Nonsmooth Optimization

Stationarity and Regularity of Real-Valued Functions

  • Alexander Y. Kruger
    • Categories Nonsmooth Optimization Tags , , , , , , , ,

    Different stationarity and regularity concepts for extended real-valued functions on metric spaces are considered in the paper. The properties are characterized in terms of certain local constants. A classification scheme for stationarity/regularity constants and corresponding concepts is proposed. The relations between different constants are established. Citation University of Ballarat, School of Information Technology and Mathematical … Read more

    A conic interior point decomposition approach for large scale semidefinite programming

  • Gema Plaza
  • Kartik Krishnan Sivaramakrishnan
  • Tamás Terlaky
    • Categories Nonsmooth Optimization, Semi-definite Programming Tags , , ,

    We describe a conic interior point decomposition approach for solving a large scale semidefinite programs (SDP) whose primal feasible set is bounded. The idea is to solve such an SDP using existing primal-dual interior point methods, in an iterative fashion between a {\em master problem} and a {\em subproblem}. In our case, the master problem … Read more

    Primal interior-point method for large sparse minimax optimization.

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags ,

    In this paper, we propose an interior-point method for large sparse minimax optimization. After a short introduction, where various barrier terms are discussed, the complete algorithm is introduced and some implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Thus nonconvex problems can be solved successfully. The results … Read more

    Trust-region interior-point method for large sparse l_1 optimization.

  • Ladislav Lukšan
  • Ctirad Matonoha
  • Jan Vlček
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags ,

    In this paper, we propose an interior-point method for large sparse l_1 optimization. After a short introduction, the complete algorithm is introduced and some implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Thus nonconvex problems can be solved successfully. The results of computational experiments given in this … Read more

    Fast Moreau Envelope Computation I: Numerical Algorithms

  • Yves Lucet
    • Categories Convex Optimization, Nonsmooth Optimization, Problem Solving Environments Tags , , , , , ,

    The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, … Read more

    Computing Proximal Points on Nonconvex Functions

  • Warren Hare
  • Claudia Sagastizábal
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

    The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with … Read more

    ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

  • Frédéric Babonneau
  • Jean-Philippe Vial
    • Categories Network Optimization, Nonsmooth Optimization Tags , ,

    This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and … Read more

    About Regularity of Collections of Sets

  • Alexander Y. Kruger
    • Categories Nonsmooth Optimization Tags , , , , , , ,

    The paper continues investigations of stationarity and regularity properties of set systems in normed spaces started in the previous paper of the author. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a set system to be regular. Citation University of Ballarat, School of … Read more

    Variable metric method for minimization of partially separable nonsmooth functions.

  • Ladislav Lukšan
  • Jan Vlček
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags ,

    In this report, we propose a new partitioned variable metric method for minimization of nonsmooth partially separable functions. After a short introduction, the complete algorithm is introduced and some implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Computational experiments given confirm efficiency and robustness of the new … Read more

    Toward a new DIRECT algorithm. A two-points based sampling method

  • Lakhdar Chiter
    • Categories Global Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    The DIRECT algorithm was motivated by a modification to Lipschitzian optimization. The algorithm begins its search by sampling the objective function at the midpoint of an interval, where this function attains its lowest value, and then divides this interval by trisecting it. One of its weakness is that if a global minimum lies at the … Read more