Convex and Nonsmooth Optimization

Characterizations of explicitly quasiconvex vector functions w.r.t. polyhedral cones

  • Christian Günther
  • Nicolae Popovici
    • Categories Generalized Convexity/Monoticity

    The aim of this paper is to present new characterizations of explicitly cone-quasiconvex vector functions with respect to a polyhedral cone of a finite-dimensional Euclidean space. These characterizations are given in terms of classical explicit quasiconvexity of certain real-valued functions, defined by composing the vector-valued function with appropriate scalarization functions, namely the extreme directions of … Read more

    Tensor Methods for Finding Approximate Stationary Points of Convex Functions

  • Geovani Nunes Grapiglia
  • Yurii Nesterov
    • Categories Convex Optimization Tags , , , ,

    In this paper we consider the problem of finding \epsilon-approximate stationary points of convex functions that are p-times differentiable with \nu-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O(\epsilon^{-1/(p+\nu-1)}) iterations to reduce the norm of the gradient of the objective below … Read more

    A family of multi-parameterized proximal point algorithms

  • Jianchao Bai
  • Ke Guo
  • Xiaokai Chang
    • Categories Convex Optimization Tags , , ,

    In this paper, a multi-parameterized proximal point algorithm combining with a relaxation step is developed for solving convex minimization problem subject to linear constraints. We show its global convergence and sublinear convergence rate from the prospective of variational inequality. Preliminary numerical experiments on testing a sparse minimization problem from signal processing indicate that the proposed … Read more

    The stochastic multi-gradient algorithm for multi-objective optimization and its application to supervised machine learning

  • Suyun Liu
  • Luis Nunes Vicente
    • Categories Convex Optimization, Multi-Criteria Optimization, Stochastic Programming Tags , , ,

    Optimization of conflicting functions is of paramount importance in decision making, and real world applications frequently involve data that is uncertain or unknown, resulting in multi-objective optimization (MOO) problems of stochastic type. We study the stochastic multi-gradient (SMG) method, seen as an extension of the classical stochastic gradient method for single-objective optimization. At each iteration … Read more

    Geometric and Metric Characterizations of Transversality Properties

  • Hoa T. Bui
  • Nguyen Duy Cuong
  • Alexander Y. Kruger
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to clarify the relations between various quantitative geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in … Read more

    Nonlinear Transversality Properties of Collections of Sets: Dual Space Necessary Characterizations

  • Nguyen Duy Cuong
  • Alexander Y. Kruger
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    This paper continues the study of ‘good arrangements’ of collections of sets in normed vector spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary and sufficient conditions … Read more

    Optimal K-Thresholding Algorithms for Sparse Optimization Problems

  • Yun-Bin Zhao
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Data-Mining Tags , , , , ,

    The simulations indicate that the existing hard thresholding technique independent of the residual function may cause a dramatic increase or numerical oscillation of the residual. This inherit drawback of the hard thresholding renders the traditional thresholding algorithms unstable and thus generally inefficient for solving practical sparse optimization problems. How to overcome this weakness and develop … Read more

    Single-Forward-Step Projective Splitting: Exploiting Cocoercivity

  • Jonathan Eckstein
  • Patrick R. Johnstone
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , ,

    This work describes a new variant of projective splitting for monotone inclusions, in which cocoercive operators can be processed with a single forward step per iteration. This result establishes a symmetry between projective splitting algorithms, the classical forward backward splitting method (FB), and Tseng’s forward-backward-forward method (FBF). Another symmetry is that the new procedure allows … Read more

    Asynchronous Stochastic Subgradient Methods for General Nonsmooth Nonconvex Optimization

  • Dan Alistarh
  • Bapi Chatterjee
  • Vyacheslav Kungurtsev
  • Malcolm Egan
    • Categories Nonsmooth Optimization

    Asynchronous distributed methods are a popular way to reduce the communication and synchronization costs of large-scale optimization. Yet, for all their success, little is known about their convergence guarantees in the challenging case of general non-smooth, non-convex objectives, beyond cases where closed-form proximal operator solutions are available. This is all the more surprising since these … Read more

    Beyond Alternating Updates for Matrix Factorization with Inertial Bregman Proximal Gradient Algorithms

  • Mahesh Chandra Mukkamala
  • Peter Ochs
    • Categories Convex and Nonsmooth Optimization Tags , , , , , , , ,

    Matrix Factorization is a popular non-convex objective, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization variables. A remedy is non-alternating schemes. However, due to a lack of Lipschitz continuity of the gradient in matrix factorization problems, convergence cannot … Read more