Smoothness Properties of a Regularized Gap Function for Quasi-Variational Inequalities

This article studies continuity and differentiability properties for a reformulation of a finite-dimensional quasi-variational inequality (QVI) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points. We therefore take a closer look at these nondifferentiability points and show, … Read more

The s-Monotone Index Selection Rule for Criss-Cross Algorithms of Linear Complementarity Problems

In this paper we introduce the s-monotone index selection rules for the well-known crisscross method for solving the linear complementarity problem (LCP). Most LCP solution methods require a priori information about the properties of the input matrix. One of the most general matrix properties often required for finiteness of the pivot algorithms (or polynomial complexity … Read more

A New Error Bound Result for Generalized Nash Equilibrium Problems and its Algorithmic Application

We present a new algorithm for the solution of Generalized Nash Equilibrium Problems. This hybrid method combines the robustness of a potential reduction algorithm and the local quadratic convergence rate of the LP-Newton method. We base our local convergence theory on an error bound and provide a new sufficient condition for it to hold that … Read more

Abstract Newtonian Frameworks and Their Applications

We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving … Read more

AN EFFICIENT ALGORITHM FOR SECOND-ORDER CONE LINEAR COMPLEMENTARITY PROBLEMS

Recently, the globally uniquely solvable (GUS) property of the linear transformation $M\in R^{n\times n}$ in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan [30] contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this … Read more

Modified alternating direction methods for the modified multiple-sets split feasibility problems

Inthispaper, weproposetwonewmultiple-setssplitfeasibilityproblem(MSFP)models, where the MSFP requires to find a point closest to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. This problem arises in image restoration, … Read more

A Newton-Fixed Point Homotopy Algorithm for Nonlinear Complementarity Problems with Generalized Monotonicity

In this paper has been considered probability-one global convergence of NFPH (Newton-Fixed Point Homotopy) algorithm for system of nonlinear equations and has been proposed a probability-one homotopy algorithm to solve a regularized smoothing equation for NCP with generalized monotonicity. Our results provide a theoretical basis to develop a new computational method for nonlinear equation systems … Read more

Interior point methods for sufficient LCP in a wide neighborhood of the central path with optimal iteration complexity

Three interior point methods are proposed for sufficient horizontal linear complementarity problems (HLCP): a large update path following algorithm, a first order corrector-predictor method, and a second order corrector-predictor method. All algorithms produce sequences of iterates in the wide neighborhood of the central path introduced by Ai and Zhang. The algorithms do not depend on … Read more

Aubin Property and Uniqueness of Solutions in Cone Constrained Optimization

We discuss conditions for the Aubin property of solutions to perturbed cone constrained programs, by using and refining results given in \cite{KlaKum02}. In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map. Moreover, we give conditions under which the critical point … Read more

STRONGLY REGULAR NONSMOOTH GENERALIZED EQUATIONS (REVISED)

This note suggests the implicit function theorem for generalized equations, unifying Robinson’s theorem for strongly regular generalized equations and Clarke’s implicit function theorem for equations with Lipschitz-continuous mappings. ArticleDownload View PDF