Nonsmooth Optimization

Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

  • Zhaosong Lu
  • Renato D.C. Monteiro
  • Ming Yuan
    • Categories Convex Optimization, Nonsmooth Optimization, Statistics Tags , , , , , ,

    In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [17], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and … Read more

    LASSO-Patternsearch Algorithm with Application to Ophthalmology and Genomic Data

  • Barbara Klein
  • Stephen Wright
  • Ronald Klein
  • Kristine Lee
  • Weiliang Shi
  • Grace Wahba
    • Categories Nonsmooth Optimization, Statistics Tags , ,

    The LASSO-Patternsearch algorithm is proposed as a two-step method to identify clusters or patterns of multiple risk factors for outcomes of interest in demographic and genomic studies. The predictor variables are dichotomous or can be coded as dichotomous. Many diseases are suspected of having multiple interacting risk factors acting in concert, and it is of … Read more

    Generating set search methods for piecewise smooth problems

  • Claudio Bogani
  • Maria Grazia Gasparo
  • Alessandra Papini
    • Categories Nonsmooth Optimization Tags , , ,

    We consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally Lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra. A generating set search method is proposed, which is named “structured” because the structure of the set of nondifferentiability near the current iterate is exploited to … Read more

    Properties of a Cutting Plane Method for Semidefinite Programming

  • John E. Mitchell
  • Kartik Krishnan Sivaramakrishnan
    • Categories Linear, Cone and Semidefinite Programming, Nonsmooth Optimization Tags , , , ,

    We analyze the properties of an interior point cutting plane algorithm that is based on a semi-infinite linear formulation of the dual semidefinite program. The cutting plane algorithm approximately solves a linear relaxation of the dual semidefinite program in every iteration and relies on a separation oracle that returns linear cutting planes. We show that … Read more

    Graph Implementations for Nonsmooth Convex Programs

  • Stephen Boyd
  • Michael Grant
    • Categories Convex Optimization, Modeling Languages and Systems, Nonsmooth Optimization Tags , ,

    We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interior-point methods for smooth or cone convex programs. Citation … Read more

    Local convergence for alternating and averaged nonconvex projections

  • Adrian Lewis
  • Jérôme Malick
  • Russell Luke
    • Categories Constrained Nonlinear Optimization, Control Applications, Nonsmooth Optimization Tags , , , , , , , ,

    The idea of a finite collection of closed sets having “strongly regular intersection” at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably “regular” (special cases being convex … Read more

    A secant method for nonsmooth optimization

  • Adil Bagirov
  • Asef Nazari
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , ,

    The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonestrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a … Read more

    Another Face of DIRECT

  • Lakhdar Chiter
    • Categories Global Optimization, Global Optimization Theory, Nonsmooth Optimization Tags , ,

    It is shown that, contrary to a claim of [D. E. Finkel, and C. T. Kelley, Additive scaling and the DIRECT algorithm, J. Glob. Optim. 36 (2006) 597-608], it is possible to divide the smallest hypercube which contains the low function value by considering hyperrectangles whose points are located on the diagonal of the center … Read more

    A Fixed-Point Continuation Method for l_1-Regularized Minimization with Applications to Compressed Sensing

  • Elaine Hale
  • Wotao Yin
  • Yin Zhang
    • Categories Applications - Science and Engineering, Convex Optimization, Nonsmooth Optimization Tags , , , ,

    We consider solving minimization problems with $\ell_1$-regularization: $$\min \|x\|_1 + \mu f(x),$$ particularly for $f(x) = \frac{1}{2}\|Ax-b\|_M^2$ where $A \in \R^{m \times n}$ with $m < n$. Our goal is to construct efficient and robust algorithms for solving large-scale problems with dense data, and our approach is based on two powerful algorithmic ideas, operator-splitting and ... Read more

    Max-min separability: incremental approach and application to supervised data classification

  • Adil Bagirov
  • Bulent Karasozen
  • Dean Webb
    • Categories Global Optimization Applications, Nonsmooth Optimization Tags , , ,

    A new algorithm for the computation of a piecewise linear function separating two finite point sets in $n$-dimensional space is developed and the algorithm is applied to solve supervised data classification problems. The algorithm computes hyperplanes incrementally and it finds as many hyperplanes as necessary to separate two sets with respect to some tolerance. An … Read more