Infinite Dimensional Optimization

Double smoothing technique for infinite-dimensional optimization problems with applications to Optimal Control.

  • Olivier Devolder
  • François Glineur
  • Yurii Nesterov
    • Categories Convex Optimization, Infinite Dimensional Optimization Tags , , , ,

    In this paper, we propose an efficient technique for solving some infinite-dimensional problems over the sets of functions of time. In our problem, besides the convex point-wise constraints on state variables, we have convex coupling constraints with finite-dimensional image. Hence, we can formulate a finite-dimensional dual problem, which can be solved by efficient gradient methods. … Read more

    Implicit Multifunction Theorems in complete metric spaces

  • Van Ngai Huynh
  • Huu Tron Nguyen
  • Michel Thera
    • Categories Infinite Dimensional Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the … Read more

    Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

  • Samir Adly
  • Abderrahim Hantoute
  • Michel Thera
    • Categories Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , , , ,

    The main objective of this paper is to provide new explicit criteria to characterize weak lower semi-continuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal … Read more

    Combinatorial Integral Approximation

  • Michael N. Jung
  • Christian Kirches
  • Sebastian Sager
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Distributed Control Tags , ,

    We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel … Read more

    Lipschitz solutions of optimal control problems with state constraints of arbitrary order

  • Joseph Frédéric Bonnans
    • Categories Control Applications, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , ,

    In this paper we generalize to an arbitrary order, under minimal hypotheses, some sufficient conditions for Lipschitz continuity of the solution of a state constrained optimal control problems. The proof combines the approach by Hager in 1979 for dealing with first-order state constraints, and the high-order alternative formulation of the optimality conditions. Citation Published as … Read more

    A continuous model for open pit mine planning

  • Felipe Álvarez
  • Andreas Griewank
  • Jorge Amaya
  • Nikolai Strogies
    • Categories Applications - OR and Management Sciences, Global Optimization, Infinite Dimensional Optimization Tags , ,

    This paper proposes a new mathematical model for the open pit mine planning problem, based on continuous functional analysis. The traditional models for this problem have been constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach which allows for a … Read more

    Optimal control of a parabolic equation with time-dependent state constraints

  • Joseph Frédéric Bonnans
  • Pascal Jaisson
    • Categories Distributed Control Tags , , , , , , ,

    In this paper we study the optimal control problem of the heat equation by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied at each time. Using for the first time the technique of alternative optimality … Read more

    Chance-constrained optimization via randomization: feasibility and optimality

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

    In this paper we study the link between a semi-infinite chance-constrained optimization problem and its randomized version, i.e. the problem obtained by sampling a finite number of its constraints. Extending previous results on the feasibility of randomized convex programs, we establish here the feasibility of the solution obtained after the elimination of a portion of … Read more

    Lecture notes: Semidefinite programs and harmonic analysis

  • Frank Vallentin
    • Categories Infinite Dimensional Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , , , ,

    Lecture notes for the tutorial at the workshop HPOPT 2008 – 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands. Citation arXiv:0809.2017v1 [math.OC] Article Download View Lecture notes: Semidefinite programs and harmonic analysis