Infinite Dimensional Optimization

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 well-posed shooting algorithm for optimal control problems with singular arcs

  • M. Soledad Aronna
  • Joseph Frédéric Bonnans
  • Pierre Martinon
    • Categories Infinite Dimensional Optimization Tags , , , , ,

    In this article we establish for the first time the well-posedness of the shooting algorithm applied to optimal control problems for which all control variables enter linearly in the Hamil- tonian. We start by investigating the case having only initial-final state constraints and free control variable, and afterwards we deal with control bounds. The shooting … Read more

    Partially affine control problems: second order conditions and a well-posed shooting algorithm

  • M. Soledad Aronna
    • Categories Infinite Dimensional Optimization Tags , , , ,

    This paper deals with optimal control problems for systems that are affine in one part of the control variables and nonlinear in the rest of the control variables. We have finitely many equality and inequality constraints on the initial and final states. First we obtain second order necessary and sufficient conditions for weak optimality. Afterwards, … Read more

    A robust Kantorovich’s theorem on inexact Newton method with relative residual error tolerance

  • Orizon Pereira Ferreira
  • Benar Fux Svaiter
    • Categories Infinite Dimensional Optimization, Nonlinear Optimization

    We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a … 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

    Quadratic order conditions for bang-singular extremals

  • M. Soledad Aronna
  • Joseph Frédéric Bonnans
  • Andrei V. Dmitruk
  • Pablo A. Lotito
    • Categories Infinite Dimensional Optimization Tags , , , ,

    This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case. Citation NUMERICAL … Read more

    There is no variational characterization of the cycles in the method of periodic projections

  • J.-B. Baillon
  • Patrick L. Combettes
  • Roberto Cominetti
    • Categories Convex Optimization, Infinite Dimensional Optimization Tags , , ,

    The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers … Read more

    Polynomial Approximations for Continuous Linear Programs

  • Dimitra Bampou
  • Daniel Kuhn
    • Categories Infinite Dimensional Optimization, Semi-definite Programming Tags , , , ,

    Continuous linear programs have attracted considerable interest due to their potential for modelling manufacturing, scheduling and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (non-separated) problem instances. In this paper we propose a more generic approximation scheme for … Read more