Infinite Dimensional Optimization

A Matrix-free Algorithm for Equality Constrained Optimization Problems with Rank-deficient Jacobians

  • Frank E. Curtis
  • Jorge Nocedal
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization, Infinite Dimensional Optimization Tags , , , , ,

    We present a line search algorithm for large-scale constrained optimization that is robust and efficient even for problems with (nearly) rank-deficient Jacobian matrices. The method is matrix-free (i.e., it does not require explicit storage or factorizations of derivative matrices), allows for inexact step computations, and is applicable for nonconvex problems. The main components of the … Read more

    An information-based approximation scheme for stochastic optimization problems in continuous time

  • Daniel Kuhn
    • Categories Infinite Dimensional Optimization, Stochastic Programming Tags , , , ,

    Dynamic stochastic optimization problems with a large (possibly infinite) number of decision stages and high-dimensional state vector are inherently difficult to solve. In fact, scenario tree based algorithms are unsuitable for problems with many stages, while dynamic programming type techniques are unsuitable for problems with many state variables. This article proposes a stage aggregation scheme … Read more

    Duality of ellipsoidal approximations via semi-infinite programming

  • Filiz Gurtuna
    • Categories Convex Optimization, Semi-infinite Programming Tags , , , , , , , , , ,

    In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results … Read more

    Convergent Network Approximation for the Continuous Euclidean Length Constrained Minimum Cost Path Problem

  • Natashia Boland
  • Ranga Muhandiramge
  • Song Wang
    • Categories Control Applications, Network Optimization, Semi-infinite Programming

    In many path planning situations we would like to find a path of constrained Euclidean length in 2D that minimises a line integral. We call this the Continuous Length-Constrained Minimum Cost Path Problem (C-LCMCPP). Generally, this will be a non-convex optimization problem, for which continuous approaches only ensure locally optimal solutions. However, network discretisations yield … Read more

    A multilevel algorithm for solving the trust-region subproblem

  • Philippe L. Toint
  • Melissa Weber-Mendonca
  • Dimitri Tomanos
    • Categories Infinite Dimensional Optimization, Optimization of Systems modeled by PDEs, Unconstrained Optimization Tags , , ,

    We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized continuous functions, in which case the different levels of discretization provide a natural multilevel context. The trust-region problem is considered at the highest … Read more

    The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases

  • Osman Guler
  • Filiz Gurtuna
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Semi-infinite Programming Tags , , , , , , , , , , ,

    A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We … Read more

    The Exact Feasibility of Randomized Solutions of Robust Convex Programs

  • Marco C. Campi
  • Simone Garatti
    • Categories Convex Optimization, Robust Optimization, Semi-infinite Programming Tags , , ,

    Robust optimization programs are hard to solve even when the constraints are convex. In previous contributions, it has been shown that approximately robust solutions (i.e. solutions feasible for all constraints but a small fraction of them) to convex programs can be obtained at low computational cost through constraints randomization. In this paper, we establish new … Read more

    Convex duality and entropy-based moment closure: Characterizing degenerate densities

  • Cory D. Hauck
  • C. David Levermore
  • André L. Tits
    • Categories Convex Optimization, Infinite Dimensional Optimization Tags , , , , , ,

    A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the … Read more

    An estimation-free, robust conditional value-at-risk portfolio allocation model

  • Carlos Jabbour
  • Javier F. Peña
  • Juan C. Vera
  • Luis F. Zuluaga
    • Categories Finance and Economics, Robust Optimization, Semi-infinite Programming

    We propose a novel optimization model for risk-averse investors to obtain robust solutions for portfolio allocation problems. Unlike related models in the literature, no historical data or statistical estimation techniques are used to compute the parameters of the model. Instead, the parameters are directly obtained from current prices of options on the assets being considered. … Read more

    Sensitivity analysis in linear semi-infinite programming via partitions

  • Miguel A. Goberna
  • Tamás Terlaky
  • Maxim I. Todorov
    • Categories Other Topics, Semi-infinite Programming Tags , , ,

    This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming problem to depend linearly on the size of the perturbations, when these perturbations are directional, involve either the cost coefficients or the right-hand-side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first … Read more