Convex and Nonsmooth Optimization

A Parallel Bundle Framework for Asynchronous Subspace Optimisation of Nonsmooth Convex Functions

  • Frank Fischer
  • Christoph Helmberg
    • Categories Convex and Nonsmooth Optimization, Parallel Algorithms Tags , ,

    An algorithmic framework is presented for optimising general convex functions by non synchronised parallel processes. Each process greedily picks a suitable adaptive subset of coordinates and runs a bundle method on a corresponding restricted problem stopping whenever a descent step is encountered or predicted decrease is reduced sufficiently. No prior knowledge on the dependencies between … Read more

    New Analysis and Results for the Conditional Gradient Method

  • Robert M. Freund
  • Paul Grigas
    • Categories Convex Optimization Tags , , , ,

    We present new results for the conditional gradient method (also known as the Frank-Wolfe method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple averaging and constant step-sizes. We also develop step-size rules and computational guarantees that depend naturally on the warm-start quality of the initial … Read more

    Randomized Block Coordinate Non-Monotone Gradient Method for a Class of Nonlinear Programming

  • Zhaosong Lu
  • Lin Xiao
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , ,

    In this paper we propose a randomized block coordinate non-monotone gradient (RBCNMG) method for minimizing the sum of a smooth (possibly nonconvex) function and a block-separable (possibly nonconvex nonsmooth) function. At each iteration, this method randomly picks a block according to any prescribed probability distribution and typically solves several associated proximal subproblems that usually have … Read more

    A Second-Order Method for Strongly Convex L1-Regularization Problems

  • Kimon Fountoulakis
  • Jacek Gondzio
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this paper a robust second-order method is developed for the solution of strongly convex l1-regularized problems. The main aim is to make the proposed method as inexpensive as possible, while even difficult problems can be efficiently solved. The proposed method is a primal-dual Newton Conjugate Gradients (pdNCG) method. Convergence properties of pdNCG are studied … Read more

    A Deterministic Rescaled Perceptron Algorithm

  • Javier F. Peña
  • Negar Soheili
    • Categories Convex Optimization Tags , ,

    The perceptron algorithm is a simple iterative procedure for finding a point in a convex cone $F$. At each iteration, the algorithm only involves a query of a separation oracle for $F$ and a simple update on a trial solution. The perceptron algorithm is guaranteed to find a feasible point in $F$ after $\Oh(1/\tau_F^2)$ iterations, … Read more

    On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

  • Oliver Stein
  • Nathan Sudermann-Merx
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

    This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from W.W. Hogan, Point-to-set maps in mathematical programming, SIAM Review, Vol. 15 (1973), 591-603, for abstract feasible set mappings under complete convexity as well as standard differentiability results … Read more

    Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems

  • Euhanna Ghadimi
  • Mikael Johansson
  • Iman Shames
  • André Teixeira
    • Categories Convex Optimization, Quadratic Programming Tags , ,

    The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm … Read more

    One condition for all: solution uniqueness and robustness of l1-synthesis and l1-analysis minimizations

  • Ming Yan
  • Wotao Yin
  • Hui Zhang
    • Categories Convex and Nonsmooth Optimization, Statistics Tags , , , , ,

    The l1-synthesis and l1-analysis models recover structured signals from their undersampled measurements. The solution of the former model is often a sparse sum of dictionary atoms, and that of the latter model often makes sparse correlations with dictionary atoms. This paper addresses the question: when can we trust these models to recover specific signals? We … Read more

    A branch and bound approach for convex semi-infinite programming

  • Le Thi Hoai An
  • Mohand Ouanes
  • A. Ismael F. Vaz
    • Categories Convex Optimization, Semi-infinite Programming

    In this paper we propose an efficient approach for globally solving a class of convex semi-infinite programming (SIP) problems. Under the objective function and constraints (w.r.t. the variables to be optimized) convexity assumption, and appropriate differentiability, we propose a branch and bound exchange type method for SIP. To compute a feasible point for a SIP … Read more