Infinite Dimensional Optimization

A Hybrid Discretization Algorithm with Guaranteed Feasibility for the Global Solution of Semi-Infinite Programs

  • Hatim Djelassi
  • Alexander Mitsos
    • Categories Semi-infinite Programming

    A discretization-based algorithm for the global solution of semi-infinite programs (SIPs) is proposed, which is guaranteed to converge to a feasible, ε-optimal solution finitely under mild assumptions. The algorithm is based on the hybridization of two existing algorithms. The first algorithm (Mitsos in Optimization 60(10–11):1291–1308, 2011) is based on a restriction of the right-hand side … Read more

    Strong Duality and Dual Pricing Properties in Semi-infinite Linear Programming–A Non-Fourier-Motzkin Elimination Approach

  • Qinghong Zhang
    • Categories Semi-infinite Programming

    The Fourier-Motzkin elimination method has been recently extended to linear inequality systems that have infinitely many inequalities. It has been used in the study of linear semi-infinite programming by Basu, Martin, and Ryan. Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang recently published in Optimization, which states … Read more

    Semi-infinite programming using high-degree polynomial interpolants and semidefinite programming

  • Dávid Papp
    • Categories Semi-definite Programming, Semi-infinite Programming, Statistics Tags , , , ,

    In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this … Read more

    Strong duality and sensitivity analysis in semi-infinite linear programming

  • Amitabh Basu
  • Kipp Martin
  • Christopher Thomas Ryan
    • Categories Semi-infinite Programming Tags , ,

    Finite-dimensional linear programs satisfy strong duality (SD) and have the “dual pricing” (DP) property. The (DP) property ensures that, given a sufficiently small perturbation of the right-hand-side vector, there exists a dual solution that correctly “prices” the perturbation by computing the exact change in the optimal objective function value. These properties may fail in semi-infinite … Read more

    Combinatorial Optimal Control of Semilinear Elliptic PDEs

  • Christoph Buchheim
  • Renke Kuhlmann
  • Christian Meyer
    • Categories Cutting Plane Approaches, Semi-infinite Programming, Systems governed by Differential Equations Optimization Tags , , , ,

    Optimal control problems (OCP) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, it results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we … Read more

    Remark on multi-target,robust linear-quadratic control problem on semi-infinite interval

  • Leonid Faybusovich
  • Thanasak Mouktonglang
    • Categories Control Applications, Infinite Dimensional Optimization Tags ,

    We consider multi-target,robust linear-quadratic control problem on semi-infinite interval. Using functional-analytic approach developed in [2], we reduce this problem to a convex optimization problem on the simplex. Explicit procedure for the reduced optimization problem is described. Citation Preprint, University of Notre Dame, August,2015 Article Download View Remark on multi-target,robust linear-quadratic control problem on semi-infinite interval

    Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

  • Patrick L. Combettes
  • Jean-Christophe Pesquet
    • Categories Infinite Dimensional Optimization, Stochastic Programming Tags , , , , ,

    We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the … Read more

    Asynchronous Block-Iterative Primal-Dual Decomposition Methods for Monotone Inclusions

  • Patrick L. Combettes
  • Jonathan Eckstein
    • Categories Convex Optimization, Infinite Dimensional Optimization Tags , , , , , ,

    We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in … Read more

    Bridging the Gap Between Multigrid, Hierarchical, and Receding-Horizon Control

  • Victor M. Zavala
    • Categories Distributed Control, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We analyze the structure of the Euler-Lagrange conditions of a lifted long-horizon optimal control problem. The analysis reveals that the conditions can be solved by using block Gauss-Seidel schemes and we prove that such schemes can be implemented by solving sequences of short-horizon problems. The analysis also reveals that a receding-horizon control scheme is equivalent … Read more

    Solving disjunctive optimization problems by generalized semi-infinite optimization techniques

  • Peter Kirst
  • Oliver Stein
    • Categories Nonsmooth Optimization, Semi-infinite Programming Tags , , , ,

    We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods, for our approach neither a conjunctive nor a disjunctive normal form is expected. Applying existing lower level reformulations for the corresponding semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally … Read more