Complementarity and Variational Inequalities

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

  • Yun-Chol Jong
    • Categories Complementarity and Variational Inequalities, Generalized Convexity/Monoticity Tags , , ,

    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

  • Florian A. Potra
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , , , ,

    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

  • Diethard Klatte
  • Bernd Kummer
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    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)

  • Alexey F. Izmailov
    • Categories Complementarity and Variational Inequalities Tags , , , , ,

    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. Article Download View STRONGLY REGULAR NONSMOOTH GENERALIZED EQUATIONS (REVISED)

    An algorithmic characterization of P-matricity

  • Ibtihel Ben Gharbia
  • Jean Charles Gilbert
    • Categories Complementarity and Variational Inequalities Tags , , , ,

    It is shown that a matrix $M$ is a P-matrix if and only if, whatever is the vector $q$, the Newton-min algorithm does not cycle between two points when it is used to solve the linear complementarity problem $0\leq x\perp (Mx+q)\geq0$. Citation Inria research report RR-8004 Article Download View An algorithmic characterization of P-matricity

    An efficient matrix splitting method for the second-order cone complementarity problem

  • Zhang Lei-Hong
  • Weihong Yang
    • Categories Complementarity and Variational Inequalities, Optimization Software and Modeling Systems, Second-Order Cone Programming Tags , , , , , , ,

    Given a symmetric and positive (semi)definite $n$-by-$n$ matrix $M$ and a vector, in this paper, we consider the matrix splitting method for solving the second-order cone linear complementarity problem (SOCLCP). The matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems (LCP), and its linear convergence … Read more

    Weighted complementarity problems – a new paradigm for computing equilibria

  • Florian A. Potra
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , ,

    This paper introduces the notion of a weighted Complementarity Problem (wCP), which consists in finding a pair of vectors $(x,s)$ belonging to the intersection of a manifold with a cone, such that their product in a certain algebra, $x\circ s$, equals a given weight vector $w$. When $w$ is the zero vector, then wCP reduces … Read more

    On the bilinearity rank of a proper cone and Lyapunov-like transformations

  • M. Seetharama Gowda
  • Jiyuan Tao
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , ,

    A real square matrix Q is a bilinear complementarity relation on a proper cone K in R^n if x in K, s in K^* with x perpendicular to s implies x^{T}Qs=0, where K^* is the dual of K. The bilinearity rank of K is the dimension of the space of all bilinear complementarity relations on … Read more

    A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones

  • Chek Beng Chua
  • Zhen Li
    • Categories Complementarity and Variational Inequalities, Convex Optimization Tags , , ,

    We present a new barrier-based method of constructing smoothing approximations for the Euclidean projector onto closed convex cones. These smoothing approximations are used in a smoothing proximal point algorithm to solve monotone nonlinear complementarity problems (NCPs) over a convex cones via the normal map equation. The smoothing approximations allow for the solution of the smoothed … Read more

    On the Convergence Properties of Non-Euclidean Extragradient Methods for Variational Inequalities with Generalized Monotone Operators

  • Cong D. Dang
  • Guanghui Lan
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    In this paper, we study a class of generalized monotone variational inequality (GMVI) problems whose operators are not necessarily monotone (e.g., pseudo-monotone). We present non-Euclidean extragradient (N-EG) methods for computing an approximate strong solution of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or H\”{o}lder continuity properties for their operators … Read more