Complementarity and Variational Inequalities

First order optimality conditions for mathematical programs with second-order cone complementarity constraints

  • Jane J. Ye
  • Jinchuan Zhou
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization, Second-Order Cone Programming Tags , , , , ,

    In this paper we consider a mathematical program with second-order cone complementarity constraints (SOCMPCC). The SOCMPCC generalizes the mathematical program with complementarity constraints (MPCC) in replacing the set of nonnegative reals by a second-order cone. We show that if the SOCMPCC is considered as an optimization problem with convex cone constraints, then Robinson’s constraint qualification … Read more

    Projected Reflected Gradient Methods for Monotone Variational Inequalities

  • Yura Malitsky
    • Categories Complementarity and Variational Inequalities Tags , , ,

    This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only … Read more

    Global convergence of sequential injective algorithm for weakly univalent vector equation: application to regularized smoothing Newton algorithm

  • Shunsuke Hayashi
    • Categories Complementarity and Variational Inequalities Tags , , ,

    It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the … Read more

    Convergence Conditions for Newton-type Methods Applied to Complementarity Systems with Nonisolated Solutions

  • Andreas Fischer
  • Alexey F. Izmailov
  • Markus Herrich
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , , , ,

    We consider a class of Newton-type methods for constrained systems of equations that involve complementarity conditions. In particular, at issue are the constrained Levenberg–Marquardt method and the recently introduced Linear Programming-Newton method, designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We … Read more

    On the Iteration Complexity of Some Projection Methods for Monotone Linear Variational Inequalities

  • Caihua Chen
  • Xiaoling Fu
  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Complementarity and Variational Inequalities Tags , , ,

    Projection type methods are among the most important methods for solving monotone linear variational inequalities. In this note, we analyze the iteration complexity for two projection methods and accordingly establish their worst-case O(1/t) convergence rates measured by the iteration complexity in both the ergodic and nonergodic senses, where t is the iteration counter. Our analysis … Read more

    On the cone eigenvalue complementarity problem for higher-order tensors

  • Hongjin He
  • Liqun Qi
  • Chen Ling
    • Categories Complementarity and Variational Inequalities, Global Optimization Tags , , , ,

    In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing … Read more

    A close look at auxiliary problem principles for equilibria

  • Giancarlo Bigi
  • Mauro Passacantando
    • Categories Complementarity and Variational Inequalities, Game Theory

    The auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a … Read more

    Globally Convergent Primal-Dual Active-Set Methods with Inexact Subproblem Solves

  • Frank E. Curtis
  • Zheng Han
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Quadratic Programming Tags , , , , ,

    We propose primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs). The iterates of the algorithms are partitions of the index set of variables, where corresponding to each partition there exist unique primal-dual variables that can be obtained by solving a (reduced) linear system. Algorithms of … Read more

    Sufficient weighted complementarity problems

  • Florian A. Potra
    • Categories Complementarity and Variational Inequalities Tags , , ,

    This paper presents some fundamental results about sufficient linear weighted complementarity problems. Such a problem depends on a nonnegative weight vector. If the weight vector is zero, the problem reduces to a sufficient linear complementarity problem that has been extensively studied. The introduction of the more general notion of a weighted complementarity problem (wCP) was … Read more

    Normally admissible stratifications and calculation of normal cones to a finite union of polyhedral sets

  • Lukáš Adam
  • Michal Cervinka
  • Miroslav Pistek
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more