Chance-Constrained Programming: Joint and Individual Constraints

Abstract

Consider a two-stage decision making framework under uncertainty, where decisions are made at all times t ∈ T. First, a set of decisions, x, of size |T| is made. Then, one of several possible scenarios, ω ∈ Ω, that represent the uncertainty of the entire spectrum t ∈ T is realized via a parameter $w_t^\omega$. Second, a new set of decisions based on this particular scenario realization ω, y^ω, each of which are of size |T|, is made. The first decision, x, is made before knowing the uncertainty. The second decision, y, is made after knowing the specific realization of the uncertainty, ω. Typically, making accurate first-stage decisions that safeguard against a wide variety of scenarios is challenging. The second-stage decisions serve as a recourse to correct or amend the first-stage decisions, however one must pay a price to use this recourse. This is a standard setting for two-stage stochastic programming.