Convex and Nonsmooth Optimization

Consistency Bounds and Support Recovery of D-stationary Solutions of Sparse Sample Average Approximations

  • Miju Ahn
    • Categories Convex and Nonsmooth Optimization Tags , , ,

    This paper studies properties of the d(irectional)-stationary solutions of sparse sample average approximation (SAA) problems involving difference-of-convex (dc) sparsity functions under a deterministic setting. Such properties are investigated with respect to a vector which satisfies a verifiable assumption to relate the empirical SAA problem to the expectation minimization problem defined by an underlying data distribution. … Read more

    A unified framework for Bregman proximal methods: subgradient, gradient, and accelerated gradient schemes

  • David H. Gutman
  • Javier F. Peña
    • Categories Convex Optimization Tags , , , ,

    We provide a unified framework for analyzing the convergence of Bregman proximal first-order algorithms for convex minimization. Our framework hinges on properties of the convex conjugate and gives novel proofs of the convergence rates of the Bregman proximal subgradient, Bregman proximal gradient, and a new accelerated Bregman proximal gradient algorithm under fairly general and mild … Read more

    First-Order Algorithms Converge Faster than (1/k)$ on Convex Problems

  • Stephen Wright
  • Ching-pei Lee
    • Categories Convex and Nonsmooth Optimization

    It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable, unconstrained convex function. In this work, we improve this rate to $o(1/k)$. We extend the result to proximal gradient and proximal coordinate … Read more

    Generating irreducible copositive matrices using the stable set problem

  • Reinier de Zeeuw
  • Peter J.C. Dickinson
    • Categories Convex Optimization, Graphs and Matroids, Linear, Cone and Semidefinite Programming Tags , , , ,

    In this paper it is considered how graphs can be used to generate copositive matrices, and necessary and sufficient conditions are given for these generated matrices to then be irreducible with respect to the set of positive semidefinite plus nonnegative matrices. This is done through combining the well known copositive formulation of the stable set … Read more

    An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems

  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
  • Weiwei Kong
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP … Read more

    A generalization of linearized alternating direction method of multipliers for solving two-block separable convex programming

  • Song Dunjiang
  • Zhao Pengjun
  • Chang Xiaokai
  • Liu Sanyang
    • Categories Convex and Nonsmooth Optimization

    The linearized alternating direction method of multipliers (ADMM), with indefinite proximal regularization, has been proved to be efficient for solving separable convex optimization subject to linear constraints. In this paper, we present a generalization of linearized ADMM (G-LADMM) to solve two-block separable convex minimization model, which linearizes all the subproblems by choosing a proper positive-definite … Read more

    Deep Unfolding of a Proximal Interior Point Method for Image Restoration

  • Carla Bertocchi
  • Émilie Chouzenoux
  • Marie-Caroline Corbineau
  • Jean-Christophe Pesquet
  • Marco Prato
    • Categories Convex and Nonsmooth Optimization, Network Optimization, Robust Optimization Tags , , , , ,

    Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and time-consuming methods. In contrast, deep learning offers very generic and efficient … Read more

    On inexact relative-error hybrid proximal extragradient, forward-backward and Tseng’s modified forward-backward methods with inertial effects

  • M. Marques Alves
  • Raul T. Marcavillaca
    • Categories Convex and Nonsmooth Optimization Tags , , , , , , , ,

    In this paper, we propose and study the asymptotic convergence and nonasymptotic global convergence rates (iteration-complexity) of an inertial under-relaxed version of the relative-error hybrid proximal extragradient (HPE) method for solving monotone inclusion problems. We analyze the proposed method under more flexible assumptions than existing ones on the extrapolation and relative-error parameters. As applications, we … Read more

    The Sard theorem for essentially smooth locally Lipschitz maps and applications in optimization

  • Xuan Duc Ha Truong
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization Tags , , , ,

    The classical Sard theorem states that the set of critical values of a $C^{k}$-map from an open set of $\R^n$ to $\R^p$ ($n\geq p$) has Lebesgue measure zero provided $k\geq n-p+1$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called “preparatory Sard theorem” for a compact countable set $I$ of $C^k$ … Read more

    A dual spectral projected gradient method for log-determinant semidefinite problems

  • Mituhiro Fukuda
  • Sunyoung Kim
  • Takashi Nakagaki
  • Makoto Yamashita
    • Categories Convex Optimization, Semi-definite Programming Tags , , , ,

    We extend the result on the spectral projected gradient method by Birgin et al in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the … Read more