%0 Journal Article
%T A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization
%A Zhijun Luo
%A Lirong Wang
%J Journal of Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/461902
%X A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved. 1. Introduction Consider the following inequality constrained optimization problems: where , are continuously differentiable. We denote To solve the problem (1), there are two type methods with superlinear convergence: sequential quadratic programming (SQP) type algorithms (see [1¨C4], etc.) and SSLE (sequential system of linear equations) type algorithms (see [5¨C9], etc.). In general, since SQP algorithms are necessary to solve one or more quadratic programming subproblems in single iteration, the computation effort is very large. SSLE algorithms were proposed to solve the problem (1), in which an iteration similar to the following linear system was considered: where is Lagrangian function, is an estimate of the Hessian of , is the current estimate of a solution , is the search direction, and is the next estimate of the Kuhn-Tucker multiplier vector associated with . Obviously, it is simpler to solve system of linear equations than to solve the QP (quadratic programming) problem with inequality constraints. In addition, parallel variable distribution (PVD) algorithm [10] is a method that distributes the variables among parallel processors. The problem is parted into many respective subproblems and each subproblem is arranged to a different processor in it. Each processor has the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions. In 2002, Sagastiz¨¢bal and Solodov [11] proposed two new variants of PVD for the constrained case. Without assuming convexity of constraints, but assuming block-separable structure, they showed that PVD subproblems can be solved inexactly by solving their quadratic programming approximations. Han et al. [12] proposed an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems with the idea in 2009, which is based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer
%U http://www.hindawi.com/journals/jmath/2014/461902/