Convex and Nonsmooth Optimization

A Subgradient Projection Method with Outer Approximation for Solving Semidefinite Programming Problems

  • Nagisa Sugishita
  • Miguel F. Anjos
    • Categories Convex Optimization, Semi-definite Programming Tags , ,

    We explore the combination of subgradient projection with outer approximation to solve semidefinite programming problems. We compare several ways to construct outer approximations using the problem structure. The resulting approach enjoys the strengths of both subgradient projection and outer approximation methods. Preliminary computational results on the semidefinite programming relaxations of graph partitioning and max-cut show … Read more

    Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

  • Takayuki Nagano
  • Bruno F. Lourenço
  • Akiko Takeda
    • Categories Cone Programming, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we … Read more

    Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization

  • Xiangyu Yang
  • Hao Wang
    • Categories Nonsmooth Optimization, Optimization in Data Science Tags , , , ,

    This paper considers the problem of minimizing the sum of a smooth function and the Schatten-\(p\) norm of the matrix. Our contribution involves proposing accelerated iteratively reweighted nuclear norm methods designed for solving the nonconvex low-rank minimization problem. Two major novelties characterize our approach. Firstly, the proposed method possesses a rank identification property, enabling the … Read more

    Composite optimization models via proximal gradient method with a novel enhanced adaptive stepsize

  • Pham Thi Hoai
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    We first consider the convex composite optimization models with the local Lipschitzness condition imposed on the gradient of the differentiable term. The classical proximal gradient method will be studied with our novel enhanced adaptive stepsize selection. To obtain the convergence of the proposed algorithm, we establish a sufficient decrease type inequality associated with our new … Read more

    Recognizing weighted means in geodesic spaces

  • Adrian Lewis
    • Categories Convex Optimization Tags , , , , ,

    Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally whether or not a given point is such an average, exactly or approximately. In nonpositively curved spaces, several averaging notions, including the usual … Read more

    The Role of Level-Set Geometry on the Performance of PDHG for Conic Linear Optimization

  • Zikai Xiong
  • Robert M. Freund
    • Categories Convex Optimization, Linear Programming, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We consider solving huge-scale instances of (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) — with heuristic enhancements and GPU implementation — has been very successful in solving huge-scale linear programming (LP) problems; however its application to more general conic convex … Read more

    Lipschitz minimization and the Goldstein modulus

  • Adrian Lewis
    • Categories Nonsmooth Optimization Tags , ,

    Goldstein’s 1977 idealized iteration for minimizing a Lipschitz objective fixes a distance – the step size – and relies on a certain approximate subgradient. That “Goldstein subgradient” is the shortest convex combination of objective gradients at points within that distance of the current iterate. A recent implementable Goldstein-style algorithm allows a remarkable complexity analysis (Zhang … Read more

    Convex optimization on CAT(0) cubical complexes

  • Adrian Lewis
    • Categories Convex Optimization Tags , , , ,

    We consider geodesically convex optimization problems involving distances to a finite set of points A in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the feasibility and projection problems for intersecting balls with centers in A. We propose a decomposition approach relying on standard Euclidean … Read more

    Subgradient Convergence Implies Subdifferential Convergence on Weakly Convex Functions: With Uniform Rates Guarantees

  • Feng Ruan
    • Categories Nonsmooth Optimization, Stochastic Programming

    In nonsmooth, nonconvex stochastic optimization, understanding the uniform convergence of subdifferential mappings is crucial for analyzing stationary points of sample average approximations of risk as they approach the population risk. Yet, characterizing this convergence remains a fundamental challenge. This work introduces a novel perspective by connecting the uniform convergence of subdifferential mappings to that of subgradient … Read more

    Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy

  • Francisco J. Aragón Artacho
  • Weiwei Cai
  • Yair Censor
  • Aviv Gibali
  • Chongyuan Shui
  • David Torregrosa-Belén
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the … Read more