variational analysis

Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability

  • Boris S. Mordukhovich
  • Jiri V. Outrata
  • Héctor Ramírez
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized di erential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related … Read more

    Partial Second-Order Subdifferentials in Variational Analysis and Optimization

  • Boris S. Mordukhovich
  • Nguyen Mau Nam
  • Nguyen Thi Yen Nhi
    • Categories Nonsmooth Optimization Tags , , , , , ,

    This paper presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial … Read more

    CHARACTERIZATIONS OF FULL STABILITY IN CONSTRAINED OPTIMIZATION

  • Boris S. Mordukhovich
  • Tyrrell Rockafellar
  • Ebrahim Sarabi
    • Categories Nonsmooth Optimization, Robust Optimization Tags , , , , , ,

    This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from both viewpoints of optimization theory and its applications. Based on second- order generalized differential tools of variational analysis, we obtain … 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

    Holder Metric Subregularity with Applications to Proximal Point Method

  • Guoyin Li
  • Boris S. Mordukhovich
    • Categories Nonsmooth Optimization Tags , , , ,

    This paper is mainly devoted to the study and applications of H\”older metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and pointbased sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact … Read more

    Slopes of multifunctions and extensions of metric regularity

  • Van Ngai Huynh
  • Alexander Y. Kruger
  • Michel Thera
    • Categories Nonsmooth Optimization Tags , , , ,

    This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and applied for characterizing metric regularity. Several kinds of local and nonlocal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated. Citation Published in Vietnam Journal of Mathematics 40:2&3(2012) … Read more

    Error bounds for vector-valued functions on metric spaces

  • Ewa M. Bednarczuk
  • Alexander Y. Kruger
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization Tags , , ,

    In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects — slopes — are introduced and a classification scheme of error bound criteria is presented. Citation Published in Vietnam … Read more

    DC approach to regularity of convex multifunctions with applications to infinite systems

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

    The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way we establish new formulas for calculating the exact regularity bound of … Read more

    Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data

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

    The paper is devoted to the subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. … Read more

    Optimization problems with value function objectives

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

    The family of optimization problems with value function objectives includes the minmax programming problem and the bilevel optimization problem. In this paper, we derive necessary optimality conditions for this class of problems. The main focus is on the case where the functions involved are nonsmooth and the constraints are the very general operator constraints. Citation … Read more