Ergodic Approach to Robust Optimization and Infinite Programming Problems
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Ergodic Approach to Robust Optimization and Infinite Programming Problems 1 ´ Pedro Perez-Aros
Received: 14 August 2019 / Accepted: 19 October 2020 / © Springer Nature B.V. 2020
Abstract In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly implies the consistency of the scenario approach for nonconvex optimization problems. Finally, we show that our method can also be used to solve infinite programming problems. Keywords Stochastic optimization · Scenario approach · Robust optimization · Infinite programming · Epi-convergence · Ergodic theorems. Mathematics Subject Classification (2010) MSC 90C15 · 90C26 · 90C90 · 60B11
1 Introduction Robust optimization (RO) corresponds to a field of mathematical programming dedicated to the study of problems with uncertainty. In this class of models, the constraint set is given by the set of points, which satisfy all (or in the presence of measurability, almost all) possible cases. Roughly speaking, an RO problem corresponds to the following mathematical optimization model min g(x) (1) s.t. x ∈ M(ξ ), almost surely ξ ∈ , where X is a Polish space, (, A, P) is a probability space, M : ⇒ X is a measurable multifunction with closed values and g : X → R ∪ {+∞} is a lower semicontinuous function. We refer to [4–6, 19, 22–24] and the references therein for more details and applications. This work was partially supported by grants Fondecyt Regular 1190110 and Fondecyt Regular 1200283. Pedro P´erez-Aros
[email protected] 1
Instituto de Ciencias de la Ingenier´ıa, Universidad de O’Higgins, Libertador Bernardo O’Higgins 611, Rancagua, Regi´on del Libertador Gral. Bernardo O’Higgins, Chile
P. P´erez-Aros
When the number of possible scenarios ξ ∈ is infinite in Problem Eq. 1, the computation of necessary and sufficient optimality conditions presents difficulties and requires a more delicate analysis than a simpler optimization problem. As far as we know, only works related to infinite programming deal directly with infinite-many constraints (see [14, 17] and the references therein for the general theory, and see [18] for a new approach based on extremal and variational principles). For that reason, it is necessary to solve an approximation of Problem Eq. 1. In this regard, the so-called scenario approach emerges as a possible solution. The scenario approach corresponds to a min-max approximation of the original robust optimization problem using a sequence of samples. It has used to provide an approximate solution to convex and nonconvex optimization problems, when the space of decision X = Rd (see, e.g., [8–10]). Furthermore, the consistency of this method has been recently provided in [7] for convex optimization problems. The inte
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