Nonsmooth Optimization

REDUCTION OF TWO-STAGE PROBABILISTIC OPTIMIZATION PROBLEMS WITH DISCRETE DISTRIBUTION OF RANDOM DATA TO MIXED INTEGER PROGRAMMING PROBLEMS

  • Andrey Kibzun
  • Vladimir I. Norkin
  • Andrey Naumov
    • Categories (Mixed) Integer Nonlinear Programming, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

    We consider models of two-stage stochastic programming with a quantile second stage criterion and optimization models with a chance constraint on the second stage objective function values. Such models allow to formalize requirements to reliability and safety of the system under consideration, and to optimize the system in extreme conditions. We suggest a method of … Read more

    A doubly stabilized bundle method for nonsmooth convex optimization

  • Welington de Oliveira
  • Mikhail V. Solodov
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , ,

    We propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations. Most bundle algorithms use a cutting-plane model of the objective function to formulate a subproblem whose solution gives the next iterate. Proximal bundle methods employ the model in the objective function of the subproblem, while level … Read more

    Orthogonal invariance and identifiability

  • Aris Daniilidis
  • Dmitriy Drusvyatskiy
  • Adrian Lewis
    • Categories Nonsmooth Optimization Tags , , , , ,

    Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann’s theorem on matrix norms is an early example. We discuss the example of “identifiability”, a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, “partial smoothness”) is the … Read more

    Abstract Newtonian Frameworks and Their Applications

  • Alexey F. Izmailov
  • A. S. Kurennoy
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , , ,

    We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving … Read more

    Variable Metric Forward-Backward algorithm for minimizing the sum of a differentiable function and a convex function

  • Émilie Chouzenoux
  • Jean-Christophe Pesquet
  • Audrey Repetti
    • Categories Nonsmooth Optimization

    We consider the minimization of a function $G$ defined on $R^N$, which is the sum of a (non necessarily convex) differentiable function and a (non necessarily differentiable) convex function. Moreover, we assume that $G$ satisfies the Kurdyka-Lojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may suffer from … Read more

    Well-posedness for Lexicographic Vector Equilibrium Problems

  • L. Q. Anh
  • Alexander Y. Kruger
  • T. Q. Duy
  • Nguyen H. Thao
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization Tags , ,

    We consider lexicographic vector equilibrium problems in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) well-posed at the reference point are established. As an application, we derive several results on well-posedness for a class of variational inequalities. CitationPublished in Constructive Nonsmooth Analysis and Related Topics (Vladimir Demyanov, Panos M. Pardalos, … Read more

    S_1/2 Regularization Methods and Fixed Point Algorithms for Affine Rank Minimization Problems

  • Peng Dingtao
  • Yu Jian
  • Naihua Xiu
    • Categories Linear, Cone and Semidefinite Programming, Nonsmooth Optimization

    The affine rank minimization problem is to minimize the rank of a matrix under linear constraints. It has many applications in various areas such as statistics, control, system identification and machine learning. Unlike the literatures which use the nuclear norm or the general Schatten $q~(0<q<1)$ quasi-norm to approximate the rank of a matrix, in this … Read more

    Partial Second-Order Subdifferentials in Variational Analysis and Optimization

  • Boris S. Mordukhovich
  • Nguyen Mau Nam
  • Nguyen Thi Yen Nhi
    • Categories Nonsmooth Optimization Tags , , , , , ,

    This paper presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial … Read more

    Clarke subgradients for directionally Lipschitzian stratifiable functions

  • Dmitriy Drusvyatskiy
  • Adrian Lewis
  • Alexander D. Ioffe
    • Categories Nonsmooth Optimization

    Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that … Read more

    Iterative Hard Thresholding Methods for $ Regularized Convex Cone Programming

  • Zhaosong Lu
    • Categories Combinatorial Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper we consider $l_0$ regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving $l_0$ regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the … Read more