normal cone

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

    On a new class of matrix support functionals with applications

  • James V. Burke
  • Tim Hoheisel
    • Categories Convex and Nonsmooth Optimization, Quadratic Programming, Semi-definite Programming Tags , , , , , ,

    A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support … Read more

    Optimality, identifiability, and sensitivity

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

    Around a solution of an optimization problem, an “identifiable” subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are “minimal”. This new notion lays a broad and intuitive variational-analytic … Read more

    Stationarity and regularity of infinite collections of sets

  • Alexander Y. Kruger
  • Marco A. López
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , , , , , , ,

    This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behaviour of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties … Read more

    Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization

  • Alexander Y. Kruger
  • Marco A. López
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , , , , , , , ,

    This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger & L�pez (2012) and is mainly focused on the application of the criteria from Kruger & L�pez (2012) to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly … Read more

    Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems

  • Xuan Duc Ha Truong
    • Categories Complementarity and Variational Inequalities, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , , , , , , , , , ,

    We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in … Read more

    Generic nondegeneracy in convex optimization

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

    We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally, for lower-C^2 functions. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. May 2010. Article Download View Generic nondegeneracy in convex optimization

    About Stationarity and Regularity in Variational Analysis

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

    Stationarity and regularity concepts for the three typical for variational analysis classes of objects — real-valued functions, collections of sets, and multifunctions — are investigated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. … Read more

    Identifying Redundant Linear Constraints in Systems of Linear Matrix Inequality Constraints

  • Shafiu Jibrin
  • Daniel Stover
    • Categories Semi-definite Programming Tags , , ,

    Semidefinite programming has been an interesting and active area of research for several years. In semidefinite programming one optimizes a convex (often linear) objective function subject to a system of linear matrix inequality constraints. Despite its numerous applications, algorithms for solving semidefinite programming problems are restricted to problems of moderate size because the computation time … Read more

    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