Research On The Algorithms For Solving Absolute Value Equations Associated With Second-Order Cone

The absolute value equation associated with second-order cone(SOCAVE for short)is a generalization of the standard absolute value equation in the frame of secondorder cone.It is an optimization problem closely related to the second-order cone complementary problem.Under suitable conditions,the SOCAVE can be equivalent to the second-order cone linear complementary problem.Therefore,the SOCAVE has extensive research significance.In this paper,we propose two algorithms for solving SOCAVE,which are the Levenberg-Marquardt(LM for short)algorithm with Armijo line search and the relaxed nonlinear PHSS-like iteration method,respectively.Firstly,combining the knowledge of Jordan algebra and the original SOCAVE,we construct a corresponding smooth function with respect to the original second-order cone absolute value equation.Using Armijo line search and the classical search direction of LM method,we propose the LM algorithm for solving SOCAVE,and obtain that the proposed algorithm is well-defined,feasible and convergent under suitable conditions.Then,based on the properties of second-order cone and nonsmooth analysis,combining with the knowledge of matrix decomposition,we propose the relaxed nonlinear PHSSlike iteration algorithm in this paper,and show that the algorithm is well-defined,feasible and convergent under certain conditions.In addition,the numerical experiments of the two algorithms are given in this paper.The numerical results show that both of the algorithms proposed in this paper are feasible and all of them show good computational performance.