizer of U : M → R on a compact Riemannian manifold M .(P)

M g 2

 2 d and M |∇θ(g)| 2 d . Since η ∈ (0, 1/2), Lemma 5 and (37) (see Remark 4) gives C 1 (µ min )|θ(r)| ≥ |r| 3/2 , when |r| < 1 and C 1 (µ min )|θ(r)| ≥ |r| η otherwise.

  Brownian motion and U a smooth mapping. Given a time t ≥ 0, a local minimizer b of U over M , a given height h > 0, denote by W the connected component of {x ∈ T d : U (x) ≤ U (b) + h} containing b and which satisfies U (b) = min W U . We assume Z t to be close to b and consider the exit time τ Lang t from W : τ Lang t inf{s ≥ 0 : Z s+t ∈ W }

  ≤ s by our choice of the convex function ϕ. Thus we have c β ≤ 1 (B(x 0 , r)) 1 + Cβ B(x 0 ,r)

Mf

  (x)(T #µ)(d ) = M f (T (x)) µ(d ).

  Tµ P(M ) = {-div M (µb) : b vector field on M , M |b(x)| 2 dµ < +∞} or Tµ P(M ) = {b vector field on M : M |b(x)| 2 dµ < +∞} = L 2 (M ; µ).

  div M (µb), div M (µb ) = M b(x), b (x) dµ.Consider a functional J : P(M ) → R, a probability µ and assume that for every b ∈ L 2 (M ; µ), the following quantity existsdJ [µ](b) = lim ε→0 1 ε J (φ b #µ) -J (µ)

6. 2

 2 On the "equivalence" between (a), (b), (c)We start by connecting (b) to (c) thanks to the previous paragraph. For this, one formally applies the previous considerations to the relative entropy functionalU β (µ) = β M U (x)µ(x)d (x) + M ρ(x) log(ρ(x)) d (x).

M

  φ(x)(p(t + h, x) -p(t, x)) d = t+h t M (-β s ∇U (x), ∇φ(x) + ∆φ(x))p(s, x) d ds.Dividing by h and letting h tend to zero, we obtainM φ(x)∂ t p(t, x) d = M (-β t ∇U (x), ∇φ(x) + ∆φ(x))p(t, x) d .Integrating by parts the right-hand side, we have for every smooth function φ,M φ(x)∂ t p(t, x) d = M φ(x) (β t div M (p(t, x)∇U (x)) + ∆p(t, x)) d ,which shows that the density function of X t satisfies the Fokker-Planck equation (b).

Often called the inverse temperature

Since β is variable, they are actually time-dependent swarm dynamics.

One may observe that the first term in[START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] is the squared norm of the Monge-Kantorovich gradient of U β evaluated at µ.

Ω(y).

γ . This is the choice made by assumption provided that k = κ -1 . One easily checks that the assumptions of Theorem 3 are satisfied, hence the result.

See, e.g.,[START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] Lemma 8.1.4]

Acknowledgements. JB acknowledges the support of ANR-3IA Artificial and Natural Intelligence Toulouse Institute, and thank Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, and ANR MasDol. JB, LM, SV acknowledges the support of ANR Chess, grant ANR-17-EURE-0010. LM and SV thank Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA8655-22-1-7016.

the authors acknowledge funding from ANR under grant ANR-17-EUR-0010 (Investissements d'Avenir program).

then

where the quantity gap(β t ) → 0 was defined in [START_REF] Bovier | Grundlehren der Mathematischen Wissenschaften[END_REF].

Remark 5 (Existence of a schedule). Assume that the function c satisfies for large β > 0,

for some exponent γ > 0 (as it is the case in Proposition 2). Then for any α ∈ (0, 1/(1 + γ)), any penalization schedule t → β(t) such that for large enough t > 0, β(t) = t α satisfies the assumptions of the convergence theorem, as it is readily checked.

Proof of Theorem 3

Proof. Recall that the evolution equation ( 16) writes

In view of ( 12), the curve of stationary measures ν t µ βt , satisfies, for each fixed t

The functionals I and J used in Section 2.4 are adapted to the time-inhomogeneous case as follows

for any admissible ρ and time t ≥ 0. Using regularity results ( 19)-( 20), the differentiation of the objective along the evolution curve t → ρ of ( 16) yields

where we have used (46) in the first equality, the evolution equation in the second-one, and finally integration by parts. Let us set v(t) = I(t, ρ). Using the functional inequality gives


Romain Silhol

Mathieu Maheu-Giroux

Nirali Soni

Arlette Simo Fotso

Nicolas Rouveau

Anthony Vautier

Clémence Doumenc-Aïdara

Christinah Kouakou N'guessan

Peter Thomas Mukandavire

Vickerman

Identifying population-specific HIV diagnosis gaps in Western Africa and assessing their impact on new infections: a modelling analysis for Côte d'Ivoire, Mali and Senegal

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

3) Contribution of undiagnosed PLHIV to HIV transmissions (tPAF, Table 2)

•

Methods

Results

Background & Aim

Conclusion References

•

  Measuring diagnosis gaps among people living with HIV (PLHIV), including key populations (KP) such as female sex workers (FSW), their clients, and men who have sex with men (MSM), and their contribution to HIV transmissions can improve HIV testing programs. • The ATLAS program 1 distributed 400,000 HIV self-tests (HIVST) over 2019-2021 in Côte d'Ivoire, Mali, and Senegal to help reduce these gaps • We used mathematical modelling to estimate HIV diagnosis gaps by risk group in the 3 countries (Figure 1), and their contribution to HIV transmission over 2012-2021 Identifying population-specific HIV diagnosis gaps in Western Africa and assessing their impact on new infections: a modelling analysis for Côte d'Ivoire, Mali and Senegal.

Figure 1 :Figure 2 :

 12 Figure 1: Map of ATLAS countries in Western Africa

  Undiagnosed PLHIV may have contributed to ~ 75% of HIV transmissions over 2012-2021 (tPAF=82% in Mali, 74% in Côte d'Ivoire, 71% in Senegal); especially due to men • Côte d'Ivoire: tPAF of undiagnosed non-KP ~ 80% • Mali: tPAF of undiagnosed FSW and clients >40% (importance of sex work in epidemic dynamics, while tPAF undiagnosed MSM ~ 5%) • Senegal: tPAF of undiagnosed MSM ~33%, and almost 25% for undiagnosed clients

Deterministic model of HIV transmission by risk groups (15-59 years old

  New efforts are needed to reduce HIV diagnosis gaps, including an increased availability of confidential HIV testing modalities, which could subsequently help further reduce HIV incidence in Western Africa R. Silhol 1 , M. Maheu-Giroux 2 , N. Soni 1 , A. Simo Fotso 3 , N. Rouveau 3 , A. Vautier 4 , C. Doumenc-Aïdara 4 , O. Geoffroy 5 , K.N. N'Guessan 5 , C. Mukandavire 6 , P. Vickerman 7 , A. Keita 8 , C.T.Ndour 9 , J. Larmarange 3 , M-C. Boily 1 , for the ATLAS Team 1 Imperial College London, United Kingdom, 2 McGill University, Canada, 3 Ceped, Université Paris Cité, IRD, Inserm, France, 4 Solthis, Sénégal, 5 Solthis, Côte d'Ivoire, 6 London School of Hygiene & Tropical Medicine, United Kingdom, 7 University of Bristol, United Kingdom, 8 Institut National de Recherche en Santé Publique (INRSP), Mali, 9 Ministère de la Santé et de l'Action Sociale, Sénégal

	each
	country
	• We used each country-specific calibrated model to estimate
	1) Fraction of undiagnosed infections in 2022
	2) Distribution per risk group of undiagnosed infections in 2022
	3) Contribution of undiagnosed PLHIV to HIV transmission by risk groups
	over 2012-2021 (i.e., transmission population-attributable fraction;
	tPAF)
	• Large fractions of infections in the 3 countries are still undiagnosed (up to ~
	50% in Mali), although there remain uncertainties around the coverage of
	HIV diagnosis among male and KP PLHIV
	• HIV testing services and approaches in Western Africa leave members of KP
	behind, especially MSM
	•

) • Parameterized and calibrated by risk group following a review of demographic, behavioural, HIV (Figure 2) and intervention data (e.g. % PLHIV ever HIV tested, % diagnosed, # annual conventional tests) in 1 Rouveau et al. BMC Public health, 2021 2 Hakim et al. PLoS One, 2018 Côte d'Ivoire Senegal Mali

Table 1 . Estimated fraction of PLHIV with an undiagnosed infection in January 2022. Median and 95%UI (2.5th and 97.5th percentiles) of estimates are shown.

 1

	Côte d'Ivoire	Mali	Senegal

Clients

37% (32-43) 57% (48-65) 23% (14-34) MSM 45% (36-58) 40% (29-52) 57% (38-74) Key populations (KP) 37% (33-43) 52% (44-59) 42% (33-53) Non-KP females 18% (14-21) 41% (37-45) 10% (7-14) Non-KP males 36% (31-42) 62% (52-69) 25% (15-35) 1)

Estimated fractions of undiagnosed infections in 2022 (Table 1)

	• A large fraction of HIV infections are undiagnosed in ATLAS countries,
	especially in Mali (~ 50%)
	• HIV diagnostic gaps are 2-3 fold larger among males than females
	• Wider gaps among KP than non-KP
	• Among FSW (vs non-KP females) particularly in Côte d'Ivoire and
	Senegal
	• Among MSM (vs non-KP males) in Côte d'Ivoire and Senegal, but not
	in Mali (coherent with 2014 data 2 among MSM and all males in Mali)

Non-

KP females Non-KP females Non-KP females Non-KP males Non-KP males Non-KP males Bi MSM

  Estimated distribution per risk group of the undiagnosed infections across countries in 2022

	(Figure 3) • Across countries: most undiagnosed infections are among males and KP • Côte d'Ivoire: over 80% of undiagnosed infections are among non-KP, due to much larger overall HIV prevalence in this country (~2.1% among adults) than in the other countries (~0.9% in Mali and ~0.3% in Senegal) • Mali: over 25% of undiagnosed infections are among FSW and their clients • Senegal: almost 50% of undiagnosed infections are among MSM Figure 3: 2) Distribution per risk group of undiagnosed infections in 2022 (disproportionate HIV prevalence and diagnosis gaps in this group)
	Côte d'Ivoire			Mali			Senegal
		41%	Exclusive MSM (2%)	49%	Exclusive MSM (1%)	FSW	15% 5% 12%	25%
	40%	Bi MSM (2%) Clients 12% 3%	22%	5%	Bi MSM (2%) Clients 21%	Clients	29%	Exclusive 14%
		FSW			FSW				MSM

Table 2 . Estimated fraction of new HIV transmissions over 2012-2021 contributed by PLHIV of different risk groups with an undiagnosed infection (tPAF). Median and 95%UI (2.5th and 97.5th percentiles) of estimates are shown. Côte d'Ivoire Mali Senegal

 2

	All


Arlette Simo Fotso

Cheryl Johnson

Anthony Vautier

Blaise Konan

Papa Moussa Kouamé

Romain Diop

Mathieu Silhol

Marie-Claude Maheu-Giroux

Nicolas Boily

Clémence Rouveau

Doumenc-Aïdara

Estimating the impact of HIV self-testing on HIV testing services, diagnoses, and treatment initiation at the population-level with routine data: the example of the ATLAS program in Côte d'Ivoire

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

However, its confidential nature and the network distribution approach (focusing on key populations and their relatives and partners) make it difficult to estimate the impact of HIVST at the population-level.

Methodology

We used quarterly programmatic data (Q3-2019 to Q1-2021) from implementing partners for the N o HIVST kits distributed through ATLAS in Côte d'Ivoire.

HIV testing services programme data (N o of conventional testing, diagnoses and and ART) was obtained from PEPFAR, aggregated by health districts (n=78, including the 9 ATLAS districts and 69 other districts not covered by ATLAS) and quarter.

Impacts of HIVST were estimated using ecological time series regression using linear mixed-models.

Results

Conclusions

The methodology used in this paper offers a promising way to use routinely collected programmatic data to estimate the effects of HIVST kit distribution in real-world programmes.

Our results show that a network-based HIVST distribution strategy focused on key populations and their relatives and partners improves diagnosis. Implementers should consider using this approach to monitor and optimize programming.

Between Q3-2019 and Q1-2021, ATLAS distributed 99,353 HIVST kits in 9/78 health districts included the analysis.

In the 9 ATLAS districts, the distribution of HIVST kits mitigated the decline in access to HIV testing (Figure b).

In the 69 districts not covered by ATLAS (Figure c), HIVST kits distributed through PEPFAR remained limited and insufficient to offset the reductions in 'conventional' testing.

There was a negative but nonsignificant signal of the effect of the number of ATLAS-distributed HIVST kits on conventional testing uptake (-190 'conventional' tests per 1,000 HIVST distributed; 95%CI: -427 to 37). However, even if only 20% of distributed kits were used, HIVST would still increase access to testing.

ATLAS HIVST kits have increased significantly the N o of HIV diagnoses (+8 diagnoses per 1,000 HIVST distributed ; 95%CI: 0 to 15).

No association with ART initiation was observed (-2 ART initiations per 1,000 HIVST distributed ; 95%CI: -8 to 5).

  Estimating the impact of HIV self-testing on HIV testing, diagnoses, and treatment initiation at the population-level with routine data : the example of the ATLAS program in Côte d'Ivoire Background HIV self-testing (HIVST) is recommended by the WHO. It has the advantage of greater autonomy and flexibility, supporting discretion, convenience and confidentiality. It makes it possible to reach untested and hard-to-reach populations.

  Arlette Simo Fotso a,b* , Cheryl Johnson c , Anthony Vautier d , Konan Blaise Kouamé e , Papa Moussa Diop d ,Romain Silhol f , Mathieu Maheu-Giroux g , Marie-Claude Boily f ,Nicolas Rouveau a , Clémence Doumenc-Aïdara d , Rachel Baggaley c , Eboi Ehui e , Joseph Larmarange a for the ATLAS team d.Programme National de Lutte contre le Sida, Abidjan, Côte d'Ivoire e.Medical Research Council Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London, United Kingdom f.Department of Epidemiology and Biostatistics, School of Population and Global Health, McGill University, Montréal, QC, H3A 1G1, Canada a.Centre Population & Développement, Université de Paris cité, IRD, Inserm, Paris, France b.French Institute for Demographic Studies, INED, Aubervilliers, France c.World Health Organization, Geneva, Switzerland d.Solidarité Thérapeutique et Initiatives pour la Santé, Solthis, Dakar, Sénégal * contact: simofotsoarlette@yahoo.fr

estimate the impacts of the ATLAS project HIVST on 'conventional' HIV testing (i.e., non-HIVST), new HIV diagnoses, and initiation of antiretroviral therapy (ART) in Côte d'Ivoire.

	Conventional testing
	decreased from 379,554
	in Q3 2019 to 268,807 in
	Q1 2021 (Figure a).

This study aims to overcome this challenge by using routine programmatic data to

Table 1 :

 1 Linear effect of N

	Outcome variables	Effect for 1000	95% CI	p-value
		HIVST
		distributed by
		ATLAS
	Conventional testing	-190	-427 to +38	0•10
	HIV diagnosis	+8	0 to +15	0•044
	ART initiations	-2	-8 to +5	0•66

o of HIVST distributed via ATLAS on 'conventional' testing, HIV diagnosis and ART initiation, in 78 PEPFAR districts in Côte d'Ivoire (Q3-2019 to Q1-2021), adjusted by quarter year and region. More information here: https://doi.org/10.1101/2022.02.08.22270670


Methods

• Scoping review of PubMed for studies using dynamic transmission models of sexually transmitted and bloodborne infections that included structural determinants

• We extracted information on structural determinants and interventions modelled (e.g., exposure definitions, populations) and methods to model their impacts (e.g., counterfactual scenarios)

1. Conceptualized exposure to structural determinants 2. Represented causal relationships over time 3. Defined counterfactual scenarios to measure the impact of structural determinants 4. Included persistent effects of exposure

Background

• Structural determinants influence HIV transmission through their effects on downstream mediators (Figure 1)

• Modelling structural determinants is important to estimate their populationlevel impacts and inform efforts towards HIV elimination

• Challenges: Limited understanding of the specific causal pathways and the magnitude of the effect of structural determinants on mediators and subsequent HIV risk

Improving our understanding of how structural determinants impact HIV epidemics: a scoping review of dynamic models to guide future research

Results -Search

• We identified 8 dynamic modelling studies of HIV, 2 of HCV and 1 that modelled HIV and HCV separately (Table 1)

• The main structural determinants were:

• Housing instability/homelessness (N studies =2)

• Incarceration (N studies =4)

• Violence (N studies =3)

• Stigma (N studies =2)

Conclusions

• We found few studies that modelled structural determinants dynamically

• Methods are diverse and could be improved

• Coherent frameworks are needed to conceptualize the synergy between including structural determinants in models and strengthened empirical analysis

• Models should use refined exposure definitions and more precise estimates of causal associations with mediators

Exposure definitions

• Most models (N=8) assumed categories of fixed lifetime exposures or current, recent, and non-recent exposure that individuals could transition between

Causal pathways:

• Structural determinants were mostly assumed to (N=7) impact HIV transmission through mediated effects on one or more proximate risk factors (Figure 2)

• e.g., condom use, number of secondary partners, treatment uptake

• In 4 studies, structural determinants were assumed to directly impact the transmission rate (i.e. no mediators)

Counterfactual definitions to measure impact

• To estimate the contribution of structural determinants or impact of interventions (e.g., decriminalization) studies mostly assumed reduced exposure in the future (N=5) or elimination of exposure (N=4) in counterfactual scenarios

Persistent effects of exposure

• Mostly assumed using categories of recent and non-recent exposure

• Models did not include delayed effects of exposure (i.e., that occur after a lag)

• Few models considered associations of past or recurring exposure with different transmission risks

Results -Appraisal of modelling studies

  James Stannah 1* , Jorge Luis Flores Anato 1 , Kate M Mitchell 2 , Joseph Larmarange 3 , Mathieu Maheu-Giroux 1 , Marie-Claude Boily 2 1 McGill University, Department of Epidemiology, Biostatistics, and Occupational Health, Montreal, Canada, 2 Imperial College London, Department of Infectious Disease Epidemiology, London, United Kingdom, 3 Université Paris Descartes, Institut de Recherche pour le Développement, Paris, France

Table 1 .

 1 Table of studies identified in the scoping review that used dynamic transmission modelling to evaluate the contribution or impact of structural determinants or interventions Structural determinants of HIV: physical, social, cultural, organizational, community, economic, legal or policy features of the environment that facilitate or limit an individual's access to HIV prevention, treatment, or care

	Aim
	Improve methods to model structural
	determinants by:
	• Identifying dynamic modelling studies that
	included structural determinants
	• Appraising methods used to model
	structural determinants and their impacts

FSW, female sex workers; HCV, hepatitis C virus; HIV, human immunodeficiency virus; IPV, intimate partner violence; N America, North America; N & C America, North and Central America; NSP, needle and syringe program; OST, opiod substitution therapy; PWID, people who inject drugs; SES, socioeconomic status.

Figure 2. Example causal pathways from models Recommendations Including structural determinants in dynamic HIV transmission models 1. Adopt precise exposure definitions

	Unsafe work	Increased	Lower	Increased HIV
	environment	violence	condom use	transmission
		Shannon, 2015
	Homelessness	Low access to entertainment	Increased sexual contacts	Increased HIV transmission
		Bhunu, 2015

• Consider severities, frequency, and duration of exposure

2. Deconstruct and estimate causal pathways

• Use quantitative and qualitative evidence of causal associations

• Consider proximate and distant mediators

3. Consider persistent effects

of exposure • Include persistent and delayed effects of exposure, and effects of past, recurrent, or acute exposure if there are associations with future transmission risks 4. Define appropriate counterfactual scenarios • Counterfactual definitions should align with the goal of modelling (estimating the contribution of determinants vs impact of interventions)

Acknowledgements

James Stannah is supported by the Fonds de Recherche de Québec en Santé (FRQS). We thank the HPTN modelling cetre, which is funded by the National Insitutes of Health through HPTN, and the Canadian Institutes of Health Research (CIHR) for partial funding of this work. Mathieu Maheu-Giroux is the Canada Research Chair in Population Health Modelling.