composite optimization

Preconditioned subgradient method for composite optimization: overparameterization and fast convergence

  • Liwei Jiang
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Optimization in Data Science Tags , , , ,

    Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution, and collaborative filtering. The subgradient method achieves local linear convergence when the composite loss is well-conditioned. However, if the smooth map is, in a … Read more

    A Simple Adaptive Proximal Gradient Method for Nonconvex Optimization

  • Ye Zilong
  • Shiqian Ma
  • Junfeng Yang
  • Danqing Zhou
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Consider composite nonconvex optimization problems where the objective function consists of a smooth nonconvex term (with Lipschitz-continuous gradient) and a convex (possibly nonsmooth) term. Existing parameter-free methods for such problems often rely on complex multi-loop structures, require line searches, or depend on restrictive assumptions (e.g., bounded iterates). To address these limitations, we introduce a novel … Read more

    A Symmetric Primal-Dual method with two extrapolation steps for Composite Convex Optimization

  • Feng Ma
    • Categories Convex Optimization Tags , , ,

    Symmetry is a recurring feature in algorithms for monotone operator theory and convex optimization, particularly in problems involving the sum of two operators, as exemplified by the Peaceman–Rachford splitting scheme. However, in more general settings—such as composite optimization problems with three convex functions or structured convex-concave saddle-point formulations—existing algorithms often exhibit inherent asymmetry. In particular, … Read more

    FISTA and Extensions – Review and New Insights

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

    The purpose of this technical report is to review the main properties of an accelerated composite gradient (ACG) method commonly referred to as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In addition, we state a version of FISTA for solving both convex and strongly convex composite minimization problems and derive its iteration complexities to generate iterates … Read more

    Analysis of the Frank-Wolfe Method for Convex Composite Optimization involving a Logarithmically-Homogeneous Barrier

  • Robert M. Freund
  • Renbo Zhao
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , , , , ,

    We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem (P): F*:= min_x f(Ax) + h(x), where f is a \theta-logarithmically-homogeneous self-concordant barrier and the function h has bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires O((Gap_0 + \theta + Var_h)\ln(\delta_0) + (\theta + Var_h)^2/\epsilon) … Read more

    A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

  • Aleksandr Y. Aravkin
  • Robert Baraldi
  • Dominique Orban
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , , ,

    We develop a trust-region method for minimizing the sum of a smooth term f and a nonsmooth term h, both of which can be nonconvex. Each iteration of our method minimizes apossibly nonconvex model of f+h in a trust region. The model coincides with f+h in value and subdifferential at the center. We establish global … 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

    Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

  • Vasileios Charisopoulos
  • Damek Davis
  • Mateo Díaz
  • Dmitriy Drusvyatskiy
  • Yudong Chen
  • Lijun Ding
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, … Read more

    Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization

  • Mahesh Chandra Mukkamala
  • Peter Ochs
  • Thomas Pock
  • Shoham Sabach
    • Categories Convex and Nonsmooth Optimization Tags , , , , , , ,

    Backtracking line-search is an old yet powerful strategy for finding better step size to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation … Read more

    Composite optimization for robust blind deconvolution

  • Vasileios Charisopoulos
  • Damek Davis
  • Mateo Díaz
  • Dmitriy Drusvyatskiy
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, sharpness, and Lipschitz continuity are all dimension independent. This phenomenon persists even when up to … Read more