Complementarity and Variational Inequalities

ON USING THE ELASTIC MODE IN NONLINEAR PROGRAMMING APPROACHES TO MATHEMATICALPROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

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

    We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sucient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and that satisfy the quadratic growth condition can … Read more

    OPTIMIZATION-BASED SIMULATION OF NONSMOOTH RIGID MULTIBODY DYNAMICS

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

    We present a time-stepping method to simulate rigid multibody dynamics with inelastic collision, contact, and friction. The method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program. We prove that a solution sequence of the method converges to the solution of a measure differential inclusion. … Read more

    Necessary and Sufficient Optimality Conditions for Mathematical Programs with Equilibrium Constraints

  • Jane J. Ye
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient or locally sufficient for optimality under some MPEC … Read more

    Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem

  • Y. Kanno
  • J.A.C. Martins
  • A. Pinto da Costa
    • Categories Complementarity and Variational Inequalities, Mechanical Engineering, Second-Order Cone Programming

    A new formulation is presented for the three-dimensional incremental quasi-static problems with unilateral frictional contact. Under the assumptions of small rotations and small strains, a Second-Order Cone Linear omplementarity Problem (SOCLCP) is formulated, which consists of complementarity conditions defined by the bilinear functions and the second-order cone constraints. The equilibrium configurations are obtained by using … Read more

    On the Global Minimization of the Value-at-Risk

  • Sven Leyffer
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Global Optimization, Nonlinear Optimization

    In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined … Read more

    Some Properties of Regularization and Penalization Schemes for MPECs

  • Daniel Ralph
  • Stephen Wright
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , ,

    Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by … Read more

    Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization

  • Robert Fourer
  • David M. Gay
    • Categories Complementarity and Variational Inequalities, Modeling Languages and Systems, Optimization Software Design Principles Tags , , , , ,

    This paper draws from our experience in developing the AMPL modeling language, to show where numerical issues have been crucial to modeling language design and where modeling language advances have strongly influenced the design of solvers. Citation Proceedings of the 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, D.F. Griffiths and G.A. Watson, eds., University … Read more

    Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

  • Hande Benson
  • Arun Sen
  • David F. Shanno
  • Robert J. Vanderbei
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Game Theory Tags , , , ,

    In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPEC’s)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties, present an example … Read more

    A Homogeneous Model for $ and *$ Nonlinear Complementarity Problems

  • Akiko Yoshise
    • Categories Complementarity and Variational Inequalities Tags , , ,

    The homogeneous model for linear programs is an elegant means of obtaining the solution or certificate of infeasibility and has importance regardless of the method used for solving the problem, interior-point methods or other methods. In 1999, Andersen and Ye generalized this model to monotone complementarity problems (CPs) and showed that most of the desirable … Read more

    A robust SQP method for mathematical programs with linear complementarity constraints

  • Xinwei Liu
  • Georgia Perakis
  • Jie Sun
    • Categories Complementarity and Variational Inequalities Tags , , , ,

    The relationship between the mathematical program with linear complementarity constraints (MPCC) and its inequality relaxation is studied. A new sequential quadratic programming (SQP) method is presented for solving the MPCC based on this relationship. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived … Read more