Convex and Nonsmooth Optimization

MGProx: A nonsmooth multigrid proximal gradient method with adaptive restriction for strongly convex optimization

  • Andersen Ang
  • Stephen A. Vavasis
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization, Nonlinear Optimization

    We study the combination of proximal gradient descent with multigrid for solving a class of possibly nonsmooth strongly convex optimization problems. We propose a multigrid proximal gradient method called MG-Prox, which accelerates the proximal gradient method by multigrid, based on using hierarchical information of the optimization problem. MGProx applies a newly introduced adaptive restriction operator … Read more

    Projection free methods on product domains

  • Immanuel M. Bomze
  • Francesco Rinaldi
  • Damiano Zeffiro
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    Projection-free block-coordinate methods avoid high computational cost per iteration and at the same time exploit the particular problem structure of product domains. Frank-Wolfe-like approaches rank among the most popular ones of this type. However, as observed in the literature, there was a gap between the classical Frank-Wolfe theory and the block-coordinate case. Moreover, most of … Read more

    Cutting plane reusing methods for multiple dual optimizations

  • Kouhei Harada
    • Categories Convex and Nonsmooth Optimization Tags ,

    We consider solving a group of dual optimization problems that share a core structure: Every primal problem of the group is obtained by the right-hand side variation of constraints in the original primal problem, while the other core part of the original primal problem, such as the objective and the left-hand side of the constraints, … Read more

    Semi-Infinite Generalized Disjunctive and Mixed Integer Convex Programs with(out) Uncertainty

  • Manish Bansal
    • Categories 0-1 Programming, Convex Optimization, Semi-infinite Programming

    In this paper, we introduce semi-infinite generalized disjunctive programs that are defined by logical propositions along with disjunctions of sets of logical equations and infinite number of algebraic inequalities. We denote these programs by SIGDPs. For SIGDPs with linear and convex inequalities, we present new reformulations: semi-infinite mixed-binary/disjunctive linear programs and semi-infinite mixed-binary/disjunctive convex programs, … Read more

    On an iteratively reweighted linesearch based algorithm for nonconvex composite optimization

  • Silvia Bonettini
  • Marco Prato
  • Simone Rebegoldi
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    In this paper we propose a new algorithm for solving a class of nonsmooth nonconvex problems, which is obtained by combining the iteratively reweighted scheme with a finite number of forward–backward iterations based on a linesearch procedure. The new method overcomes some limitations of linesearch forward–backward methods, since it can be applied also to minimize … Read more

    A Novel Stepsize for Gradient Descent Method

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

    In this paper, we propose a novel stepsize for the classical gradient descent scheme to solve unconstrained nonlinear optimization problems. We are concerned with the convex and smooth objective without the globally Lipschitz gradient condition. Our new method just needs the locally Lipschitz gradient but still gets the rate $O(\frac{1}{k})$ of $f(x^k)-f_*$ at most. By … Read more

    Barzilai-Borwein-like rules in proximal gradient schemes for ℓ1−regularized problems

  • Serena Crisci
  • Simone Rebegoldi
  • Gerardo Toraldo
  • Marco Viola
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , ,

    We propose a novel steplength selection rule in proximal gradient methods for minimizing the sum of a differentiable function plus an ℓ1-norm penalty term. The proposed rule modifies one of the classical Barzilai-Borwein steplength, extending analogous results obtained in the context of gradient projection methods for constrained optimization. We analyze the spectral properties of the … Read more

    A Semismooth Conjugate Gradients Method — Theoretical Analysis

  • Franz Bethke
  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , ,

    In large scale applications, deterministic and stochastic variants of Cauchy’s steepest descent method are widely used for the minimization of objectives that are only piecewise smooth. In this paper we analyse a  deterministic descent method based on the generalization of rescaled conjugate gradients proposed by Philip Wolfe in 1975 for objectives that are convex. Without … Read more

    The Jordan algebraic structure of the rotated quadratic cone

  • Baha Alzalg
  • Karima Tamsaouete
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Quadratic Programming

    In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is … Read more

    A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares

  • Dominique Orban
  • Aleksandr Y. Aravkin
  • Robert Baraldi
    • Categories Data Science Algorithms, Nonlinear Systems and Least-Squares, Nonsmooth Optimization Tags , , , , ,

    We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term \(f(x) = \frac{1}{2} \|F(x)\|_2^2\) and a nonsmooth term \(h\). Both \(f\) and \(h\) may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of \(h\) using a first-order method such as … Read more