Showing papers by "Jong-Shi Pang published in 1994"

Error bounds for analytic systems and their applications

Abstract: Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush--Kuhn--Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0---1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.

Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory

Abstract: This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we ...

A Trust Region Method for Constrained Nonsmooth Equations

Abstract: In this paper, we develop and analyze the convergence of a fairly general trust region method for solving a system of nonsmooth equations subject to some linear constraints. The method is based on the existence of an iteration function for the nonsmooth equations and involves the solution of a sequence of sub- problems defined by this function. A particular realization of the method leads to an arbitrary-norm trust region method. Applications of the latter method to the nonlinear complementarity and related problems are discussed. Sequential convergence of the method and its rate of convergence are established under certain regularity conditions similar to those used in the NE/SQP method [14] and its generalization [16]. Some computational results are reported.

Global error bounds for convex quadratic inequality systems

Abstract: A global error bound is given for any system of convex quadratic inequalities in terms of a residual function of the system. The residual function consists of the norm of the violation vector plus its l/(2 d )-th power, where d is called the degree of singularity of the system. When the system satisfies the Slater constraint qualification, the error bound recovers a result of Luo and Luo [2] with d= 0. The global error bound of Mangasarian and Shiau [10] for monotone linear complementarity problems is also a direct consequence in which d = 1. In general, the degree of singularity is bounded by the number of constraints in the system and the error bound is valid under no constraint qualification on the system. Finally, we show by examples that the error bound is best possible.

On the Boundedness and Stability of Solutions to the Affine Variational Inequality Problem

Abstract: This paper investigates the boundedness and stability of solutions to the affine variational inequality problem. The concept of a solution ray to a variational inequality defined by an affine, mapping and on a closed convex set is introduced and characterized; the connection of such a ray with the boundedness of the solution set of the given problem is explained. In the case of the monotone affine variational inequality, a complete description of the solution set is obtained which leads to a simplified characterization of the boundedness of this set as well as to a new error bound result for approximate solutions to such a variational problem. The boundedness results are then combined with certain degree- theoretic arguments to establish the stability of the solution set of an affine variational inequality problem.

Serial and Parallel Computation of Karush–Kuhn–Tucker Points via Nonsmooth Equations

Abstract: In this paper, a robust, iterative algorithm is presented for computing Karush–Kuhn–Tucker (KKT) points of nonlinear programs. This algorithm is a variation of the NE/SQP method for solving the nonlinear complementarily problem; it makes use of a merit function that combines the original objective function of the nonlinear program with a residual function of the KKT conditions formulated as a system of nonsmooth equations. Global convergence and a Q-quadratic rate of convergence of the algorithm are established under some standard constraint qualifications in nonlinear programming theory, but without the strict complementarily condition. Parallel implementations of the algorithm are discussed.