Convergence Analysis Of (Stochastic) Mathematical Program Problems With Second-order-cone Complementarity Constraints And Non-Lipschitz Objective

The mathematical program with second-order-cone complementarity constraints(MP-SOCCC)problems are a kind of mathematical program with equilibrium constraints problems.Their constraints include second-order cone complementarity constraints,which are widely used in many fields such as support vector machines,network flow problems,and robust optimization.They are one of the hotspots of research in optimization circles today.On the basis of Euclidean Jordan algebra theory,many scholars have carried out indepth and systematic research on the theory,algorithm,and application of the mathematical program with second-order-cone complementarity constraints problems.In these research results,the objective function in the mathematical program with second-order-cone complementarity constraints problems is usually continuously differentiable,or locally Lipschitz continuous.However,in practical problems,it is often encountered that the objective function has a non-Lippschitz objective.For this reason,this article considers the mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective.Since the objective function of the mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective considered in this paper does not satisfy the condition of continuous differentiability,even the local Lipschitz continuity cannot be satisfied.Therefore,the method of solving mathematical program with second-order-cone complementarity constraints problems cannot be directly applied to mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective.For this reason,this paper uses the local Lipschitz continuous function to approximate the non-Lipschitz objective,and uses the second-order cone complementarity function and its smoothing function to propose an approximation problem for solving the mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective.In addition,the convergence of the global optimal solution sequence and the stable point sequence of the approximation problem is theoretically proved.However,in practical applications,there are often many uncertain factors,such as demand,supply,weather conditions,etc.These factors cannot be ignored,otherwise it will cause unpredictable consequences or losses,and even lead to decision-making errors.For this reason,this paper further considers the stochastic mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective.Based on the research on the deterministic mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective,we use the local Lipschitz continuous function to approximate the non-Lippschitz objective,the second-order cone complementary function and its smoothing function,and the sample average approximation method to obtain the SAA approximation problem of stochastic mathematical program problems with second-order-cone complementarity constraints and non-Lipschitz objective.Furthermore,since the SAA approximation problem is usually non-convex,we give the convergence result of the stable point sequence of the SAA approximation problem.