Complementarity and Variational Inequalities

The Principle of Hamilton for Mechanical Systems with Impacts and Unilateral Constraints

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , ,

    An action integral is presented for Hamiltonian mechanics in canonical form with unilateral constraints and/or impacts. The transition conditions on generalized impulses and the energy are presented as variational inequalities, which are obtained by the application of discontinuous transversality conditions. The energetical behavior for elastic, plastic and blocking type impacts are analyzed. A general impact … Read more

    Solving Bilevel Mixed Integer Program by Reformulations and Decomposition

  • Yu An
  • Bo Zeng
    • Categories (Mixed) Integer Linear Programming, Complementarity and Variational Inequalities Tags , , ,

    In this paper, we study bilevel mixed integer programming (MIP) problem and present a novel computing scheme based on reformulations and decomposition strategy. By converting bilevel MIP into a constrained mathematical program, we present its single-level reformulations that are friendly to perform analysis and build insights. Then, we develop a decomposition algorithm based on column-and-constraint … Read more

    Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-type Constraints and a Regularization Method

  • Oleg Burdakov
  • Christian Kanzow
  • Alexandra Schwartz
    • Categories Complementarity and Variational Inequalities, Global Optimization, Nonlinear Optimization Tags , , , , , , ,

    Optimization problems with cardinality constraints are very dicult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in … Read more

    Justification of Constrained Game Equilibrium Models

  • Igor V. Konnov
    • Categories Basic Sciences Applications, Complementarity and Variational Inequalities, Game Theory

    We consider an extension of a noncooperative game where players have joint binding constraints. In this model, the constrained equilibrium can not be implemented within the same noncooperative framework and requires some other additional regulation procedures. We consider several approaches to resolution of this problem. In particular, a share allocation method is presented and substantiated. … Read more

    Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities

  • Shu Lu
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , ,

    Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build … Read more

    Individual confidence intervals for true solutions to stochastic variational inequalities

  • Amarjit Budhiraja
  • Shu Lu
  • Lamm Michael
    • Categories Complementarity and Variational Inequalities, Stochastic Programming

    Stochastic variational inequalities (SVI) provide a means for modeling various optimization and equilibrium problems where data are subject to uncertainty. Often it is necessary to estimate the true SVI solution by the solution of a sample average approximation (SAA) problem. This paper proposes three methods for building confidence intervals for components of the true solution, … Read more

    Calmness of linear programs under perturbations of all data: characterization and modulus

  • M.J. Cánovas
  • Abderrahim Hantoute
  • J. Parra
  • F.J. Toledo
    • Categories Complementarity and Variational Inequalities, Convex Optimization, Linear Programming Tags , , ,

    This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations … Read more

    On Auction Models of Conflict with Network Applications

  • Igor V. Konnov
    • Categories Complementarity and Variational Inequalities, Telecommunications, Transportation

    We consider several models of complex systems with active elements and show that the auction mechanism appears very natural in attaining proper equilibrium states, even in comparison with game theory ones. In particular, network equilibria are treated as implementation of the auction principle. An additional example of resource allocation in wireless communication networks is also … Read more

    On Calmness of the Argmin Mapping in Parametric Optimization Problems

  • Diethard Klatte
  • Bernd Kummer
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Semi-infinite Programming Tags , , ,

    Recently, Canovas et. al. (2013) presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semi-infinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework … Read more

    Superlinearly convergent smoothing Newton continuation algorithms for variational inequalities over definable sets

  • Chek Beng Chua
  • Le Thi Khanh Hien
    • Categories Complementarity and Variational Inequalities Tags , , ,

    In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li to extend various smoothing Newton continuation algorithms to variational inequalities over general closed convex sets X. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly … Read more