constraint qualifications

A New Relaxation Scheme for Mathematical Programs with Equilibrium Constraints

  • Michael Ulbrich
  • Sonja Veelken
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , ,

    We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently non-degenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The … Read more

    Strong Duality and Minimal Representations for Cone Optimization

  • Levent Tunçel
  • Henry Wolkowicz
    • Categories Convex Optimization, Semi-definite Programming Tags , , ,

    The elegant results for strong duality and strict complementarity for linear programming, \LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a … Read more

    Metric regularity and systems of generalized equations

  • Andrei V. Dmitruk
  • Alexander Y. Kruger
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization Tags , , , ,

    The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity, metric regularity along a subspace, strong metric regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version … Read more

    On Augmented Lagrangian methods with general lower-level constraints

  • Roberto Andreani
  • Ernesto G. Birgin
  • José Mario Martínez
  • María Laura Schuverdt
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved … Read more

    Perturbation analysis of second order programming problems

  • Joseph Frédéric Bonnans
  • Héctor Ramírez
    • Categories Linear, Cone and Semidefinite Programming, Second-Order Cone Programming Tags , , , ,

    We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions. Citation Research Report 5293 (August … 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

    Universal Duality in Conic Convex Optimization

  • Dianne P. O'Leary
  • Simon P. Schurr
  • André L. Tits
    • Categories Linear, Cone and Semidefinite Programming Tags , , , ,

    Given a primal-dual pair of linear programs, it is well known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +infinity and -infinity. In contrast, for optimization problems over nonpolyhedral convex cones, … 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