convergence analysis

An Enhanced Spatial Branch-and-Bound Method in Global Optimization with Nonconvex Constraints

  • Peter Kirst
  • Oliver Stein
  • Paul Steuermann
    • Categories Global Optimization Tags , , ,

    We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian-Fromovitz directions to feasible points whose … Read more

    Greedy approximation in convex optimization

  • Vladimir Temlyakov
    • Categories Convex Optimization Tags , , ,

    We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. … Read more

    Greedy expansions in convex optimization

  • Vladimir Temlyakov
    • Categories Convex Optimization Tags , , ,

    This paper is a follow up to the previous author’s paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach spaces greedy algorithms — Weak Chebyshev Greedy … Read more

    Algorithms for Bilevel Pseudomonotone Variational Inequality Problems

  • Bui Van Dinh
  • Le Dung Muu
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , , , ,

    We propose easily implementable algorithms for minimizing the norm with pseudomonotone variational inequality constraints. This bilevel problem arises in the Tikhonov regularization method for pseudomonone variational inequalities. Since the solution set of the lower variational inequality is not given explicitly, the available methods of mathematical programming and variational inequality can not be applied directly. With … Read more

    A Bilevel Direct Search Method for Leader-Follower Optimization Problems and Applications

  • Gui-Hua Lin
  • Dali Zhang
    • Categories Game Theory, Other Topics, Stochastic Programming Tags , , , ,

    In the paper, we propose a bilevel direct search method for solving a type of leader-follower problems with each decision maker’s objective being a “black-box” function. First, we give a description for a leader-follower optimization problem. Then, we investigate a bilevel direct search method including two algorithms for combinatorially solving the upper and lower level … Read more

    Two stage stochastic equilibrium problems with equilibrium constraints: modeling and numerical schemes

  • Huifu Xu
  • Dali Zhang
    • Categories Complementarity and Variational Inequalities, Game Theory, Stochastic Programming Tags , , ,

    This paper presents a two stage stochastic equilibrium problem with equilibrium constraints(SEPEC) model. Some source problems which motivate the model are discussed. Monte Carlo sampling method is applied to solve the SEPEC. The convergence analysis on the statistical estimators of Nash equilibria and Nash stationary points are presented. Article Download View Two stage stochastic equilibrium … Read more

    The mesh adaptive direct search algorithm for periodic variables

  • Charles Audet
  • Sébastien Le Digabel
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization Tags , , , ,

    This work analyzes constrained black box optimization in which the functions defining the problem are periodic with respect to some or all the variables. We show that the natural strategy of mapping trial points into the interval defined by the period in the Mesh Adaptive Direct Search (MADS) framework can be easily done in practice, … Read more

    Parallel Space Decomposition of the Mesh Adaptive Direct Search algorithm

  • Charles Audet
  • John E. Dennis, Jr.
  • Sébastien Le Digabel
    • Categories Applications - OR and Management Sciences, Nonsmooth Optimization, Parallel Algorithms Tags , , , ,

    This paper describes a parallel space decomposition PSD technique for the mesh adaptive direct search MADS algorithm. MADS extends a generalized pattern search for constrained nonsmooth optimization problems. The objective of the present work is to obtain good solutions to larger problems than the ones typically solved by MADS. The new method PSD-MADS is an … Read more

    Two theoretical results for sequential semidefinite programming

  • Rodrigo Garcés
  • Walter Gómez
  • Florian Jarre
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems Tags , , , , , ,

    We examine the local convergence of a sequential semidefinite programming approach for solving nonlinear programs with nonlinear semidefiniteness constraints. Known convergence results are extended to slightly weaker second order sufficient conditions and the resulting subproblems are shown to have local convexity properties that imply a weak form of self-concordance of the barrier subproblems. Citation Preprint, … Read more

    Convergence Analysis of an Interior-Point Method for Nonconvex Nonlinear Programming

  • Hande Benson
  • Arun Sen
  • David F. Shanno
    • Categories Constrained Nonlinear Optimization Tags , ,

    In this paper, we present global and local convergence results for an interior-point method for nonlinear programming. The algorithm uses an $\ell_1$ penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method [6]. … Read more