Nonsmooth Optimization

Solving disjunctive optimization problems by generalized semi-infinite optimization techniques

  • Peter Kirst
  • Oliver Stein
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , , , ,

    We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods, for our approach neither a conjunctive nor a disjunctive normal form is expected. Applying existing lower level reformulations for the corresponding semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally … Read more

    Optimality and complexity for constrained optimization problems with nonconvex regularization

  • Wei Bian
  • Xiaojun Chen
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set and the objective function has a nonsmooth, nonconvex regularizer. Such regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding and non-Lipschitz $L_p$ penalties as special cases. Using the theory of the generalized directional … Read more

    Nonsmooth Methods for Control Design with Integral Quadratic Constraints

  • Pierre Apkarian
  • Dominikus Noll
  • Olivier Prot
    • Categories Control Applications, Nonsmooth Optimization Tags , , , , ,

    We develop an optimization technique to compute local solutions to synthesis problems subject to integral quadratic constraints (IQCs). We use the fact that IQCs may be transformed into semi-infinite maximum eigenvalue constraints over the frequency axis and approach them via nonsmooth optimization methods. We develop a suitable spectral bundle method and prove its convergence in … Read more

    Metric subregularity of composition set-valued mappings with applications to fixed point theory

  • Marius Durea
  • Radu Strugariu
    • Categories Nonsmooth Optimization Tags

    In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at … Read more

    Trust-region methods without using derivatives: Worst case complexity and the non-smooth case

  • R. Garmanjani
  • Luis Nunes Vicente
  • Diogo Júdice
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , ,

    Trust-region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts. In particular, they have been studied and applied for problems without using derivatives. The analysis of trust-region derivative-free methods has focused on global convergence, and they have been proved to generate a sequence of … Read more

    LP formulations for mixed-integer polynomial optimization problems

  • Daniel Bienstock
  • Gonzalo Muñoz
    • Categories (Mixed) Integer Nonlinear Programming, Nonsmooth Optimization Tags

    We present polynomial-time algorithms for constrained optimization problems overwhere the intersection graph of the constraint set has bounded tree-width. In the case of binary variables we obtain exact, polynomial-size linear programming formulations for the problem. In the mixed-integer case with bounded variables we obtain polynomial-size linear programming representations that attain guaranteed optimality and feasibility bounds. … Read more

    Regularity of collections of sets and convergence of inexact alternating projections

  • Alexander Y. Kruger
  • Nguyen H. Thao
    • Categories Nonsmooth Optimization Tags , , ,

    We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed. Article Download View Regularity of collections of sets … Read more

    An optimal subgradient algorithm for large-scale bound-constrained convex optimization

  • Masoud Ahookhosh
  • Arnold Neumaier
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    This paper shows that the OSGA algorithm — which uses first-order information to solve convex optimization problems with optimal complexity — can be used to efficiently solve arbitrary bound-constrained convex optimization problems. This is done by constructing an explicit method as well as an inexact scheme for solving the bound-constrained rational subproblem required by OSGA. … Read more

    An optimal subgradient algorithm for large-scale convex optimization in simple domains

  • Masoud Ahookhosh
  • Arnold Neumaier
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , ,

    This paper shows that the optimal subgradient algorithm, OSGA, proposed in \cite{NeuO} can be used for solving structured large-scale convex constrained optimization problems. Only first-order information is required, and the optimal complexity bounds for both smooth and nonsmooth problems are attained. More specifically, we consider two classes of problems: (i) a convex objective with a … Read more

    Solving ill-posed bilevel programs

  • Alain Zemkoho
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , , ,

    This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to … Read more