Infinite Dimensional Optimization

Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints

  • Hatim Djelassi
  • Alexander Mitsos
  • Moll Glass
    • Categories Global Optimization Applications, Global Optimization Theory, Semi-infinite Programming

    Discretization-based algorithms are proposed for the global solution of mixed-integer nonlinear generalized semi-infinite (GSIP) and bilevel (BLP) programs with lower-level equality constraints coupling the lower and upper level. The algorithms are extensions, respectively, of the algorithm proposed by Mitsos and Tsoukalas (J Glob Optim 61(1):1–17, 2015. https://doi.org/10.1007/s10898-014-0146-6) and by Mitsos (J Glob Optim 47(4):557–582, 2010. … Read more

    A transformation-based discretization method for solving general semi-infinite optimization problems

  • Karl-Heinz Küfer
  • Jan Schwientek
  • Tobias Seidel
    • Categories Semi-infinite Programming Tags , , , ,

    Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the $x$-dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In … Read more

    Balancing Communication and Computation in Distributed Optimization

  • Albert S. Berahas
  • Raghu Bollapragada
  • Nitish Shirish Keskar
  • Ermin Wei
    • Categories Convex Optimization, Distributed Control, Network Optimization Tags , , , ,

    Methods for distributed optimization have received significant attention in recent years owing to their wide applicability in various domains including machine learning, robotics and sensor networks. A distributed optimization method typically consists of two key components: communication and computation. More specifically, at every iteration (or every several iterations) of a distributed algorithm, each node in … Read more

    A simplex method for uncapacitated pure-supply infinite network flow problems

  • Marina A. Epelman
  • Christopher Thomas Ryan
  • Robert L. Smith
    • Categories Infinite Dimensional Optimization, Network Optimization Tags , ,

    We provide a simplex algorithm for a structured class of uncapacitated countably-infinite network flow problems. Previous efforts required explicit capacities on arcs with uniformity properties that facilitate duality arguments. By contrast, this paper takes a “primal” approach by devising a simplex method that provably converges to optimal value using arguments based on convergence of spanning … Read more

    Revisiting Approximate Linear Programming Using a Saddle Point Approach

  • Qihang Lin
  • Selvaprabu Nadarajah
  • Negar Soheili
    • Categories Dynamic Programming, Semi-infinite Programming, Stochastic Programming Tags , , , , ,

    Approximate linear programs (ALPs) are well-known models for computing value function approximations (VFAs) of intractable Markov decision processes (MDPs) arising in applications. VFAs from ALPs have desirable theoretical properties, define an operating policy, and provide a lower bound on the optimal policy cost, which can be used to assess the suboptimality of heuristic policies. However, … Read more

    Decomposition Algorithms for Distributionally Robust Optimization using Wasserstein Metric

  • Fengqiao Luo
  • Sanjay Mehrotra
    • Categories Data-Mining, Robust Optimization, Semi-infinite Programming Tags , , , ,

    We study distributionally robust optimization (DRO) problems where the ambiguity set is de ned using the Wasserstein metric. We show that this class of DRO problems can be reformulated as semi-in nite programs. We give an exchange method to solve the reformulated problem for the general nonlinear model, and a central cutting-surface method for the convex case, … Read more

    Optimal scenario generation and reduction in stochastic programming

  • René Henrion
  • Werner Romisch
    • Categories Semi-infinite Programming, Stochastic Programming Tags , , ,

    Scenarios are indispensable ingredients for the numerical solution of stochastic optimization problems. Earlier approaches for optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based on distances containing minimal information, i.e., on data appearing … Read more

    From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

  • Daniel Kuhn
  • Tobias Sutter
  • John Lygeros
  • Peyman Mohajerin Esfahani
    • Categories Dynamic Programming, Infinite Dimensional Optimization, Stochastic Programming Tags , , , ,

    We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of … Read more

    Distributionally Robust Stochastic Optimization with Dependence Structure

  • Anton J. Kleywegt
  • Rui Gao
    • Categories Infinite Dimensional Optimization, Robust Optimization, Stochastic Programming

    Distributionally robust stochastic optimization (DRSO) is a framework for decision-making problems under certainty, which finds solutions that perform well for a chosen set of probability distributions. Many different approaches for specifying a set of distributions have been proposed. The choice matters, because it affects the results, and the relative performance of different choices depend on … Read more

    The structure of the infinite models in integer programming

  • Amitabh Basu
  • Michele Conforti
  • Marco Di Summa
  • Joseph Paat
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches, Semi-infinite Programming

    The infinite models in integer programming can be described as the convex hull of some points or as the intersection of half-spaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for finite dimensional corner polyhedra. One consequence is that nonnegative continuous functions suffice to … Read more