Nonlinear Optimization

A Filter Algorithm for Nonlinear Semidefinite Programming

  • Walter Gómez
  • Héctor Ramírez
    • Categories Nonlinear Optimization, Semi-definite Programming

    This paper proposes a filter method for solving nonlinear semidefinite programming problems. Our method extends to this setting the filter SQP (sequential quadratic programming) algorithm, recently introduced for solving nonlinear programming problems, obtaining their respective global convergence results. Citation CMM-B-06/10 – 171 Centre for Mathematical Modelling, UMR 2071, Universidad de Chile-CNRS. Casilla 170-3 Santiago 3, … Read more

    The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

  • Andrei V. Dmitruk
  • Alexander Kaganovich
    • Categories Constrained Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , , ,

    We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle. Citation A.V. Dmitruk, A.M. Kaganovich. The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle, Systems & Control … Read more

    Using Simplex Gradients of Nonsmooth Functions in Direct Search Methods

  • Ana Luisa Custódio
  • John E. Dennis, Jr.
  • Luis Nunes Vicente
    • Categories Unconstrained Optimization Tags , , , , ,

    It has been shown recently that the efficiency of direct search methods that use opportunistic polling in positive spanning directions can be improved significantly by reordering the poll directions according to descent indicators built from simplex gradients. The purpose of this paper is twofold. First, we analyze the properties of simplex gradients of nonsmooth functions … Read more

    Global Convergence of General Derivative-Free Trust-Region Algorithms to First and Second Order Critical Points

  • Andrew R. Conn
  • Katya Scheinberg
  • Luis Nunes Vicente
    • Categories Unconstrained Optimization Tags , , ,

    In this paper we prove global convergence for first and second-order stationarity points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of linear or quadratic models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds but, … Read more

    Simplex-type algorithm for optimizing a pseudolinear quadratic fractional function over a polytope

  • Miklós Ujvári
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity

    Recently Cambini and Carosi described a characterization of pseudolinearity of quadratic fractional functions. A reformulation of their result was given by Rapcsák. Using this reformulation, in this paper we describe an alternative proof of the Cambini–Carosi Theorem. Our proof is shorter than the proof given by Cambini–Carosi and less involved than the proof given by … Read more

    A New Class of Interior Proximal Methods for Optimization over the Positive Orthant

  • Paulo Roberto Oliveira
  • Sissy da S. Souza
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    In this work we present a family of variable metric interior proximal methods for solving optimization problems under nonnegativity constraints. We define two algorithms, in the inexact and exact forms. The kernels are metrics generated by diagonal matrices in each iteration and the regularization parameters are conveniently chosen to force the iterates to be interior … Read more

    A Brief History of Filter Methods

  • Roger Fletcher
  • Sven Leyffer
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    We consider the question of global convergence of iterative methods for nonlinear programming problems. Traditionally, penalty functions have been used to enforce global convergence. In this paper we review a recent alternative, so-called filter methods. Instead of combing the objective and constraint violation into a single function, filter methods view nonlinear optimization as a biobjective … 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

    The complexity of optimizing over a simplex, hypercube or sphere: a short survey

  • Etienne de Klerk
    • Categories Global Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems have many applications. We review known approximation results as well as negative (inapproximability) results from the recent literature. Citation CentER Discussion paper 2006-85 Tilburg University THe Netherlands Article Download View The complexity … Read more

    Large Scale Portfolio Optimization with Piecewise Linear Transaction Costs

  • Potaptchik Marina
  • Levent Tunçel
  • Henry Wolkowicz
    • Categories Finance and Economics, Nonsmooth Optimization, Quadratic Programming Tags , , ,

    We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model of the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle … Read more