Decision Making in an Uncertain Environment: the Scenario based Optimization Approach

Abstract

A central issue arising in financial, engineering and, more generally, in many applicative endeavors is to make a decision in spite of an uncertain environment. Along a robust approach, the decision should be guaranteed to work well in all possible realizations of the uncertainty. A less restrictive approach consists instead of requiring that the risk of failure associated to the decision should be small in some – possibly probabilistic – sense. From a mathematical viewpoint, the latter formulation leads to a chance-constrained optimization program, i.e. to an optimization program subject to constraints in probability. Unfortunately, however, both the robust approach as well as the chance-constrained approach are computationally intractable in general. In this paper, we present a computationally efficient methodology for dealing with uncertainty in optimization based on sampling a finite number of instances (or scenarios) of the uncertainty. In particular, we consider uncertain programs with convexity structure, and show that the scenario-based solution is, with high confidence, a feasible solution for the chance-constrained problem. The proposed approach represents a viable way to address general convex decision making problems in a risk-adjusted sense.

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