Complementarity and Variational Inequalities

Stochastic Mathematical Programs with Equilibrium Constraints, Modeling and Sample Average Approximation

  • Alexander Shapiro
  • Huifu Xu
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , , ,

    In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate piecewise structure and directional differentiability of both — the lower level equilibrium solution and objective integrant. We show almost sure convergence of optimal values, optimal solutions … Read more

    Lowner’s Operator and Spectral Functions in Euclidean Jordan Algebras

  • Jie Sun
  • Defeng Sun
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    We study analyticity, differentiability, and semismoothness of Lowner’s operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra … Read more

    Interior Methods for Mathematical Programs with Complementarity Constraints

  • Sven Leyffer
  • Gabriel Lopez-Calva
  • Jorge Nocedal
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then … Read more

    Convergence Analysis of an Interior-Point Method for Mathematical Programs with Equilibrium Constraints

  • Arun Sen
  • David F. Shanno
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    We prove local and global convergence results for an interior-point method applied to mathematical programs with equilibrium constraints. The global result shows the algorithm minimizes infeasibility regardless of starting point, while one result proves local convergence when penalty functions are exact; another local result proves convergence when the solution is not even a KKT point. … Read more

    An Algorithm for Perturbed Second-order Cone Programs

  • Yu Xia
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization, Second-Order Cone Programming Tags , , ,

    The second-order cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Q-quadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples … Read more

    The Q Method for Symmetric Cone Programming

  • Farid Alizadeh
  • Yu Xia
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We extend the Q method to the symmetric cone programming. An infeasible interior point algorithm and a Newton-type algorithm are given. We give convergence results of the interior point algorithm and prove that the Newton-type algorithm is good for Citation AdvOl-Report#2004/18 McMaster University, Advanced Optimization Laboratory Hamilton, Ontario, Canada October 2004 Article Download View The … Read more

    Leader-Follower Equilibria for Electric Power and NO_x Allowances Markets

  • Yihsu Chen
  • Sven Leyffer
  • Todd S. Munson
  • Benjamin Hobbs
    • Categories Complementarity and Variational Inequalities, Finance and Economics Tags , ,

    This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medium-sized firms are treated as Cournot followers with … Read more

    Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

  • Akiko Yoshise
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in Faybusovich (1997) depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang (1998), where a vast set of conclusions concerning continuous trajectories is shown for monotone … Read more

    Stochastic Programming with Equilibrium Constraints

  • Alexander Shapiro
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , , ,

    In this paper we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems. Citation School of Industrial … Read more

    GLOBAL CONVERGENCE OF AN ELASTIC MODE APPROACH FOR A CLASS OF MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

  • Mihai Anitescu
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Mechanical Engineering Tags , , ,

    We prove that any accumulation point of an elastic mode approach, applied to the optimization of a mixed P variational inequality, that approximately solves the relaxed subproblems is a C-stationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addition, the accumulation point satis es the MPCC-LICQ constraint quali cation and if … Read more