The Merit Function Method For Solving The Weighted Second-order Cone Complementarity Problems

The second-order cone weighted complementarity problem(SOCWCP)is an important class of cone optimization problems,which has a wide range of applications in the fields of economics,engineering,chemistry,mechanics,and has attracted much attention in recent years.SOCWCP reduces to the second-order cone complementarity problem(SOCCP)when the weighted vector is zero.Problems such as Fisher's market equilibrium in economics can be solved by establishing a weighted complementarity problem(WCP)model,which is more effective than establishing a complementarity problem(CP)model.Therefore,it is of great theoretical and practical significance to study the theory and algorithm of WCP.So far,there are only a few papers about the WCP.In this paper,we present the merit function methods for solving SOCWCP.The main results are as follows:1.A class of parametric merit functions is proposed for the SOCWCP,discuss its smoothness,and derive the computable formula for its Jacobian matrix.Based on the class of merit functions,we reformulate the SOCWCP as an unconstrained optimization problem,and present a descent method to solve it.Some numerical results show that the proposed method can stably solve the SOCWCP.2.A nonmontone derivative-free descent method is proposed for solving the linear SOCWCP(LSOCWCP).We construct a new merit function for the LSOCWCP,which is used to transform the original problem into an unconstrained minimization problem,and the boundedness of its level set is discussed.The method uses nonmonotone line search technique to improve the performance of the method.Under the mild assumption,we show that the search direction of the method satisfies descent condition,and the proposed method is globally convergent.The numerical results verify the effectiveness of the method.3.By combining the nonmonotone line search with derivative-free descent algorithm,a derivative-free nonmonotone line search descent algorithm for solving SOCWCP is proposed.A new merit function is constructed to transform SOCWCP into an equivalent unconstrained minimization problem.Some properties of the merit function are studied,such as smoothness and boundedness of level set,and the merit function is shown to provide a global error bound for SOCWCP.The algorithm is globally convergent under appropriate assumptions,and numerical results are given to indicate its good performance.