Nonsmooth Optimization

Optimality conditions for the nonlinear programming problems on Riemannian manifolds

  • Zhang Lei-Hong
  • Weihong Yang
  • Song Ruyi
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    In recent years, many traditional optimization methods have been successfully generalized to minimize objective functions on manifolds. In this paper, we first extend the general traditional constrained optimization problem to a nonlinear programming problem built upon a general Riemannian manifold $\mathcal{M}$, and discuss the first-order and the second-order optimality conditions. By exploiting the differential geometry … Read more

    A quasi-Newton proximal splitting method

  • Stephen Becker
  • Jalal Fadili
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization

    A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration … Read more

    Sparse Approximation via Penalty Decomposition Methods

  • Zhaosong Lu
  • Yong Zhang
    • Categories (Mixed) Integer Nonlinear Programming, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-“norm” of a vector being a part of constraints or objective function. In particular, we first study the first-order optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of … Read more

    Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential.

  • Dmitriy Drusvyatskiy
  • Adrian Lewis
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    We prove that uniform second order growth, tilt stability, and strong metric regularity of the subdifferential — three notions that have appeared in entirely different settings — are all essentially equivalent for any lower-semicontinuous, extended-real-valued function. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. May … Read more

    Error Forgetting of Bregman Iteration

  • Stanley Osher
  • Wotao Yin
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    This short article analyzes an interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method after a change of variable, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b. The procedure obtains its solution by solving a sequence of unconstrained subproblems of minimizing J(x)+1/2||Ax-b^k||^2, where b^k … Read more

    Packing Ellipsoids with Overlap

  • Caroline Uhler
  • Stephen Wright
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact … Read more

    Learning how to play Nash, potential games and alternating minimization method for structured nonconvex problems on Riemannian manifolds

  • J. X. da Cruz Neto
  • Paulo Roberto Oliveira
  • Antoine Soubeyran
  • Pedro A. Soares Jr.
    • Categories Applications - Science and Engineering, Game Theory, Nonsmooth Optimization

    In this paper we consider minimization problems with constraints. We show that if the set of constaints is a Riemannian manifold of non positive curvature and the objective function is lower semicontinuous and satisfi es the Kurdyka-Lojasiewicz property, then the alternating proximal algorithm in Euclidean space is naturally extended to solve that class of problems. We … Read more

    On Differentiability Properties of Player Convex Generalized Nash Equilibrium Problems

  • Nadja Harms
  • Oliver Stein
  • Christian Kanzow
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonsmooth Optimization Tags , , , , ,

    This article studies differentiability properties for a reformulation of a player convex generalized Nash equilibrium problem as a constrained and possibly nonsmooth minimization problem. By using several results from parametric optimization we show that, apart from exceptional cases, all locally minimal points of the reformulation are differentiability points of the objective function. This justifies a … Read more

    Nonsmooth cone-constrained optimization with applications to semi-infinite programming

  • Boris S. Mordukhovich
  • Nghia Tran
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , ,

    The paper is devoted to the study of general nonsmooth problems of cone-constrained optimization (or conic programming) important for various aspects of optimization theory and applications. Based on advanced constructions and techniques of variational analysis and generalized differentiation, we derive new necessary optimality conditions (in both “exact” and “fuzzy” forms) for nonsmooth conic programs, establish … Read more

    Reweighted $\ell_1hBcMinimization for Sparse Solutions to Underdetermined Linear Systems

  • Duan Li
  • Yun-Bin Zhao
    • Categories Basic Sciences Applications, Linear Programming, Nonsmooth Optimization Tags , , , , ,

    Numerical experiments have indicated that the reweighted $\ell_1$-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. Thus it is important to carry out a further investigation of this class of methods. In this paper, we point out that reweighted $\ell_1$-methods are intrinsically associated with the minimization of the so-called merit … Read more