Convex and Nonsmooth Optimization

Generalized Bundle Methods for Sum-Functions with Easy” Components: Applications to Multicommodity Network Design

  • Antonio Frangioni
  • Enrico Gorgone
    • Categories Convex and Nonsmooth Optimization, Linear Programming, Network Optimization Tags , , , ,

    We propose a modification to the (generalized) bundle scheme for minimization of a convex nondifferentiable sum-function in the case where some of the components are “easy”, that is, they are Lagrangian functions of explicitly known convex programs with “few” variables and constraints. This happens in many practical cases, particularly within applications to combinatorial optimization. In … Read more

    Integration formulas via the Legendre-Fenchel Subdifferential of nonconvex functions

  • Rafael Correa
  • Garcia Yboon
  • Abderrahim Hantoute
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the epsilon-subdifferential and the Legendre-Fenchel subdefferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction … Read more

    Positive polynomials on unbounded equality-constrained domains

  • Javier F. Peña
  • Juan C. Vera
  • Luis F. Zuluaga
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of … Read more

    Dependence of bilevel programming on irrelevant data

  • Stephan Dempe
  • Sebastian Lohse
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    In 1997, Macal and Hurter have found that adding a constraint to the lower level problem, which is not active at the computed global optimal solution, can destroy global optimality. In this paper this property is reconsidered and it is shown that this solution remains locally optimal under inner semicontinuity of the original solution set … Read more

    Implementation of a block-decomposition algorithm for solving large-scale conic semidefinite programming problems

  • Renato D.C. Monteiro
  • Camilo Ortiz
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , ,

    In this paper, we consider block-decomposition first-order methods for solving large-scale conic semidefinite programming problems. Several ingredients are introduced to speed-up the method in its pure form such as: an aggressive choice of stepsize for performing the extragradient step; use of scaled inner products in the primal and dual spaces; dynamic update of the scaled … Read more

    An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , , , ,

    This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) method. Iteration-complexity results are established for the A-HPE method, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE method are … Read more

    Structured Sparsity via Alternating Direction Methods

  • Donald Goldfarb
  • Zhiwei (Tony) Qin
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

    We consider a class of sparse learning problems in high dimensional feature space regularized by a structured sparsity-inducing norm which incorporates prior knowledge of the group structure of the features. Such problems often pose a considerable challenge to optimization algorithms due to the non-smoothness and non-separability of the regularization term. In this paper, we focus … Read more

    Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces

  • Peng Jian-Wen
  • Xinmin Yang
  • Zhou Zhi-Ang
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization Tags

    In this paper, firstly, a generalized subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established. Citation {\bf AMS 2010 Subject Classifications:} 90C26, 90C29, 90C30 Article Download View Optimality conditions … Read more

    Hierarchical Classification via Orthogonal Transfer

  • Mingrui Wu
  • Lin Xiao
  • Dengyong Zhou
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization Tags , , ,

    We consider multiclass classification problems where the set of labels are organized hierarchically as a category tree. We associate each node in the tree with a classifier and classify the examples recursively from the root to the leaves. We propose a hierarchical Support Vector Machine (SVM) that encourages the classifier at each node of the … Read more

    An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections

  • Dirk A. Lorenz
  • Marc E. Pfetsch
  • Andreas M. Tillmann
    • Categories Convex Optimization, Global Optimization Theory, Nonsmooth Optimization Tags , , , , ,

    We propose a new subgradient method for the minimization of convex functions over a convex set. Common subgradient algorithms require an exact projection onto the feasible region in every iteration, which can be efficient only for problems that admit a fast projection. In our method we use inexact adaptive projections requiring to move within a … Read more