Stochastic $R_0$ matrix linear complementarity problems: the Fischer–Burmeister function-based expected residual minimization

Abstract

Stochastic $R_0$ matrix, which is a generalization of $R_0$ matrix, is instrumental in the study of the stochastic linear complementarity problem (SLCP). In this paper, we focus on the expected residual minimization (ERM) of the SLCP via the Fischer–Burmeister (FB) function, which enjoys better properties of the continuity and differentiability than the min-function-based ERM when the involved matrix is a stochastic $R_0$ matrix. First, we prove that the solution set of the FB-function-based ERM is nonempty and bounded if and only if the involved matrix is a stochastic $R_0$ matrix. Second, we show that its objective function is continuously differentiable in ${\mathbb {R}}^n$, which makes it possible for us to design the gradient-type algorithm. Finally, we implement the numerical experiments generated randomly via sample average approximation, and the numerical results indicate that the optimal solutions of the FB-function-based ERM are better than that of the min-function-based ERM in terms of preserving the nonnegativity of the involved linear function, especially when the involved matrix is a constant matrix.