Constrained Nonlinear Optimization

Spectral properties of Barzilai-Borwein rules in solving singly linearly constrained optimization problems subject to lower and upper bounds

  • Serena Crisci
  • Federica Porta
  • Luca Zanni
  • Valeria Ruggiero
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In 1988, Barzilai and Borwein published a pioneering paper which opened the way to inexpensively accelerate first-order methods. More in detail, in the framework of unconstrained optimization, Barzilai and Borwein developed two strategies to select the steplength in gradient descent methods with the aim of encoding some second-order information of the problem without computing and/or … Read more

    An accelerated inexact proximal point method for solving nonconvex-concave min-max problems

  • Renato D.C. Monteiro
  • Weiwei Kong
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    Abstract This paper presents a quadratic-penalty type method for solving linearly-constrained composite nonconvex-concave min-max problems. The method consists of solving a sequence of penalty subproblems which, due to the min-max structure of the problem, are potentially nonsmooth but can be approximated by smooth composite nonconvex minimization problems. Each of these penalty subproblems is then solved … Read more

    Using interior point solvers for optimizing progressive lens models with spherical coordinates

  • Glòria Casanellas
  • Jordi Castro
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

    Designing progressive lenses is a complex problem that has been previously solved by formulating an optimization model based on Cartesian coordinates. In this work a new progressive lens model using spherical coordinates is presented, and interior point solvers are used to solve this new optimization model. Although this results in a highly nonlinear, nonconvex, continuous … Read more

    Solving Chance-Constrained Problems via a Smooth Sample-Based Nonlinear Approximation

  • James R. Luedtke
  • Alejandra Pena-Ordieres
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization, Stochastic Programming Tags , , , , , ,

    We introduce a new method for solving nonlinear continuous optimization problems with chance constraints. Our method is based on a reformulation of the probabilistic constraint as a quantile function. The quantile function is approximated via a differentiable sample average approximation. We provide theoretical statistical guarantees of the approximation, and illustrate empirically that the reformulation can … Read more

    A Proximal Interior Point Algorithm with Applications to Image Processing

  • Émilie Chouzenoux
  • Marie-Caroline Corbineau
  • Jean-Christophe Pesquet
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , , , ,

    In this article, we introduce a new proximal interior point algorithm (PIPA). This algorithm is able to handle convex optimization problems involving various constraints where the objective function is the sum of a Lipschitz differentiable term and a possibly nonsmooth one. Each iteration of PIPA involves the minimization of a merit function evaluated for decaying … Read more

    Stability Analysis for a Class of Sparse Optimization Problems

  • J.L. Xu
  • Yun-Bin Zhao
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Linear Programming

    The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The $\ell_{0}$-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The $\ell_{1}$-minimization method is one of the plausible approaches for solving the $\ell_{0}$-minimization problems, and … Read more

    Self-Concordance and Matrix Monotonicity with Applications to Quantum Entanglement Problems

  • Leonid Faybusovich
  • Cunlu Zhou
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , ,

    Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant … Read more

    An Alternating Manifold Proximal Gradient Method for Sparse PCA and Sparse CCA

  • Shixiang Chen
  • Shiqian Ma
  • Lingzhou Xue
  • Hui Zou
    • Categories Constrained Nonlinear Optimization, Statistics

    Sparse principal component analysis (PCA) and sparse canonical correlation analysis (CCA) are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as an optimization problem with nonsmooth objective and nonconvex constraints. Since non-smoothness and nonconvexity bring numerical difficulties, most algorithms suggested in the literature either solve … Read more

    An Augmented Lagrangian algorithm for nonlinear semidefinite programming applied to the covering problem

  • Ernesto G. Birgin
  • Walter Gómez
  • Gabriel Haeser
  • Leonardo M. Mito
  • Daiana O. Santos
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Semi-definite Programming Tags , , ,

    In this work we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done in order to allow exploiting the … Read more

    Inertial Block Mirror Descent Method for Non-Convex Non-Smooth Optimization

  • Nicolas Gillis
  • Le Thi Khanh Hien
  • Panagiotis Patrinos
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    In this paper, we propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. We use the general framework of Bregman distance functions to compute the proximal maps. Our method not only allows using two different extrapolation points to evaluate gradients and adding the inertial force, but also takes advantage … Read more