Semi-infinite Programming

How to Solve a Semi-infinite Optimization Problem

  • Oliver Stein
    • Categories Semi-infinite Programming Tags , , , , , , , ,

    After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. … 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

    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

    A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

  • Moritz Diehl
  • Oliver Stein
  • Boris Houska
  • Paul Steuermann
    • Categories Complementarity and Variational Inequalities, Robust Optimization, Semi-infinite Programming Tags , , , ,

    This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be … 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

    A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

  • Amitabh Basu
  • Robert Hildebrand
  • Matthias Köppe
  • Marco Molinaro
    • Categories Cutting Plane Approaches, Semi-infinite Programming Tags , , ,

    We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornu\’ejols and Molinaro for k=2. Article Download View A … Read more

    Convexity Conditions of Kantorovich Function and Related Semi-infinite Linear Matrix Inequalities

  • Yun-Bin Zhao
    • Categories Convex Optimization, Robust Optimization, Semi-infinite Programming Tags , , , ,

    The Kantorovich function $(x^TAx)( x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only … Read more

    Multiobjective DC Programming with Infinite Convex Constraints

  • Mark Goh
  • Shaojian Qu
  • Soonyi Wu
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization, Semi-infinite Programming Tags , , , ,

    In this paper new results are established in multiobjective DC programming with infinite convex constraints ($MOPIC$ for abbr.) that are defined on Banach space (finite or infinite) with objectives given as the difference of convex functions subject to infinite convex constraints. This problem can also be called multiobjective DC semi-infinite and infinite programming, where decision … Read more