Boris S. Mordukhovich

New Fractional Error Bounds for Nonconvex Polynomial Systems with Applications to Holderian Stability in Optimization and Spectral Theory of Tensors

  • Guoyin Li
  • Boris S. Mordukhovich
  • Tiên-So'n Pham
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    In this paper we derive new fractional error bounds for nonconvex polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The results obtained do not require any regularity assumptions and resolve, in particular, some open questions posed in the literature. The developed techniques are … Read more

    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

    Second-order variational analysis and characterizations of tilt-stable optimal solutions in finite and infinite dimensions

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

    The paper is devoted to developing second-order tools of variational analysis and their applications to characterizing tilt-stable local minimizers of constrained optimization problems in finite-dimensional and infinite-dimensional spaces. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization. Based on second-order generalized differentiation, we obtain qualitative and quantitative … 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

    Approximate Maximum Principle for Discrete Approximations of Optimal Control Systems with Nonsmooth Objectives and Endpoint Constraints

  • Boris S. Mordukhovich
  • Ilya Shvartsman
    • Categories Control Applications, Convex and Nonsmooth Optimization Tags , , ,

    The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an intermediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of … 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

    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

    Necessary optimality conditions in pessimistic bilevel programming

  • Stephan Dempe
  • Boris S. Mordukhovich
  • Alain Zemkoho
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    This paper is devoted to the so-called pessimistic version of bilevel programming programs. Minimization problems of this type are challenging to handle partly because the corresponding value functions are often merely upper (while not lower) semicontinuous. Employing advanced tools of variational analysis and generalized differentiation, we provide rather general frameworks ensuring the Lipschitz continuity of … 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