Complementarity and Variational Inequalities

Solving Linear Programs with Complementarity Constraints using Branch-and-Cut

  • John E. Mitchell
  • Jong-Shi Pang
  • Bin Yu
    • Categories Branch and Cut Algorithms, Complementarity and Variational Inequalities Tags , ,

    A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadratic programs, and problems involving equilibrium constraints. The … Read more

    Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

  • Radu Ioan Bot
  • Ernö Robert Csetnek
  • Nimit Nimana
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , , ,

    We consider the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. In order to solve this problem, we propose an algorithm which combines the gradient method with a penalization technique. Moreover, we insert in our algorithm an inertial term, which is able to take … Read more

    A Study of the Difference-of-Convex Approach for Solving Linear Programs with Complementarity Constraints

  • Francisco Jara-Moroni
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    This paper studies the difference-of-convex (DC) penalty formulations and the associated difference-of-convex algorithm (DCA) for computing stationary solutions of linear programs with complementarity constraints (LPCCs). We focus on three such formulations and establish connections between their stationary solutions and those of the LPCC. Improvements of the DCA are proposed to remedy some drawbacks in a … Read more

    Evaluating the effect of environmental regulations on a closed-loop supply chain network: a variational inequality approach

  • Elisabetta Allevi
  • Igor V. Konnov
  • Adriana Gnudi
  • Giorgia Oggioni
    • Categories Complementarity and Variational Inequalities, Supply Chain Management

    Global climate change has encouraged international and regional adoption of pollution taxes and carbon emission reduction policies. Europe has taken the leadership in environmental regulations by introducing the European Union Emissions Trading System (EU-ETS) in 2005 and by developing and promoting a set of policies destined to lower carbon emissions from transport sectors. These environmental … Read more

    Decomposition Method for Oligopolistic Competitive Models with Common Pollution Regulation

  • Elisabetta Allevi
  • Igor V. Konnov
  • Adriana Gnudi
  • Giorgia Oggioni
    • Categories Complementarity and Variational Inequalities

    We consider the general problem of industrial production in a set of countries subject to a common environmental regulation that limits the emissions of specific sectors. Due to these restrictions, the problem is treated as a generalized non-cooperative game where players (countries) have joint (environmental) constraints caused by the necessity of a common and compulsory … Read more

    Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

  • Christian Kirches
  • Felix Lenders
  • Paul Manns
    • Categories Approximation Algorithms, Complementarity and Variational Inequalities, Systems governed by Differential Equations Optimization

    We extend recent work on mixed-integer nonlinear optimal control prob- lems (MIOCPs) to the case of integer control functions subject to constraints. Promi- nent examples of such systems include problems with restrictions on the number of switches permitted, or problems that minimize switch cost. We extend a theorem due to [Sager et al., Math. Prog. … Read more

    Approximations and Generalized Newton Methods

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

    We study local convergence of generalized Newton methods for both equations and inclusions by using known and new approximations and regularity properties at the solution. Including Kantorovich-type settings, our goal are statements about all (not only some) Newton sequences with appropriate initial points. Our basic tools are results of Klatte-Kummer (2002) and Kummer (1988, 1995), … Read more

    Gap functions for quasi-equilibria

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

    An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool … Read more

    The Lyapunov rank of an improper cone

  • Michael Orlitzky
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , , ,

    Let K be a closed convex cone with dual K^* in a finite-dimensional real inner-product space V. The complementarity set of K is C(K) = { (x, s) in K × K^* | = 0 }. We say that a linear transformation L : V -> V is Lyapunov-like on K if = 0 for all (x, … Read more

    A Practical Scheme to Compute Pessimistic Bilevel Optimization Problem

  • Bo Zeng
    • Categories Complementarity and Variational Inequalities, Game Theory Tags ,

    In this paper, we present a new computation scheme for pessimistic bilevel optimization problem, which so far does not have any computational methods generally applicable yet. We first develop a tight relaxation and then design a simple scheme to ensure a feasible and optimal solution. Then, we discuss using this scheme to compute linear pessimistic … Read more