Nonsmooth Optimization

Implementing Algorithms for Signal and Image Reconstruction on Graphical Processing Units

  • Sangkyun Lee
  • Stephen Wright
    • Categories Nonsmooth Optimization, Parallel Algorithms Tags , , ,

    Several highly effective algorithms that have been proposed recently for compressed sensing and image processing applications can be implemented efficiently on commodity graphical processing units (GPUs). The properties of algorithms and application that make for efficient GPU implementation are discussed, and computational results for several algorithms are presented that show large speedups over CPU implementations. … Read more

    Lipschitz behavior of the robust regularization

  • Adrian Lewis
  • C.H. Jeffrey Pang
    • Categories Control Applications, Nonsmooth Optimization, Robust Optimization Tags , , , , , , , ,

    To minimize or upper-bound the value of a function “robustly”, we might instead minimize or upper-bound the “epsilon-robust regularization”, defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius … Read more

    Impulsive Optimal Control of Hybrid Finite-Dimensional Lagrangian Systems

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Control Applications, Nonsmooth Optimization Tags , ,

    The scope of this dissertation addresses numerical and theoretical issues in the impulsive control of hybrid finite-dimensional Lagrangian systems. In order to treat these aspects, a modeling framework is presented based on the measure-differential inclusion representation of the Lagrangian dynamics. The main advantage of this representation is that it enables the incorporation of set-valued force … Read more

    Necessary Conditions for the Impulsive Optimal Control of Multibody Mechanical Systems

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , , ,

    In this work, necessary conditions for the impulsive optimal control of multibody mechanical systems are stated. The conditions are obtained by the application subdifferential calculus techniques to extended-valued lower semi-continuous generalized Bolza functional that is evaluated on multiple intervals. Contrary to the approach in literature so far, the instant of possibly impulsive transition is considered … Read more

    Dynamic Subgradient Methods

  • Grégory Emiel
  • Claudia Sagastizábal
    • Categories Cutting Plane Approaches, Nonsmooth Optimization

    Lagrangian relaxation is commonly used to generate bounds for mixed-integer linear programming problems. However, when the number of dualized constraints is very large (exponential in the dimension of the primal problem), explicit dualization is no longer possible. In order to reduce the dual dimension, different heuristics were proposed. They involve a separation procedure to dynamically … Read more

    An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise

  • Junfeng Yang
  • Wotao Yin
  • Yin Zhang
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , ,

    We extend a recently proposed alternating minimization algorithm to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1-norm. We derive the algorithm by … Read more

    On the String Averaging Method for Sparse Common Fixed Points Problems

  • Yair Censor
  • Alexander Segal
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , ,

    We study the common fixed points problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We propose a definition of sparseness of a family of operators and investigate a string-averaging algorithmic scheme that favorably handles the common fixed points problem when … Read more

    Parallel Space Decomposition of the Mesh Adaptive Direct Search algorithm

  • Charles Audet
  • John E. Dennis, Jr.
  • Sébastien Le Digabel
    • Categories Applications - OR and Management Sciences, Nonsmooth Optimization, Parallel Algorithms Tags , , , ,

    This paper describes a parallel space decomposition PSD technique for the mesh adaptive direct search MADS algorithm. MADS extends a generalized pattern search for constrained nonsmooth optimization problems. The objective of the present work is to obtain good solutions to larger problems than the ones typically solved by MADS. The new method PSD-MADS is an … Read more

    OrthoMADS: A deterministic MADS instance with orthogonal directions

  • Mark A. Abramson
  • Charles Audet
  • John E. Dennis, Jr.
  • Sébastien Le Digabel
    • Categories Applications - OR and Management Sciences, Nonsmooth Optimization Tags , , , ,

    he purpose of this paper is to introduce a new way of choosing directions for the mesh adaptive direct search (Mads) class of algorithms. The advantages of this new OrthoMads instantiation of Mads are that the polling directions are chosen deterministically, ensuring that the results of a given run are repeatable, and that they are … Read more

    A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

  • Defeng Sun
  • Kim-Chuan Toh
  • Xinyuan Zhao
    • Categories Nonsmooth Optimization, Semi-definite Programming Tags , , ,

    We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive … Read more