An SAA approach for solving a class of stochastic inverse optimal value problems
Optimization Eruditorum, Volume 2, Issue 1, April 2025, Pages 32–45
YUE LU
School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
ZHI-QIANG HU
School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
DONG-YANG XUE
School of Mechanical Engineering, Tianjin University of Commerce, Tianjin 300134, China
Abstract
In this paper, we consider a class of stochastic inverse optimal value problems, in which the forward problem is a linear programming problem (LP), and the data in its constraints are affected by a random variable. The corresponding inverse optimal value problem can be reformulated as a mathematical program with stochastic linear complementarity constraints (MPSLCC). By employing the techniques of sample average approximation (SAA), we construct a series of smooth SAA subproblems and transform them into nonlinear programming problems by utilizing the smooth Fischer-Burmeister function for linear complementarity constraints. In addition, we prove that the sequence of global minimizer (KKT point) of these SAA subproblems converges with probability one (w.p.1) to a global minimizer (an S-stationary point) of MPSLCC under mild conditions. Finally, some numerical experiments are presented to show the availability of our method for solving the given stochastic inverse optimal value problems.
Cite this Article as
Yue Lu, Zhi-Qiang Hu, and Dong-Yang Xue, An saa approach for solving a class of stochastic inverse optimal value problems, Optimization Eruditorum, 2(1), 32–45, 2025