Complementarity and Variational Inequalities

On QPCCs, QCQPs and Completely Positive Programs

  • Lijie Bai
  • John E. Mitchell
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming

    This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a quadratically constrained quadratic program (QCQP), a quadratic program with complementarity constraints (QPCC), and rank constrained semidefinite programs. Our results … Read more

    A Relaxed-Projection Splitting Algorithm for Variational Inequalities in Hilbert Spaces

  • Yunier Bello-Cruz
  • R. Diaz Millan
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each … Read more

    On the irreducibility, Lyapunov rank, and automorphisms of speical Bishop-Phelps cones

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

    Motivated by optimization considerations, we consider special Bishop-Phelps cones in R^n which are of the form {(t,x): t \geq ||x||} for some norm on R^(n-1). We show that for n bigger than 2, such cones are always irreducible. De fining the Lyapunov rank of a proper cone K as the dimension of the Lie algebra of … Read more

    Complementarity Formulations of l0-norm Optimization Problems

  • Mingbin Feng
  • John E. Mitchell
  • Jong-Shi Pang
  • Xin Shen
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , ,

    In a number of application areas, it is desirable to obtain sparse solutions. Minimizing the number of nonzeroes of the solution (its l0-norm) is a difficult nonconvex optimization problem, and is often approximated by the convex problem of minimizing the l1-norm. In contrast, we consider exact formulations as mathematical programs with complementarity constraints and their … Read more

    Local Convergence of the Method of Multipliers for Variational and Optimization Problems under the Sole Noncriticality Assumption

  • Alexey F. Izmailov
  • A. S. Kurennoy
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local superlinear convergence is established under the … Read more

    A hybrid proximal extragradient self-concordant primal barrier method for monotone variational inequalities

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
  • Mauricio Romero
    • Categories Complementarity and Variational Inequalities Tags ,

    In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of “R. D. C. Monteiro and B. F. Svaiter, Iteration-Complexity of a Newton … Read more

    A game-theoretic approach to computation offloading in mobile cloud computing

  • Valeria Cardellini
  • Francisco Facchinei
  • Veronica Piccialli
  • Vittoria De Nitto Persone'
  • Valerio Di Valerio
  • Vincenzo Grassi
  • Francesco Lo Presti
    • Categories Applications - OR and Management Sciences, Complementarity and Variational Inequalities, Game Theory

    We consider a three-tier architecture for mobile and pervasive computing scenarios, consisting of a local tier of mobile nodes, a middle tier (cloudlets) of nearby computing nodes, typically located at the mobile nodes access points but characterized by a limited amount of resources, and a remote tier of distant cloud servers, which have practically infinite … Read more

    Inverse Parametric Optimization with an Application to Hybrid System Control

  • Paul J. Goulart
  • Andreas B. Hempel
  • John Lygeros
    • Categories Complementarity and Variational Inequalities, Control Applications, Convex Optimization Tags ,

    We present a number of results on inverse parametric optimization and its application to hybrid system control. We show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem. We exploit these results in … Read more

    A direct splitting method for nonsmooth variational inequalities

  • Yunier Bello-Cruz
  • R. Diaz Millan
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonsmooth Optimization

    We propose a direct splitting method for solving nonsmooth variational inequality problems in Hilbert spaces. The weak convergence is established, when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which explores strongly the structure of the operator … Read more

    An approximation scheme for a class of risk-averse stochastic equilibrium problems

  • Juan Pablo Luna
  • Claudia Sagastizábal
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , ,

    We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions … Read more