Wanco: Weak Adversarial Networks for Constrained Optimization Problems

Abstract

This paper focuses on integrating neural networks and adversarial training to develop a framework algorithm for constrained optimization problems. For such problems, we first transform them into minimax problems using the augmented Lagrangian method and then use two (or several) deep neural networks to represent the primal and dual variables, respectively. The parameters in the neural networks are then trained by an adversarial process. Compared to penalty-based deep learning methods, the proposed architecture exhibits enhanced insensitivity to constraint value scales and enforces constraints more effectively through Lagrange multipliers. Extensive examples for optimization problems with scalar constraints, nonlinear vector constraints, partial differential equation constraints, and inequality constraints are considered to show the capability and robustness of the proposed method, with applications ranging from Ginzburg--Landau energy minimization problems, partition problems, fluid-solid topology optimization, to obstacle problems.

Keywords: Deep neural networks, Constrained optimization, Augmented Lagrangian method, Adversarial neural networks