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text:
Siham Tassouli
email: siham.tassouli@centralesupelec.fr

Abdel Lisser
email: abdel.lisser@centralesupelec.fr

A neural network approach to solve geometric programs with joint probabilistic constraints

Keywords: Stochastic geometric programming, Joint probabilistic constraints, Biconvex optimisation, Dynamical neural network, Lyapunov theory, Partial KKT system

In this paper we study a dynamical neural network approach for solving geometric programs with joint probabilistic constraints (GPPC for short) with normally distributed coefficients and independent matrix row vectors. We proof the convergence and the stability of our neural network in the sense of Lyapunov.

Finally, we use the proposed method to solve a geometric transportation problem.

Introduction

Geometric programming (GP for short) is a special class of nonlinear programming problems. It is used to minimize or maximize functions expressed by posynomials subject to constraints of the same type.

In the paper, we study the following stochastic geometric programming prob-lem:

min t∈R M ++ E[Σ i∈I0 c i M j=1 t aij j ], s.t. P(Σ i∈I k c i M j=1 t aij j ≤ 1, k = 1, ...., K) ≥ 1 -. (1)
Where c i , i ∈ I k is a K × n are uncorrelated normally distributed random variables i.e c i ∼ N (E i , σ 2 i ). The coefficients a ij , i ∈ I k , j = 1, ..., M are deterministic, and 1 -is a given probability level with ∈ (0, 0.5].

Geometric programming was first introduced in 1967 by Duffin et al. [START_REF] Perlumutter | Geometric programming-theory and application[END_REF]. Since then, GP was widely used in several fields. Chiang et al. [START_REF] Chiang | Power control by geometric programming[END_REF] use geometric programming to solve power control problems in wireless cellular or ad hoc networks. Li & Chen [START_REF] Li | Optimal channel doping profile of two-dimensional metal-oxide-semiconductor field-effect transistors via geometric programming[END_REF] deal with the problem of choosing channel doping profile in semiconductor via geometric programming. Hoburg & Abbeel [4] formulate conceptual-stage aircraft design problems as geometric problems. Vanderhaegen et al [START_REF] Vanderhaegen | Automated design of operational transconductance amplifiers using reversed geometric programming[END_REF] design operational amplifiers using reversed geometric programming.

Dupacova [START_REF] Dupacová | Stochastic geometric programming with an application[END_REF] studies geometric programming for metal cutting optimization problems. Singh et al. [START_REF] Singh | A geometric programming-based worst case gate sizing method incorporating spatial correlation[END_REF] use a posynomial delay model to give a worst case design scheme for spatial gate sizing. Geometric programming has also been used in business for profit maximization problem , see Kojić & Lukač. [START_REF] Kojić | Solving profit maximization problem in case of the cobb-douglas production function via weighted ag inequality and geometric programming[END_REF]. Liu et al. [START_REF] Liu | An improved geometric programming approach for optimization of biochemical systems[END_REF] address the optimization of biochemical systems using an improved geometric programming approach. Li et al. [START_REF] Li | A geometric programming approach to the optimization of mechatronic systems in early design stages[END_REF] use geometric programming as a solver to optimize mechatronic system design.

To solve geometric optimisation problems several approaches were introduced.

A differential evolution algorithm is used by Wang et al. [START_REF] Wang | A differential evolution algorithm for solving geometric programming problems[END_REF] to solve a deterministic geometric program. Liu et al. [START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF] discuss joint stochastic geometric programs, where the stochastic parameters are normally distributed and pairwise independent. They propose two approaches, namely piecewise linear approximation and a sequential convex optimization algorithm. Chiang & Boyd [START_REF] Mung | Geometric programming duals of channel capacity and rate distortion[END_REF] show that the Lagrange dual problems of the channel capacity problem are actually geometric programs and hence give an upper and a lower bounds for the initial problem. Perelman & Amin [START_REF] Sela Perelman | Control of tree water networks: A geometric programming approach[END_REF] approximate the network control problem of tree water supply systems using GP then transform the control problem into a convex optimization problem using a logarithmic transformation of the objective function and the constraints. To overcome the uncertainty in the geometric problem data, Hsiung et al. [START_REF] Hsiung | Tractable approximate robust geometric programming[END_REF] use robust geometric programming.

The method consists in approximating the constraints of the robust model by a piecewise-linear convex approximation. Khanjani-Shiraz et al. [START_REF] Khanjani-Shiraz | Copula theory approach to stochastic geometric programming[END_REF] study a stochastic geometric programming problem with joint chance constraints with elliptically distributed random parameters where the constraints are dependent.

They use the variable transformation, proposed by Cheng & Lisser in [START_REF] Cheng | A second-order cone programming approach for linear programs with joint probabilistic constraints[END_REF] and Jia et al. in [START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF], to come up with lower and upper bounds.

Several papers in the literature use artificial neural networks to solve optimisation problems. Villarrubia et al. [START_REF] Villarrubia | Artificial neural networks used in optimization problems[END_REF] propose to approximate the objective function by neural network for nonlinear geometric problem. Nazemi & Tahmasbi [START_REF] Nazemi | A high performance neural network model for solving chance constrained optimization problems[END_REF] introduce a dynamical neural network for solving individual chance constrained optimization problems.

Our paper is organised as follows. In Section 2 a biconvex deterministic equivalent of problem (1) is given together with a related partial KKT system. Section 3 proposes a dynamical neural network to solved the biconvex problem and discusses its stability and convergence. Finally, numerical experiments are given in Section 4.

Deterministic equivalent problem and partial KKT System

Deterministic equivalent problem

Problem (1) can be rewritten by introducing auxiliary variables y k , k = 1, ..., K as follows [START_REF] Liu | Stochastic geometric optimization with joint probabilistic constraints[END_REF]

min t∈R M ++ E[Σ i∈I0 c i M j=1 t aij j ], s.t. P(Σ i∈I k c i M j=1 t aij j ≤ 1) ≥ y k , k = 1, ..., K, (2)
K k=1 y k ≥ 1 -, 0 ≤ y k ≤ 1, k = 1, ..., K.
We introduce h k , k = 1, .., K defined by 2) is then given by:

h k = Σ i∈I k c i M j=1 t aij j . Note that the mean of h k is hk = Σ i∈I k E i M j=1 t aij j and var(h k ) = Σ i∈I k σ 2 i M j=1 t aij j . A deterministic equivalent problem of (
60 min t∈R M ++ Σ i∈I0 E i M j=1 t aij j , s.t. hk + φ -1 (y k ) var(h k ) ≤ 1, k = 1, ..., K, K k=1 y k ≥ 1 -, (3)
0 ≤ y k ≤ 1, k = 1, ..., K,
where φ -1 (y k ) is the quantile of the standard normal distribution. An equivalent biconvex problem for (3) can be obtained with the log-transformation r j = log(t j ), j = 1, ..., M and x k = log(y k ), k = 1, ..., K :

65 min r∈R M Σ i∈I0 E i exp(Σ M j=1 a ij r j ), s.t. Σ i∈I k (E i exp(Σ M j=1 a ij r j )) (4)
+ Σ i∈I k σ 2 i exp(Σ M j=1 (2a ij r j + log(φ -1 (e x k ) 2 ))) ≤ 1, k = 1, ..., K, Σ K k=1 x k ≥ log(1 -), x k ≤ 0, k = 1, ..., K. 2.2. Partial KKT System Let f (r) = Σ i∈I0 E i exp(Σ M j=1 a ij r j ), l k (r, x) = Σ i∈I k (E i exp(Σ M j=1 a ij r j )) + Σ i∈I k σ 2 i exp(Σ M j=1 (2a ij r j + log(φ -1 (e x k ) 2 ))) -1, h(r, x) = log(1 -) -Σ K k=1 x k
and g k (r, x) = x k , we can then write problem (4) as follows:

min r∈R M f (r), s.t. l k (r, x) ≤ 0, k = 1, .., K, h(r, x) ≤ 0, (5)
g k (r, x) ≤ 0, k = 1, .., K.
The feasible set of ( 5) is denoted by

U = {(r, x) ∈ R m ×R k |l k (r, x) ≤ 0, h(r, x) ≤ 0 and g k (r, x) ≤ 0, k = 1, .., K}. Let U(r) = {x ∈ R k |l k (r, x) ≤ 0, h(r, x) ≤ 0 and g k (r, x) ≤ 0, k = 1, .., K} and
U(x) = {r ∈ R m |l k (r, x) ≤ 0, h(r, x) ≤ 0 and g k (r, x) ≤ 0, k = 1, .
., K}, we define the following sub-problems:

when x is fixed find r such that min f (r), s.t. r ∈ U(x), (6)
and when r is fixed find x such that

x ∈ U(r).

Definition 1. (r * , x * ) is called a partial optimum of (5) if f (r * ) ≤ f (r), ∀r ∈ U(x * ). This means that r * is an optimal solution for (6) when x = x * and x * is feasible for (7) when r = r * .

Remark 1. In biconvex optimisation problems, there always exists a partial optimal solution [START_REF] Jiang | Partial exactness for the penalty function of biconvex programming[END_REF].

Let (r * , x * ) ∈ R m × R K , if there exists λ (1) , λ (2) , µ (1)
i , µ

(1) i and γ

i , i = 1, .., K such that:

∇f (r * ) + K i=1 µ (1) i ∇ r l i (r * , x * ) + λ (1) ∇ r h(r * , x * ) + K i=1 γ (1) i ∇ r g i (r * , x * ) = 0, (8) µ (1) ≥ 0, µ (1)
i l i (r * , x * ) = 0, λ (1) ≥ 0, λ (1) h(r * , x * ) = 0, γ (1) i ≥ 0, γ (1)
i g i (r * , x * ) = 0, i = 1, .., K, (9) K i=1 µ (2) i ∇ x l i (r * , x * ) + λ (2) ∇ x h(r * , x * ) + K i=1 γ (2) i ∇ x g i (r * , x * ) = 0, (10) µ (2) ≥ 0, µ (2)
i l i (r * , x * ) = 0, λ (2) ≥ 0, λ (2) h(r * , x * ) = 0, γ (2) i ≥ 0, γ (2)
i g i (r * , x * ) = 0, i = 1, .., K, [START_REF] Wang | A differential evolution algorithm for solving geometric programming problems[END_REF] then (r * , x * ) is called a partial KKT point of [START_REF] Vanderhaegen | Automated design of operational transconductance amplifiers using reversed geometric programming[END_REF], where λ (1) , λ (2) , µ

(1) i , µ (2) i , γ (1)
i and γ

(2) i are the Lagrange multipliers associated with ∇ r h, ∇ x h, ∇ r l i , ∇ x l i , ∇ r g i and ∇ x g i i = 1, .., K, respectively. Let (r * , x * ) ∈ R M × R K be a partial solution of [START_REF] Vanderhaegen | Automated design of operational transconductance amplifiers using reversed geometric programming[END_REF], with respect to partial slater constraints qualification [START_REF] Jiang | Partial exactness for the penalty function of biconvex programming[END_REF] at (r * , x * ). Then (r * , x * ) is a partial KKT point of ( 5) if and only if the partial KKT system ( 8)-( 11) holds with λ (1) = λ (2) , µ (1) = µ (2) and γ (1) = γ (2) [START_REF] Jiang | Partial exactness for the penalty function of biconvex programming[END_REF][START_REF] Tassouli | A neural network approach to solve linear programs with joint probabilistic constraints[END_REF].

Dynamical neural network approach

In this section, we propose a dynamic neural network model to solve problem

(5). Let l(r, x) =      l 1 (r, x) . . . l K (r, x)      and g(r, x) =      g 1 (r, x) . . . g K (r, x)     
.

The dynamical equation of the neural network is given by

dr dt = -(∇f (r) + ∇ r l(r, x) T (µ + l(r, x)) + + ∇ r h(r, x) T (λ + h(r, x)) + +∇ r g(r, x) T (γ + g(r, x)) + ), ( 12
) dx dt = -(∇ x l(r, x) T (µ + l(r, x)) + + ∇ x h(r, x) T (λ + h(r, x)) + +∇ x g(r, x) T (γ + g(r, x)) + ), ( 13
) dµ dt = (µ + l(r, x)) + -µ, (14)
dλ dt = (λ + h(r, x)) + -λ, (15)
dγ dt = (γ + g(r, x)) + -γ. (16)
We denote z = (r, x, µ, λ, γ) and define

Φ(z) =           -(∇f (r) + ∇rl(r, x) T (µ + l(r, x)) + + ∇rh(r, x) T (λ + h(r, x)) + + +∇rg(r, x) T (γ + g(r, x)) + ) -(∇xl(r, x) T (µ + l(r, x)) + + ∇xh(r, x) T (λ + h(r, x)) + + ∇xg(r, x) T (γ + g(r, x)) + ) (µ + l(r, x)) + -µ (λ + h(r, x)) + -λ (γ + g(r, x)) + -γ          
We can rewrite the neural network defined in ( 12)-( 16) as

   dz dt = κΦ ( z), z(t 0 ) = z 0 ,
where κ is a scale parameter and indicates the convergence rate of the neural network ( 12)-( 16). For the sake of simplicity, we set κ = 1. The interconnections between the different inputs of the neural network ( 12)-( 16) are represented in Figure 1. 12)-( 16)

Now, we study the convergence and the stability of the proposed dynamical neural network.

Theorem 2. Let (r * , x * , µ * , λ * , γ * ) an equilibrium point of the neural network defined by ( 12)-( 16), then (r * , x * ) is a partial KKT point of [START_REF] Vanderhaegen | Automated design of operational transconductance amplifiers using reversed geometric programming[END_REF]. On the other hand, if (r * , x * ) ∈ R M × R K is a partial KKT point of (5), then there exists µ * ≥ 0, λ * ≥ 0 and γ * ≥ 0 such that (r * , x * , µ * , λ * , γ * ) is an equilibrium point of the Neural Network ( 12)-( 16).

Proof. Let (r * , x * , µ * , λ * , γ * ) an equilibrium point of the neural network defined by ( 12)-( 16). Then, dr * dt = 0, dx * dt = 0, dµ * dt = 0, dλ * dt = 0 and dγ * dt = 0. We have

-(∇f (r * ) + ∇ r l(r * , x * ) T (µ * + l(r * , x * )) + + ∇ r h(r * , x * ) T (λ * + h(r * , x * )) + + ∇ r g(r * , x * ) T (γ * + g(r * , x * )) + ) = 0, -(∇ x l(r * , x * ) T (µ * +l(r * , x * )) + +∇ x h(r * , x * ) T (λ * +h(r * , x * )) + +∇ x g(r * , x * ) T (γ * + g(r * , x * )) + ) = 0,
(µ * +l(r * , x * )) + -µ * = 0, (λ * +h(r * , x * )) + -λ * = 0 and (γ * +g(r * , x * )) + -γ * = 0

We note that (µ * + l(r * , x * )) + -µ * = 0 if and only if µ * ≥ 0, l(r * , x * ) ≤ 0 and µ * T l(r * , x * ) = 0. By the same approach we have, λ * ≥ 0, h(r * , x * ) ≤ 0, λ * T h(r * , x * ) = 0, γ * ≥ 0, g(r * , x * ) ≤ 0 and γ * T g(r * , x * ) = 0.

Therefore, we obtain the partial KKT system [START_REF] Dupacová | Stochastic geometric programming with an application[END_REF].The converse part of the theorem is straightforward.

In order to prove the stability and the convergence of the neural network, we show first that the jacobian matrix ∇Φ(z) is negative semidefinite matrix.

Lemma 3. The jacobian matrix ∇Φ(z) is negative semidefinite matrix.

Proof. We assume the existence of 0 < p, q < K such that (µ -l) + = (µ 1 + l 1 (r, x), µ 2 + l 2 (r, x), ....., µ p + l p (r, x), 0, ...., 0

K-p ),
(γ -g) + = (γ 1 + g 1 (r, x), γ 2 + g 2 (r, x), ....., γ q + g q (r, x), 0, ...., 0

K-q ).
Without loss of generality we consider the case where λ + h(r, x) = 0. We represent the jacobian matrix ∇Φ as follows

∇Φ(z) =            A 1 A 2 A 3 A 4 A 5 B 1 B 2 B 3 B 4 B 5 C 1 C 2 C 3 C 4 C 5 D 1 D 2 D 3 D 4 D 5 E 1 E 2 E 3 E 4 E 5            ,
where

A 1 = -(∇ 2 f (r) + p i=1 ((µ i + l i )∇ 2 r l p i (r, x)) + ∇ r l p (r, x) T ∇ r l p (r, x)) + (λ + h)∇ 2 r h(r, x)+∇ r h(r, x) T ∇ r h(r, x))+ q i=1 ((γ i +g i )∇ 2 r g q i (r, x))+∇ r g q (r, x) T ∇ r g q (r, x)), A 2 = -( p i=1 ((µ i +l i )∇ x ∇ r l p i (r, x))+∇ x l p (r, x) T ∇ r l p (r, x))+(λ+h)∇ x ∇ r h(r, x)+ ∇ x h(r, x) T ∇ r h(r, x)) + q i=1 ((γ i + g i )∇ x ∇ r g q i (r, x)) + ∇ x g q (r, x) T ∇ r g q (r, x)), B 1 = -( p i=1 ((µ i +l i )∇ r ∇ x l p i (r, x))+∇ r l p (r, x) T ∇ x l p (r, x))+(λ+h)∇ r ∇ x h(r, x)+ ∇ r h(r, x) T ∇ x h(r, x)) + q i=1 ((γ i + g i )∇ r ∇ x g q i (r, x)) + ∇ r g q (r, x) T ∇ x g q (r, x)), B 2 = -( p i=1 ((µ i + l i )∇ 2 x l p i (r, x)) + ∇ x l p (r, x) T ∇ x l p (r, x)) + (λ + h)∇ 2 x h(r, x) + ∇ x h(r, x) T ∇ x h(r, x)) + q i=1 ((γ i + g i )∇ 2 x g q i (r, x)) + ∇ x g q (r, x) T ∇ x g q (r, x)), A 3 = -C 1 = -∇ r l p (r, x) T , A 4 = -D 1 = -∇ r h(r, x) T , A 5 = -E 1 = -∇ r g q (r, x) T , B 3 = -C 2 = -∇ x l p (r, x) T , B 4 = -D 2 = -∇ x h(r, x) T , B 5 = -E 2 = -∇ x g q (r, x) T , C 3 = -S p =   O p×p O p×(K-p) O (K-p)×p I (K-p)×(K-p)   , E 5 = -S q =   O q×q O q×(K-q) O (K-q)×q I (K-q)×(K-q)   , C 4 = 0, C 5 = 0, D 3 = 0, D 4 = 0, D 5 = 0, E 3 = 0 and E 4 = 0.
Since g, h and l are twice differentiable, by Schwarz's theorem, we have

∇ r ∇ x g p i (x, y) = ∇ x ∇ r g p i (x, y), ∀i ∈ [1, p], ∇ r ∇ x l q i (x, y) = ∇ x ∇ r l q i (x, y), ∀i ∈ [1, q] and ∇ r ∇ x h(x, y) = ∇ x ∇ r h(x, y). It follows that A 2 = B T 1 ∇Φ(z) becomes ∇Φ(z) =            A 1 B T 1 A 3 A 4 A 5 B 1 B 2 B 3 B 4 B 5 -A 3 -B 3 -S p 0 0 -A 4 -B 4 0 0 0 -A 5 -B 5 0 0 -S q            .
It is easy to show that -S p and -S q are negative semidefinite. It follows that

S =      -S p 0 0 0 0 0 0 0 -S q      is negative semidefinite.
Since the function f is convex and twice differentiable, then ∇ 2 f (x) is positive semidefinite. Additionally, g, h and l are biconvex and twice differentiable, then ∇ 2 x g p i (x, y), ∇ 2 y g p i (x, y), i = 1, ...q; ∇ 2 x h(x, y),∇ 2 y h(x, y) and ∇ 2 x l p i (x, y), ∇ 2 y g l i (x, y), i = 1, ...q are positive semidefinite. It follows that A 1 and B 2 are negative semidefinite and then A =

  A 1 B T 1 B 1 B 2   is negative semidefinite. Let B =   A 3 A 4 A 5 B 3 B 4 B 5   , ∇Φ(z) can be written as ∇Φ(z) =   A B -B T S   .
Since S and A are negative semidefinite, the conclusion follows. The proof where λ + h(r, x) = 0 follows along the same lines.

Definition 2. A mapping F : R n -→ R n is said to be monotonic if: the total wall area and the floor area are considered as random variables. We assume that these random variables are independent. We consider a stochastic 205 geometric transportation problem adapted from [START_REF] Lisser | Rectangular chance constrained geometric optimization[END_REF] given by:

min x -1 1 x -1 2 x -1 3 , s.t. P( 1 A wall (2x 3 x 2 + 2x 1 x 3 ) ≤ 1, 1 A f loor x 1 x 2 ≤ 1) ≥ 1 -. αx -1 1 x 2 ≤ 1, (20)
βx -1 2 x 2 ≤ 1, γx -1 2 x 3 ≤ 1, δx -1 3 x 2 ≤ 1,
To solve the joint probabilistic problem (20), we give the following equivalent deterministic problem:

min x -1 1 x -1 2 x -1 3 , s.t. P( 1 A wall (2x 3 x 2 + 2x 1 x 3 ) ≤ 1) ≥ y 1 , P( 1 A f loor x 1 x 2 ≤ 1) ≥ y 2 , αx -1 1 x 2 ≤ 1, βx -1 2 x 2 ≤ 1, γx -1 2 x 3 ≤ 1, (21)
δx -1 3 x 2 ≤ 1,
y 1 y 2 ≥ 1 -, 0 ≤ y 1 , y 2 ≤ 1.
For the computations we set = 0.1, 1 A wall ∼ N (0.05, 0.01), 1 A f loor ∼ N (0.5, 0.01) and α = β = γ = δ = 0.5. In addition to problem [START_REF] Tassouli | A neural network approach to solve linear programs with joint probabilistic constraints[END_REF], we solve a deterministic problem with the mean values of 1 A wall and 1 A f loor as values of 1 A wall and 1 A f loor , respectively. We solve the problem obtained by replacing the joint constraints by individual ones. The results are recapitulated in Table 1. Column one gives the objective value obtained by the deterministic approach, column two gives the number of violated scenarios (VS) over 100 scenarios generated and column three the correspondent CPU time. The left columns represent the same information for the remaining approaches.

We observe that the deterministic problem gives the lowest value for the objective function. We notice that the number of violated scenarios (VS) in Figure 3 is equal to 100. The joint constraints approach ensures the best robustness compared to the two other problems as only two scenarios are violated, see Fig- In order to evaluate the performances of our approach on large size instances, we introduce a more general formulation that is defined as follows, x -1 i ,

s.t. P(Σ m-1 j=1 ( m -1 A wallj x 1 m i=1,i =j x i ) ≤ 1, 1 A f loor m j=2 x j ≤ 1, (22) 1 γ i,j x i x -1 j ≤ 1, 1 ≤ i = j ≤ M ) ≥ 1 -.
As for the numerical experiments, we set = 0.05, The results for different values of m for problem [START_REF] Tyrrell | Variational Analysis[END_REF] are given in Table 2, where Column one gives the data of the problem i.e (2, 4) means the problem is composed of 2 variables and 4 constraints. Columns two and three show the optimal value obtained by the expected value approach and the number of VS over 100 generated scenarios, respectively. Column four gives the associated CPU time.

Columns five, six, seven and eight give the objective value of the individual constraints problem, the number of VS, the gap with the deterministic problem and the CPU Time, respectively. We note that the gap is computed compared to the deterministic approach and computed as follows gap = [ abs ((objective value of the deterministic program -objective value of the individual program) / objective value of the deterministic program)].The last four columns give the optimal value obtained by the joint constraints problem, the number of VS, the gap with the deterministic approach and the CPU time. We observe that the problem with joint constraints remains the approach that covers well the risk region, the number of VS for joint problem is lower than those for the deterministic and the individual approaches i.e for m = 6 eleven scenarios were violated over 100 scenarios compared to 59 for the individual constraints and 100 for the deterministic program see Figures 6789. We observe also that the joint approach is a time consuming method i.e for m = 6 the convergence takes 2113.29 seconds. In fact, the ODE solver is time consuming when the size of the problem increases.

conclusion

In this paper, we propose a dynamical neural network based on the partial KKT system to solve joint probabilistic geometric programs. The problem is first transformed into a deterministic program using the independence of the random variables. A biconvex equivalent is then given using a logarithmic transformation. Finally, a neural network is constructed based on the partial KKT To evaluate the performances of our dynamical neural network, we study a shape optimisation problem. The results show the robustness of the proposed 260 approach. We use a standard ODE solver iteratively which leads to a significant CPU time. We believe that an ODE solver based on machine learning would decrease drastically the CPU time of our approach.

Figure 1 :

 1 Figure 1: Feedback interconnection of neural network (12)-[START_REF] Khanjani-Shiraz | Copula theory approach to stochastic geometric programming[END_REF]

Figure 2 :

 2 Figure 2: Shape of the box[START_REF]Geometric Programming[END_REF]

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Out of 100 scenarios the constraints were violated 100 times

1 A

 1 f loor ∼ N (1.0/20.0, 0.1), 1 A wall j follows a normal distribution of mean randomly drawn from [1.0/60.0, 1.0/40.0] and of standard deviation randomly drawn from [0.3, 0.5] and γ i,j ∼ N (0.5, 0.1).

Figure 6 :Figure 7 :

 67 Figure 6: Out of 100 scenarios the constraints were violated 100 times

Figure 8 :

 8 Figure 8: Out of 100 scenarios the constraints were violated 11 times

Table 1 :

 1 Results of the box shaping problem

	Deterministic Approach	Individual constraints		Joint constraints
	Obj value VS CPU Time Obj value VS CPU Time Obj value VS CPU Time
	0.125	100 0.01	0.131	8	0.06	0.132	2	8.75

Table 2 :

 2 Results of the generalised transportation problem

(x -y) T (F (x) -F (y)) ≥ 0, ∀x, y ∈ R n . Lemma 4. [START_REF] Tyrrell | Variational Analysis[END_REF]. A differentiable mapping F : R n -→ R n is monotonic, if and only if the jacobian matrix ∇F (x), ∀x ∈ R n , is positive semidefinite.

We show in the following theorem that the neural network ( 12)-( 16) is stable in the sense of Lyapunov and globally convergent.

Theorem 5. The neural network ( 12)-( 16) is stable in the Lyapunov sense and converges to (r * , x * , µ * , λ * , γ * ), where (r * , x * ) is a partial KKT point of problem [START_REF] Vanderhaegen | Automated design of operational transconductance amplifiers using reversed geometric programming[END_REF].

Proof. Let y * = (r * , x * , µ * , λ * , γ * ) an equilibrium point for the neural network ( 12)-( 16). We define the following Lyapunov function,

We have

Since dΦ dt = ∇Φ ∇y dy dt = ∇Φ(y)Φ(y), then

By Lemma 3., we have Φ T (∇Φ(y))Φ ≤ 0 and Φ T (∇Φ(y) T )Φ ≤ 0.

Based on Theorem 2. and Lemma 4., we prove that (y -y * ) T (Φ(y)) = (y -

We conclude that dV (y(t)) dt ≤ 0 and then the neural network ( 12)-( 16) is stable at y * in the sense of Lyapunov.

Since V (y) ≥ 1 2 y -y * 2 , there exists a convergent subsequence (y(t) t≥0 ) such that lim t-→∞ y(t) = ŷ, and dV (ŷ(t))

LaSalle's invariance principle we can see that any solution starting from a certain y 0 converges to the largest invariant set contained in M .

Notice that

It follows that ŷ is an equilibrium point of the neural network ( 12)-( 16).

Now we consider a new Lyapunov function defined by V (y) = Φ(y)

then lim t-→∞ y -ŷ = 0 and lim t-→∞ y(t) = ŷ. We conclude that the neural network ( 12)-( 16) is convergent in the sense of Lyupanov to an equilibrium point ŷ = (r, x, μ, λ, γ) where (r, x) is a partial KKT point of problem (5).

Numerical experiments

To evaluate the performances of our approach, we study in this section a transportation problem where the objective is to find the optimal shape of a transportation box subject to some geometric constraints. We generalise the three-dimension problem in order to evaluate the performances of the approach on large size instances.

We use Python to implement our dynamic neural network.The random instances are generated by numpy.random, the ODE systems are solved using solve ivp of scipy.integrate, the deterministic equivalent programs are solved by the package gekko and the gradients and partial derivatives are computed using autograd.grad and autograd.jacobian.

We run our algorithms on Intel(R) Core(TM) i7-10610U CPU @ 1.80GHz.

The transportation problem consisits in shifting the grain from a warehouse to a factory in an open rectangular box as shown by Figure 2 of length x 1 meters, width x 2 meters and height x 3 meters. The aim is to maximize the volume x 1 x 2 x 3 subject to two constraints on the floor area and the wall area of the rectangular box in order to meet the shape of a given truck. The limits on