Convex and Nonsmooth Optimization

Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

  • Daniel Kuhn
  • Peyman Mohajerin Esfahani
  • Viet Anh Nguyen
    • Categories Convex Optimization, Robust Optimization Tags , ,

    We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a p-dimensional Gaussian random vector from n independent samples. The proposed model minimizes the worst case (maximum) of Stein’s loss across all normal reference distributions within a prescribed Wasserstein distance from the normal distribution … Read more

    A family of spectral gradient methods for optimization

  • Yu-Hong Dai
  • Yakui Huang
  • Xin-Wei Liu
    • Categories Convex Optimization, Unconstrained Optimization

    We propose a family of spectral gradient methods, whose stepsize is determined by a convex combination of the short Barzilai-Borwein (BB) stepsize and the long BB stepsize. It is shown that each member of the family shares certain quasi-Newton property in the sense of least squares. The family also includes some other gradient methods as … Read more

    Parallel and Distributed Successive Convex Approximation Methods for Big-Data Optimization

  • Gesualdo Scutari
  • Ying Sun
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    Recent years have witnessed a surge of interest in parallel and distributed optimization methods for large-scale systems. In particular, nonconvex large-scale optimization problems have found a wide range of applications in several engineering fields. The design and the analysis of such complex, large-scale, systems pose several challenges and call for the development of new optimization … Read more

    Robust-to-Dynamics Optimization

  • Amir Ali Ahmadi
  • Oktay Gunluk
    • Categories Convex Optimization, Robust Optimization, Semi-infinite Programming Tags , , ,

    A robust-to-dynamics optimization (RDO) problem} is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a feasible set $\Omega\subseteq\mathbb{R}^n$), and (ii) a dynamical system (a map $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$). Its goal is to minimize $f$ over the set $\mathcal{S}\subseteq\Omega$ of initial conditions that forever remain in $\Omega$ under … Read more

    A second order dynamical approach with variable damping to nonconvex smooth minimization

  • Radu Ioan Bot
  • Ernö Robert Csetnek
  • Szilard Csaba Laszlo
    • Categories Nonsmooth Optimization Tags , , ,

    We investigate a second order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov’s accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the … Read more

    Variational Analysis and Optimization of Sweeping Processes with Controlled Moving Sets

  • Boris S. Mordukhovich
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    This paper briefly overviews some recent and very fresh results on a rather new class of dynamic optimization problems governed by the so-called sweeping (Moreau) processes with controlled moving sets. Uncontrolled sweeping processes have been known in dynamical systems and applications starting from 1970s while control problems for them have drawn attention of mathematicians, applied … Read more

    On proximal point-type algorithms for weakly convex functions and their connection to the backward Euler method

  • Tim Hoheisel
  • Maxime Laborde
  • Adam Oberman
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this article we study the connection between proximal point methods for nonconvex optimization and the backward Euler method from numerical Ordinary Differential Equations (ODEs). We establish conditions for which these methods are equivalent. In the case of weakly convex functions, for small enough parameters, the implicit steps can be solved using a strongly convex … Read more

    Gradient Sampling Methods for Nonsmooth Optimization

  • James V. Burke
  • Frank E. Curtis
  • Adrian Lewis
  • Michael L. Overton
  • Lucas E. A. Simões
    • Categories Nonsmooth Optimization

    This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various … Read more

    Golden Ratio Algorithms for Variational Inequalities

  • Yura Malitsky
    • Categories Complementarity and Variational Inequalities, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. … Read more

    An algorithm to compute the Hoffman constant of a system of linear constraints

  • Javier F. Peña
  • Juan C. Vera
  • Luis F. Zuluaga
    • Categories Convex Optimization, Polyhedra Tags , ,

    We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the … Read more