Constrained Nonlinear Optimization

A Matrix-free Algorithm for Equality Constrained Optimization Problems with Rank-deficient Jacobians

  • Frank E. Curtis
  • Jorge Nocedal
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization, Infinite Dimensional Optimization Tags , , , , ,

    We present a line search algorithm for large-scale constrained optimization that is robust and efficient even for problems with (nearly) rank-deficient Jacobian matrices. The method is matrix-free (i.e., it does not require explicit storage or factorizations of derivative matrices), allows for inexact step computations, and is applicable for nonconvex problems. The main components of the … Read more

    A Primal-Dual Augmented Lagrangian

  • Philip E. Gill
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both … Read more

    Concave programming for minimizing the zero-norm over polyhedral sets

  • Francesco Rinaldi
  • Fabio Schoen
  • Marco Sciandrone
    • Categories Constrained Nonlinear Optimization, Data-Mining Tags , ,

    Given a non empty polyhedral set, we consider the problem of finding a vector belonging to it and having the minimum number of nonzero components, i.e., a feasible vector with minimum zero-norm. This nonsmooth combinatorial optimization problem is NP-Hard and arises in various fields such as machine learning, pattern recognition, signal processing. We propose two … Read more

    On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

  • Marco D'Apuzzo
  • Valentina De Simone
  • Daniela di Serafino
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. … Read more

    Disjunctive Cuts for Non-Convex Mixed Integer Quadratically Constrained Programs

  • Pierre Bonami
  • Jon Lee
  • Anureet Saxena
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Global Optimization Tags , , , ,

    This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non-convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, … Read more

    A scaling algorithm for polynomial constraint satisfaction problems

  • Ferenc Domes
  • Arnold Neumaier
    • Categories Constrained Nonlinear Optimization Tags , ,

    Good scaling is an essential requirement for the good behavior of many numerical algorithms. In particular, for problems involving multivariate polynomials, a change of scale in one or more variable may have drastic effects on the robustness of subsequent calculations. This paper surveys scaling algorithms for systems of polynomials from the literature, and discusses some … Read more

    Rigorous enclosures of ellipsoids and directed Cholesky factorizations

  • Ferenc Domes
  • Arnold Neumaier
    • Categories Constrained Nonlinear Optimization Tags , , , , , , , , ,

    This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems. A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. The Cholesky factorization can be used to transform a strictly convex quadratic constraint into a norm inequality, for … Read more

    Formulation of Oligopolistic Competition in AC Power Networks: An NLP Approach

  • Miguel F. Anjos
  • Guillermo Bautista
  • Anthony Vannelli
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Game Theory Tags , , , , ,

    In this paper, oligopolistic competition in a centralized power market is characterized by a multi-leader single-follower game, and formulated as a nonlinear programming (NLP) problem. An AC network is used to represent the transmission system and is modeled using rectangular coordinates. The follower is composed of a set of competitive suppliers, demands, and the system … Read more

    Numerical Study of Affine Supply Function Equilibrium in AC Network-Constrained Markets

  • Miguel F. Anjos
  • Guillermo Bautista
  • Anthony Vannelli
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Game Theory Tags , , , , , , ,

    An affine supply function equilibrium (SFE) approach is used to discuss voltage constraints and reactive power issues in the modeling of strategic behavior. Generation companies (GenCos) can choose their bid parameters with no restrictions for both energy and spinning reserves. The strategic behavior of generators is formulated as a multi-leader single-follower game. Each GenCo is … Read more

    LANCELOt_simple, a simple interface to LANCELOT B

  • Nicholas I. M. Gould
  • Dominique Orban
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Optimization Software and Modeling Systems Tags , ,

    We describe LANCELOT_simple, an interface to the LANCELOT B nonlinear optimization package within the GALAHAD} library (Gould, Orban, Toint, 2003) which ignores problem structure. The result is an easy-to-use Fortran 90 subroutine, with a small number of intuitively interpretable arguments. However, since structure is ignored, the means of presenting problems to the solver limited and … Read more