Uncertain convex programs: randomized solutions and confidence levels
 Giuseppe Calafiore,
 M.C. Campi
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Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chanceconstrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chanceconstrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a prespecified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations.
In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.
 Title
 Uncertain convex programs: randomized solutions and confidence levels
 Journal

Mathematical Programming
Volume 102, Issue 1 , pp 2546
 Cover Date
 200501
 DOI
 10.1007/s101070030499y
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
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 Authors

 Giuseppe Calafiore ^{(1)}
 M.C. Campi ^{(2)}
 Author Affiliations

 1. Dipartimento di Automatica e Informatica, Politecnico di Torino, corso Duca degli Abruzzi, 24, 10129, Torino, Italy
 2. Dipartimento di Automatica per l’Automazione, Università di Brescia, via Branze 38, 25123, Brescia, Italy