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Distributionally Robust Geometric Joint Chance-Constrained Optimization: Neurodynamic Approaches

Ange Valli, Siham Tassouli, Abdel Lisser

TL;DR

A neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods is proposed.

Abstract

This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.

Distributionally Robust Geometric Joint Chance-Constrained Optimization: Neurodynamic Approaches

TL;DR

A neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods is proposed.

Abstract

This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.
Paper Structure (18 sections, 11 theorems, 49 equations, 7 figures, 7 tables, 1 algorithm)

This paper contains 18 sections, 11 theorems, 49 equations, 7 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement and reformulation
  3. Uncertainty Sets with First Two Order Moments
  4. Case with Jointly Independent Row Vectors
  5. Case with Jointly Dependent Row Vectors.
  6. Uncertainty Sets with Known First Order Moment and Nonnegative Support
  7. Case with Jointly Independent Row Vectors.
  8. Case with Jointly Dependent Row Vectors.
  9. A dynamical recurrent neural network for and ( )
  10. A two-time scale neurodynamic duplex for ( )
  11. Numerical experiments
  12. Uncertainty Sets with First Two Order Moments
  13. A three-dimension shape optimization problem
  14. Multidimensional shape optimization problem
  15. Uncertainty Sets with Known First Order Moment and Nonnegative Support
  16. ...and 3 more sections

Key Result

Theorem 1

Given Assumption assmp1, (JCP) is equivalent to

Figures (7)

  • Figure 1: A block diagram for the neural network (\ref{['eq:conditional_KKT_1']}-\ref{['eq:conditional_KKT_2']})
  • Figure 2: A block diagram depicting a duplex neurodynamic system with a two-timescale configuration
  • Figure 3: A block diagram of the neurodynamic duplex for the neural network (\ref{['conditional_NN1']})-(\ref{['conditional_NN2']})
  • Figure 4: 3D-box shape rao2009geometric
  • Figure 5: Transient behaviors of the state variables
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Definition 1
  • Lemma 4
  • ...and 7 more