Nonsmooth Optimization

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

    KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

  • Stephan Dempe
  • Alain Zemkoho
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both … Read more

    Level Bundle Methods for Constrained Convex Optimization with Various Oracles

  • Welington de Oliveira
  • Wim van Ackooij
    • Categories Nonsmooth Optimization

    We propose restricted memory level bundle methods for minimizing constrained convex nonsmooth optimization problems whose objective and constraint functions are known through oracles (black-boxes) that might provide inexact information. Our approach is general and covers many instances of inexact oracles, such as upper, lower and on-demand oracles. We show that the proposed level bundle methods … Read more

    Nonsmooth Optimization Using Uncontrolled Inexact Information

  • Welington de Oliveira
  • Jérôme Malick
  • Sofia Zaourar
    • Categories Nonsmooth Optimization Tags , , , ,

    We consider convex nonsmooth optimization problems whose objective function is known through a (fine) oracle together with some additional (cheap but poor) information – formalized as a second coarse oracle with uncontrolled inexactness. It is the case when the objective function is itself the output of an optimization solver, using a branch-and-bound procedure, or decomposing … Read more

    Robust convex relaxation for the planted clique and densest k-subgraph problems

  • Brendan Ames
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and l1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our … Read more

    GENERALIZATIONS OF THE DENNIS-MOR\’E THEOREM II

  • Asen L. Dontchev
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    This paper is a continuation of our previous paper were we presented generalizations of the Dennis-Mor\’e theorem to characterize q-superliner convergences of quasi-Newton methods for solving equations and variational inequalities in Banach spaces. Here we prove Dennis-Mor\’e type theorems for inexact quasi-Newton methods applied to variational inequalities in finite dimensions. We first consider variational inequalities … Read more

    Second-order growth, tilt stability, and metric regularity of the subdifferential

  • Dmitriy Drusvyatskiy
  • Boris S. Mordukhovich
  • Nghia Tran
    • Categories Nonsmooth Optimization Tags , , , , ,

    This paper sheds new light on several interrelated topics of second-order variational analysis, both in finite and infinite-dimensional settings. We establish new relationships between second-order growth conditions on functions, the basic properties of metric regularity and subregularity of the limiting subdifferential, tilt-stability of local minimizers, and positive definiteness/semidefiniteness properties of the second-order subdifferential (or generalized … Read more

    REDUCTION OF TWO-STAGE PROBABILISTIC OPTIMIZATION PROBLEMS WITH DISCRETE DISTRIBUTION OF RANDOM DATA TO MIXED INTEGER PROGRAMMING PROBLEMS

  • Andrey Kibzun
  • Vladimir I. Norkin
  • Andrey Naumov
    • Categories (Mixed) Integer Nonlinear Programming, Nonsmooth Optimization, Stochastic Programming Tags , , , , ,

    We consider models of two-stage stochastic programming with a quantile second stage criterion and optimization models with a chance constraint on the second stage objective function values. Such models allow to formalize requirements to reliability and safety of the system under consideration, and to optimize the system in extreme conditions. We suggest a method of … Read more

    A doubly stabilized bundle method for nonsmooth convex optimization

  • Welington de Oliveira
  • Mikhail V. Solodov
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , ,

    We propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations. Most bundle algorithms use a cutting-plane model of the objective function to formulate a subproblem whose solution gives the next iterate. Proximal bundle methods employ the model in the objective function of the subproblem, while level … Read more

    Orthogonal invariance and identifiability

  • Aris Daniilidis
  • Dmitriy Drusvyatskiy
  • Adrian Lewis
    • Categories Nonsmooth Optimization Tags , , , , ,

    Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann’s theorem on matrix norms is an early example. We discuss the example of “identifiability”, a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, “partial smoothness”) is the … Read more