Nonsmooth Optimization

The Principle of Hamilton for Mechanical Systems with Impacts and Unilateral Constraints

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , ,

    An action integral is presented for Hamiltonian mechanics in canonical form with unilateral constraints and/or impacts. The transition conditions on generalized impulses and the energy are presented as variational inequalities, which are obtained by the application of discontinuous transversality conditions. The energetical behavior for elastic, plastic and blocking type impacts are analyzed. A general impact … Read more

    Local Linear Convergence of Forward–Backward under Partial Smoothness

  • Jalal Fadili
  • Jingwei Liang
  • Gabriel Peyré
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    In this paper, we consider the Forward–Backward proximal splitting algorithm to minimize the sum of two proper closed convex functions, one of which having a Lipschitz–continuous gradient and the other being partly smooth relatively to an active manifold $\mathcal{M}$. We propose a unified framework in which we show that the Forward–Backward (i) correctly identifies the … Read more

    Discrete Approximations of a Controlled Sweeping Process

  • Giovanni Colombo
  • René Henrion
  • Boris S. Mordukhovich
  • Nguyen Dinh Hoang
    • Categories Nonsmooth Optimization Tags , , , , ,

    The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhedral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of … Read more

    Global convergence of splitting methods for nonconvex composite optimization

  • Guoyin Li
  • Ting Kei Pong
    • Categories Nonsmooth Optimization

    We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. … Read more

    On Lipschitz optimization based on gray-box piecewise linearization

  • Torsten Bosse
  • Sabrina Fiege
  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , ,

    We address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function. They possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term. This overestimating local model is continuous but generally nonconvex. It can be generated … Read more

    A Class of Randomized Primal-Dual Algorithms for Distributed Optimization

  • Jean-Christophe Pesquet
  • Audrey Repetti
    • Categories Convex Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , , , ,

    Based on a preconditioned version of the randomized block-coordinate forward-backward algorithm recently proposed in [Combettes,Pesquet,2014], several variants of block-coordinate primal-dual algorithms are designed in order to solve a wide array of monotone inclusion problems. These methods rely on a sweep of blocks of variables which are activated at each iteration according to a random rule, … Read more

    Splitting methods with variable metric for KL functions

  • Pierre Frankel
  • Juan Peypouquet
  • Guillaume Garrigos
    • Categories Nonsmooth Optimization, Other Topics

    We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satis es the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f and obtain new convergence rates both for the values and the iterates. The analysis covers alternating … Read more

    An inertial Tseng’s type proximal algorithm for nonsmooth and nonconvex optimization problems

  • Radu Ioan Bot
  • Ernö Robert Csetnek
    • Categories Nonsmooth Optimization Tags , , , , ,

    We investigate the convergence of a forward-backward-forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth function with a smooth one in the absence of convexity. The convergence is obtained provided an appropriate regularization of the objective satisfies the Kurdyka-\L{}ojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. Article … Read more

    A Quasi-Newton Algorithm for Nonconvex, Nonsmooth Optimization with Global Convergence Guarantees

  • Frank E. Curtis
  • Xiaocun Que
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Unconstrained Optimization Tags , , , , ,

    A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy … Read more

    A proximal multiplier method for separable convex minimization

  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
  • Orlando Sarmiento
    • Categories Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization

    In this paper, we propose an inexact proximal multiplier method using proximal distances for solving convex minimization problems with a separable structure. The proposed method unified the work of Chen and Teboulle (PCPM method), Kyono and Fukushima (NPCPMM) and Auslender and Teboulle (EPDM) and extends the convergence properties for a class of phi-divergence distances. We … Read more