In this paper, we consider a class of stochastic inverse linear semidefinite optimal value problems, in which the forward problem is a linear semidefinite programming problem (LSDP), and the data in its constraints is affected by a random variable. Under some mild assumptions for LSDP, the corresponding inverse optimal value problem can be reformulated as a mathematical program with stochastic linear semidefinite complementarity constraints (MPSLSDCC). By employing the techniques of sample average approximation (SAA), we construct a series of smooth SAA subproblems and transform them into nonlinear semidefinite programming problems by utilizing the smooth Fischer-Burmeister function for linear semidefinite complementarity constraints. In addition, we prove that the sequence of global minimizer (respectively, KKT point) of these SAA subproblems converge with probability one (w.p.1) to a global minimizer (respectively, an S-stationary point) of MPSLSDCC under mild assumptions. Finally, some numerical experiments are presented to show the ability of our method for solving the given stochastic linear semidefinite inverse optimal value problems.
Lu et al. (Mon,) studied this question.