variational inequality

A von Neumann Alternating Method for Finding Common Solutions to Variational Inequalities

  • Yair Censor
  • Aviv Gibali
  • Simeon Reich
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , , , , , ,

    Modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space. Citation Nonlinear Analysis Series A: Theory, Methods & … Read more

    Proximal Methods with Bregman Distances to Solve VIP on Hadamard manifolds

  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Global Optimization, Global Optimization Theory Tags , , , ,

    We present an extension of the proximal point method with Bregman distances to solve Variational Inequality Problems (VIP) on Hadamard manifolds (simply connected finite dimensional Riemannian manifold with nonpositive sectional curvature). Under some natural assumption, as for example, the existence of solutions of the (VIP) and the monotonicity of the multivalued vector field, we prove … Read more

    A Class of Dantzig-Wolfe Type Decomposition Methods for Variational Inequality Problems

  • Juan Pablo Luna
  • Claudia Sagastizábal
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags ,

    We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with set-valued or nonmonotone operators, as well as various kinds of approximations in the subproblems of the functions and derivatives in the single-valued … Read more

    On the O(1/t) convergence rate of alternating direction method

  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , ,

    The old alternating direction method (ADM) has found many new applications recently, and its empirical efficiency has been well illustrated in various fields. However, the estimate of ADM’s convergence rate remains a theoretical challenge for a few decades. In this note, we provide a uniform proof to show the O(1/t) convergence rate for both the … Read more

    On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

  • Bingsheng He
    • Categories Complementarity and Variational Inequalities, Convex Optimization Tags , ,

    Recently, Nemirovski’s analysis indicates that the extragradient method has the $O(1/t)$ convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, we have developed a class of Fej\’er monotone projection and contraction methods. Until now, only convergence results are available to these projection and contraction methods, though … Read more

    An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , , , ,

    This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) method. Iteration-complexity results are established for the A-HPE method, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE method are … Read more

    Iteration-Complexity of a Newton Proximal Extragradient Method for Monotone Variational Inequalities and Inclusion Problems

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , , , , , ,

    In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient framework has been used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth … Read more

    Variational Convergence of Bifunctions: Motivating Applications

  • Alejandro Jofré
  • Roger J.B. Wets
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Game Theory Tags , , , , , , , , ,

    It’s shown that a number of variational problems can be cast as finding the maxinf-points (or minsup-points) of bivariate functions, coveniently abbreviated to bifunctions. These variational problems include: linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, non-cooperative games, Walras and Nash equilibrium problems. One can then appeal to the theory of lopsided convergence … Read more

    Convergence analysis of primal-dual algorithms for total variation image restoration

  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , ,

    Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primal-dual algorithms and shows that the involved parameters of those primal-dual algorithms (including the step sizes) can be significantly enlarged if … Read more

    A quasi-Newton strategy for the sSQP method for variational inequality and optimization problems

  • Damián Fernández
    • Categories Constrained Nonlinear Optimization, Quadratic Programming Tags , , ,

    The quasi-Newton strategy presented in this paper preserves one of the most important features of the stabilized Sequential Quadratic Programming method, the local convergence without constraint qualifications assumptions. It is known that the primal-dual sequence converges quadratically assuming only the second-order sufficient condition. In this work, we show that if the matrices are updated by … Read more