parametric optimization

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

    Lectures on Parametric Optimization: An Introduction

  • Georg Still
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , ,

    The report aims to provide an overview over results from Parametric Optimization which could be called classical results on the subject. Parametric Optimization considers optimization problems depending on a parameter and describes how the feasible set, the value function, and the local or global minimizers of the program depend on changes in the parameter. After … 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

    A predictor-corrector path-following algorithm for dual-degenerate parametric optimization problems

  • Johannes Jäschke
  • Vyacheslav Kungurtsev
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    Most path-following algorithms for tracing a solution path of a parametric nonlinear optimization problem are only certifiably convergent under strong regularity assumptions about the problem functions, in particular, the linear independence of the constraint gradients at the solutions, which implies a unique multiplier solution for every nonlinear program. In this paper we propose and prove … Read more

    KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

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

    For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both … Read more

    On Differentiability Properties of Player Convex Generalized Nash Equilibrium Problems

  • Nadja Harms
  • Oliver Stein
  • Christian Kanzow
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonsmooth Optimization Tags , , , , ,

    This article studies differentiability properties for a reformulation of a player convex generalized Nash equilibrium problem as a constrained and possibly nonsmooth minimization problem. By using several results from parametric optimization we show that, apart from exceptional cases, all locally minimal points of the reformulation are differentiability points of the objective function. This justifies a … Read more

    Generalized Support Set Invariancy Sensitivity Analysis

  • Alireza Ghaffari Hadigheh
  • Tamás Terlaky
    • Categories Linear Programming Tags , , , , , ,

    Support set invariancy sensitivity analysis deals with finding the range of the parameter variation where there are optimal solutions with the same positive variables for all parameter values throughout this range. This approach to sensitivity analysis has been studied for Linear Optimization (LO) and Convex Quadratic Optimization (CQO) problems, when they are in standard form. … Read more

    Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors

  • Alireza Ghaffari Hadigheh
  • Oleksandr Romanko
  • Tamás Terlaky
    • Categories Finance and Economics, Linear Programming, Quadratic Programming Tags , , , , , ,

    In this paper we study the behavior of Convex Quadratic Optimization problems when variation occurs simultaneously in the right-hand side vector of the constraints and in the coefficient vector of the linear term in the objective function. It is proven that the optimal value function is piecewise-quadratic. The concepts of transition point and invariancy interval … Read more

    Sensitivity analysis in linear optimization: Invariant support set intervals

  • Alireza Ghaffari Hadigheh
  • Tamás Terlaky
    • Categories Linear Programming Tags , , , ,

    Sensitivity analysis is one of the most nteresting and preoccupying areas in optimization. Many attempts are made to investigate the problem’s behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties in sensitivity analysis. In this … Read more