A Smooth C-Function-Based ERM Method for Stochastic Symmetric Cone Linear Complementarity Problems

This paper considers the stochastic symmetric cone linear complementarity problem (S-SCLCP), which includes the stochastic linear complementarity problem and the stochastic second-order cone linear complementarity problem as special cases. We propose a new expected residual minimization (ERM) formulation for S-SCLCP and apply the Monte Carlo technique to generate the corresponding approximation problem. Different from existing ERM formulations for stochastic complementarity problems, the proposed ERM formulation is based on a smooth C-function and a variant merit function. Relying on the Euclidean Jordan algebra associated with symmetric cones, we address several important issues, including coerciveness, existence of solutions, global convergence, and exponential convergence rate. Furthermore, we present some numerical examples and the practical applications of ERM schemes in solving an uncertain Nash–Cournot game and a stochastic optimal power flow problem in the radial network, demonstrating the effectiveness of this method. Funding: This work was partially supported by the National Natural Science Foundation of China [Grants 12371305 and 12571321], the Shandong Provincial Natural Science Foundation [Grants ZR2023MA020, ZR2022MA038, and ZR2024MA004], the Henan Province Natural Science Foundation [Grant 252300423005], the Key Scientific Research Projects of Higher Education of Henan Province [Grant 26A110018], and the National Science Foundation of the United States [Grants DMS-2309549 and DMS-2514001].