In this paper, we consider complementarity problem associated with circular cone, which is a type of nonsymmetric cone complementarity problem. The main purpose of this paper is to show the readers how to construct complementarity functions for such nonsymmetric cone complementarity problem, and propose a few merit functions for solving such a complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also show that these merit functions provide an error bound for the circular cone complementarity problem. These results ensure that the sequence generated by descent methods has at least one accumulation point, and build up a theoretical basis for designing the merit function method for solving circular cone complementarity problem.

This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.

In the paper [J.-S. Chen, S. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, 40 (2008) 389404], the authors proposed a derivative-free descent algorithm for nonlinear complementarity problems (NCPs) by the generalized FischerBurmeister merit function: p (a, b)= 1 2 [(a, b) p(a+ b)] 2, and observed that the choice of the parameter p has a great influence on the numerical performance of the algorithm. In this paper, we analyze the phenomenon theoretically for a derivative-free descent algorithm which is based on a penalized form of p and uses a different direction from that of Chen and Pan. More specifically, we show that the algorithm proposed is globally convergent and has a locally R-linear convergence rate, and furthermore, its convergence rate will become worse when the parameter p decreases

We propose a primal-dual continuation approach for the capacitated multi-facility Weber problem (CMFWP) based on its nonlinear second-order cone program (SOCP) reformulation. The main idea of the approach is to reformulate the CMFWP as a nonlinear SOCP with a nonconvex objective function, and then introduce a logarithmic barrier term and a quadratic proximal term into the objective to construct a sequence of convexified subproblems. By this, this class of nondifferentiable and nonconvex optimization problems is converted into the solution of a sequence of nonlinear convex SOCPs. In this paper, we employ the semismooth Newton method proposed in Kanzow etal. (SIAM Journal of Optimization 20:297320, 2009) to solve the KKT system of the resulting convex SOCPs. Preliminary numerical results are reported for eighteen test instances, which indicate that the continuation approach is

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over R n. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the FischerBurmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation