Nonlinear Optimization

Copositivity-based approximations for mixed-integer fractional quadratic optimization

  • Paula Alexandra Amaral
  • Immanuel M. Bomze
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    We propose a copositive reformulation of the mixed-integer fractional quadratic problem (MIFQP) under general linear constraints. This problem class arises naturally in many applications, e.g., for optimizing communication or social networks, or studying game theory problems arising from genetics. It includes several APX-hard subclasses: the maximum cut problem, the $k$-densest subgraph problem and several of … Read more

    Minimal Points, Variational Principles, and Variable Preferences in Set Optimization

  • Boris S. Mordukhovich
  • Antoine Soubeyran
  • Bao Q. Truong
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization, Nonlinear Optimization

    The paper is devoted to variational analysis of set-valued mappings acting from quasimetric spaces into topological spaces with variable ordering structures. Besides the mathematical novelty, our motivation comes from applications to adaptive dynamical models of behavioral sciences. We develop a unified dynamical approach to variational principles in such settings based on the new minimal point … Read more

    A note on Fejér-monotone sequences in product spaces and its applications to the dual convergence of augmented Lagrangian methods

  • M. Marques Alves
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    In a recent Math. Program. paper, Eckstein and Silva proposed a new error criterion for the approximate solutions of augmented Lagrangian subproblems. Based on a saddle-point formulation of the primal and dual problems, they proved that dual sequences generated by augmented Lagrangians under this error criterion are bounded and that theirs limit points are dual … Read more

    Assessing the reliability of general-purpose Inexact Restoration methods

  • Ernesto G. Birgin
  • Luís Felipe Bueno
  • José Mario Martínez
    • Categories Constrained Nonlinear Optimization Tags , ,

    Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper [N. Banihashemi … Read more

    Parallel Algorithms for Big Data Optimization

  • Francisco Facchinei
  • Simone Sagratella
  • Gesualdo Scutari
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Parallel Algorithms Tags , , , ,

    We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss-Seidel (i.e., sequential) ones, as … Read more

    Generalized Inexact Proximal Algorithms: Habit’s/ Routine’s Formation with Resistance to Change, following Worthwhile Changes

  • Glaydston Bento
  • Antoine Soubeyran
    • Categories Applications - Science and Engineering, Nonsmooth Optimization, Unconstrained Optimization Tags

    This paper shows how, in a quasi metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,…). More precisely, the new perturbation term represents an index of resistance to change, defined as a “curved enough” function of the quasi distance … Read more

    Dynamic scaling in the Mesh Adaptive Direct Search algorithm for blackbox optimization

  • Charles Audet
  • Sébastien Le Digabel
  • Christophe Tribes
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    Blackbox optimization deals with situations in which the objective function and constraints are typically computed by launching a time-consuming computer sim- ulation. The subject of this work is the Mesh Adaptive Direct Search (MADS) class of algorithms for blackbox optimization. We propose a way to dynamically scale the mesh, which is the discrete spatial structure … Read more

    On the update of constraint preconditioners for regularized KKT systems

  • Stefania Bellavia
  • Valentina De Simone
  • Daniela di Serafino
  • Benedetta Morini
    • Categories Nonlinear Optimization, Quadratic Programming Tags , , ,

    We address the problem of preconditioning sequences of regularized KKT systems, such as those arising in Interior Point methods for convex quadratic programming. In this case, Constraint Preconditioners (CPs) are very effective and widely used; however, when solving large-scale problems, the computational cost for their factorization may be high, and techniques for approximating them appear … Read more

    Direct search based on probabilistic descent

  • Serge Gratton
  • Clément W. Royer
  • Luis Nunes Vicente
  • Z. Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional … Read more

    Parallel Multi-Block ADMM with o(1/k) Convergence

  • Wei Deng
  • Ming-Jun Lai
  • Zhimin Peng
  • Wotao Yin
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , ,

    This paper introduces a parallel and distributed extension to the alternating direction method of multipliers (ADMM). The algorithm decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This Jacobian-type algorithm is well suited for distributed computing and is particularly attractive for solving certain large-scale problems. This paper introduces … Read more