optimality conditions

Optimality Conditions and Duality for Nonsmooth Multiobjective Optimization Problems with Cone Constraints and Applications

  • Jiawei Chen
  • Zhong-Ping Wan
  • Zheng Yue
    • Categories Nonsmooth Optimization, Other Topics Tags , , , , , ,

    In this work, a nonsmooth multiobjective optimization problem involving generalized invexity with cone constraints and Applications (for short, (MOP)) is considered. The Kuhn-Tucker necessary and sufficient conditions for (MOP) are established by using a generalized alternative theorem of Craven and Yang. The relationship between weakly efficient solutions of (MOP) and vector valued saddle points of … Read more

    Derivative-free methods for nonlinear programming with general lower-level constraints

  • Maria A. Diniz-Ehrhardt
  • José Mario Martínez
  • Lucas G. Pedroso
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. … Read more

    A new sequential optimality condition for constrained optimization and algorithmic consequences

  • Roberto Andreani
  • José Mario Martínez
  • Benar Fux Svaiter
    • Categories Nonlinear Optimization Tags , , ,

    Necessary first-order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. These conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qual- i cations. A new strong sequential optimality condition is introduced in the present paper. A proof that a well established Augmented Lagrangian … Read more

    On approximate KKT condition and its extension to continuous variational inequalities

  • Gabriel Haeser
  • María Laura Schuverdt
    • Categories Nonlinear Optimization Tags , ,

    In this work we introduce a necessary natural sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality problem without constraint quali cations, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition … Read more

    Standard Bi-Quadratic Optimization Problems and Unconstrained Polynomial Reformulations

  • Immanuel M. Bomze
  • Liqun Qi
  • Chen Ling
  • Xinzhen Zhang
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory Tags , , , ,

    A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, … Read more

    On sequential optimality conditions for smooth constrained optimization

  • Roberto Andreani
  • Gabriel Haeser
  • José Mario Martínez
    • Categories Nonlinear Optimization Tags , , ,

    Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed. ArticleDownload View PDF

    Optimal structure of gas transmission trunklines

  • Jean André
  • Joseph Frédéric Bonnans
    • Categories Constrained Nonlinear Optimization, Facility Planning and Design Tags , , , ,

    In this paper, we consider the optimal design of a straight pipeline system. Suppose a gas pipeline is to be designed to transport a specified flowrate from the entry point to the gas demand point. Physical and contractual requirements at supply and delivery nodes are known as well as the costs to buy and lay … Read more

    An elementary proof of optimality conditions for linear programming

  • Anders Forsgren
    • Categories Linear Programming Tags ,

    We give an elementary proof of optimality conditions for linear programming. The proof is direct, built on a straightforward classical perturbation of the constraints, and does not require either the use of Farkas’ lemma or the use of the simplex method. CitationTechnical Report TRITA-MAT-2008-OS6, Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, … Read more

    Duality of ellipsoidal approximations via semi-infinite programming

  • Filiz Gurtuna
    • Categories Convex Optimization, Semi-infinite Programming Tags , , , , , , , , , ,

    In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results … Read more

    Relaxing the Optimality Conditions of Box QP

  • Samuel Burer
  • Jieqiu Chen
    • Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming Tags , , ,

    We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first- and second-order necessary optimality conditions. We compare these relaxations with a basic semidefinite relaxation due to Shor, particularly in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger. We … Read more