Constrained Nonlinear Optimization

A Regularized SQP Method with Convergence to Second-Order Optimal Points

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

    Regularized and stabilized sequential quadratic programming methods are two classes of sequential quadratic programming (SQP) methods designed to resolve the numerical and theoretical difficulties associated with ill-posed or degenerate nonlinear optimization problems. Recently, a regularized SQP method has been proposed that provides a strong connection between augmented Lagrangian methods and stabilized SQP methods. The method … Read more

    Constant rank constraint qualifications: a geometric introduction

  • Roberto Andreani
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , ,

    Constraint qualifications (CQ) are assumptions on the algebraic description of the feasible set of an optimization problem that ensure that the KKT conditions hold at any local minimum. In this work we show that constraint qualifications based on the notion of constant rank can be understood as assumptions that ensure that the polar of the … Read more

    Complementarity Formulations of l0-norm Optimization Problems

  • Mingbin Feng
  • John E. Mitchell
  • Jong-Shi Pang
  • Xin Shen
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , ,

    In a number of application areas, it is desirable to obtain sparse solutions. Minimizing the number of nonzeroes of the solution (its l0-norm) is a difficult nonconvex optimization problem, and is often approximated by the convex problem of minimizing the l1-norm. In contrast, we consider exact formulations as mathematical programs with complementarity constraints and their … Read more

    Rounding on the standard simplex: regular grids for global optimization

  • Immanuel M. Bomze
  • E. Alper Yildirim
  • Stefan Gollowitzer
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory

    Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all $\ell^p$-norms for $p\ge 1$. We show that the minimal $\ell^p$-distance to the regular grid on the standard simplex can exceed one, even for very fine … Read more

    The Euclidean distance degree of an algebraic variety

  • Jan Draisma
  • Emil Horobet
  • Giorgio Ottaviani
  • Bernd Sturmfels
  • Rekha R. Thomas
    • Categories Constrained Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational … Read more

    A new formulation of protein evolutionary models that account for structural constraints

  • Andrew J. Bordner
  • Hans D Mittelmann
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags , , , ,

    Despite the importance of a thermodynamically stable structure with a conserved fold for protein function, almost all evolutionary models neglect site-site correlations that arise from physical interactions between neighboring amino acid sites. This is mainly due to the difficulty in formulating a computationally tractable model since rate matrices can no longer be used. Here we … Read more

    String-Averaging Projected Subgradient Methods for Constrained Minimization

  • Yair Censor
  • Alexander J. Zaslavski
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , , ,

    We consider constrained minimization problems and propose to replace the projection onto the entire feasible region, required in the Projected Subgradient Method (PSM), by projections onto the individual sets whose intersection forms the entire feasible region. Specifically, we propose to perform such projections onto the individual sets in an algorithmic regime of a feasibility-seeking iterative … Read more

    A Flexible Inexact Restoration Method and Application to Optimization with Multiobjective Constraints under Weighted-Sum Scalarization

  • Luís Felipe Bueno
  • Gabriel Haeser
  • José Mario Martínez
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    We introduce a new flexible Inexact-Restoration (IR) algorithm and an application to problems with multiobjective constraints (MOCP) under the weighted-sum scalarization approach. In IR methods each iteration has two phases. In the first phase one aims to improve the feasibility and, in the second phase, one minimizes a suitable objective function. This is done in … Read more

    Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization

  • Saeed Ghadimi
  • Guanghui Lan
  • Hongchao Zhang
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a certain non-differentiable (but convex) component. In order to solve these problems, we propose a randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are … Read more

    A Sequential Quadratic Optimization Algorithm with Rapid Infeasibility Detection

  • James V. Burke
  • Frank E. Curtis
  • Hao Wang
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    We present a sequential quadratic optimization (SQO) algorithm for nonlinear constrained optimization. The method attains all of the strong global and fast local convergence guarantees of classical SQO methods, but has the important additional feature that fast local convergence is guaranteed when the algorithm is employed to solve infeasible instances. A two-phase strategy, carefully constructed … Read more