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

We provide an affirmative answer to an question that the Fischer-Burmeister complementarity function associated with symmetric cones, named the FB SC complementarity function, is globally Lipschitz continuous and strongly semismooth everywhere for Hn and Qn. This is achieved with the help of embedding Hn and Qn into certain Sm.

We consider an extended second-order cone linear complementarity problem (SOCLCP), including the generalized SOCLCP, the horizontal SOCLCP, the vertical SOCLCP, and the mixed SOCLCP as special cases. In this paper, we present some simple second-order cone constrained and unconstrained reformulation problems, and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended SOCLCP. Particularly, we develop a proximal gradient descent method for solving the second-order cone constrained problems. This method is very simple and at each iteration makes only one Euclidean projection onto second-order cones. We establish global convergence and, under a local Lipschitzian error bound assumption, linear rate of convergence. Numerical comparisons are made with the limited-memory BFGS method for the

For the symmetric cone complementarity problem, we show that each stationary point of the unconstrained minimization reformulation based on the FischerBurmeister merit function is a solution to the problem, provided that the gradient operators of the mappings involved in the problem satisfy column monotonicity or have the Cartesian P 0-property. These results answer the open question proposed in the article that appeared in Journal of Mathematical Analysis and Applications 355 (2009) 195215.

We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions t on H H with the parameter t[0, 2). We show that the squared norm of t with t(0, 2) is a continuously F (rchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.