optimal value function

Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations

  • Falk M. Hante
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach … Read more

    Hadamard Directional Diff erentiability of the Optimal Value of a Linear Second-order Conic Programming Problem

  • Duan Qingsong
  • Liwei Zhang
  • Sainan Zhang
    • Categories Second-Order Cone Programming, Stochastic Programming Tags , , ,

    In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the primal problem and the Lagrange dual problem. We demonstrate that the optimal value function can be expressed as … Read more

    Estimates of generalized Hessians for optimal value functions in mathematical programming

  • Alain Zemkoho
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    The \emph{optimal value function} is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the \emph{cost/revenue} generated while \emph{minimizing/maximizing} a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity … Read more

    Pessimistic bilevel linear optimization

  • Stephan Dempe
  • Susanne Franke
  • Guimei Luo
    • Categories Nonsmooth Optimization Tags , , , ,

    In this paper, we investigate the pessimistic bilevel linear optimization problem (PBLOP). Based on the lower level optimal value function and duality, the PBLOP can be transformed to a single-level while nonconvex and nonsmooth optimization problem. By use of linear optimization duality, we obtain a tractable and equivalent transformation and propose algorithms for computing global … Read more

    Stability and genericity for semi-algebraic compact programs

  • Gue Myung Lee
  • Tiên-So'n Pham
    • Categories Constrained Nonlinear Optimization, Global Optimization Tags , , , , , , ,

    In this paper we consider the class of polynomial optimization problems with inequality and equality constraints, in which every problem of the class is obtained by perturbations of the objective function, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties (e.g., strong H\”older stability with explicitly determined exponents, semicontinuity, … 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

    New optimality conditions for the semivectorial bilevel optimization problem

  • Stephan Dempe
  • Alain Zemkoho
  • Nazih Gadhi
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    The paper is concerned with the optimistic formulation of a bilevel optimization problem with multiobjective lower-level problem. Considering the scalarization approach for the multiobjective program, we transform our problem into a scalar-objective optimization problem with inequality constraints by means of the well-known optimal value reformulation. Completely detailed rst-order necessary optimality conditions are then derived in … Read more

    Sensitivity analysis in linear semi-infinite programming via partitions

  • Miguel A. Goberna
  • Tamás Terlaky
  • Maxim I. Todorov
    • Categories Other Topics, Semi-infinite Programming Tags , , ,

    This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming problem to depend linearly on the size of the perturbations, when these perturbations are directional, involve either the cost coefficients or the right-hand-side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first … Read more