first-order method

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

    First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points

  • Zhichao Jia
  • Benjamin Grimmer
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show a simple first-order method finds a feasible, ϵ-stationary point at a convergence rate of O(ϵ−4) without relying on compactness or Constraint Qualification (CQ). When CQ holds, this convergence is measured by … Read more

    A first-order augmented Lagrangian method for constrained minimax optimization

  • Zhaosong Lu
  • Sanyou Mei
    • Categories Constrained Nonlinear Optimization, Robust Optimization Tags , , ,

    \(\) In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method recently developed in [26] by the authors. Under some suitable assumptions, … Read more

    Accelerated first-order methods for a class of semidefinite programs

  • Alex L. Wang
  • Fatma Kılınç-Karzan
    • Categories Convex and Nonsmooth Optimization, Quadratic Programming, Semi-definite Programming Tags , , , ,

    This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs , is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard … Read more

    Limited-memory Common-directions Method for Large-scale Optimization: Convergence, Parallelization, and Distributed Optimization

  • Ching-pei Lee
  • Po-Wei Wang
  • Chih-Jen Lin
    • Categories Parallel Algorithms, Unconstrained Optimization Tags , , ,

    In this paper, we present a limited-memory common-directions method for smooth optimization that interpolates between first- and second- order methods. At each iteration, a subspace of a limited dimension size is constructed using first-order information from previous iterations, and an ef- ficient Newton method is deployed to find an approximate minimizer within this subspace. With … Read more