An Implementable SAA Nonlinear Lagrange Algorithm for Constrained Minimax Stochastic Optimization Problems

This paper proposes an implementable SAA (sample average approximation) nonlinear Lagrange algorithm for the constrained minimax stochastic optimization problem based on the sample average approximation method. A computable nonlinear Lagrange function with sample average approximation functions of original functions is minimized and the Lagrange multiplier is updated based on the sample average approximation functions of original functions in the algorithm. And it is shown that the solution sequences obtained by the novel algorithm for solving subproblem converge to their true counterparts with probability one as the sample size approximates infinity under some moderate assumptions. Finally, numerical experiments are carried out for solving some typical test problems and the obtained numerical results preliminarily demonstrate that the proposed algorithm is promising.

A Nonlinear Lagrange Algorithm for Stochastic Minimax Problems Based on Sample Average Approximation Method

An implementable nonlinear Lagrange algorithm for stochastic minimax problems is presented based on sample average approximation method in this paper, in which the second step minimizes a nonlinear Lagrange function with sample average approximation functions of original functions and the sample average approximation of the Lagrange multiplier is adopted. Under a set of mild assumptions, it is proven that the sequences of solution and multiplier obtained by the proposed algorithm converge to the Kuhn-Tucker pair of the original problem with probability one as the sample size increases. At last, the numerical experiments for five test examples are performed and the numerical results indicate that the algorithm is promising.