On the computational complexity and generalization properties of multi-stage and stage-wise coupled scenario programs

Abstract

We discuss the computational complexity and feasibility properties of scenario based techniques for uncertain optimization programs. We consider different solution alternatives ranging from the standard scenario approach to recursive variants, and compare feasibility as a function of the total computation burden. We identify trade-offs between the different methods depending on the problem structure and the desired probability of constraint satisfaction. Our motivation for this work stems from the applicability and complexity reduction when making decisions by means of recursive algorithms. We illustrate our results on an example from the area of approximate dynamic programming. Index TermsScenario approach, randomized optimization, uncertain systems, approximate dynamic programming.

Exploiting structure of chance constrained programs via submodularity

“…When the constrains are linear in x and there are no binary variables, i.e., when b = 0, specifically, when f (x, y, w) = [10] with [25], discussed in Lemma 5 in Appendix B, yields that if (1) is tight for the class of convex fully-supported problems, see [6]. Using this result, we provide a slight extension of [14] to the mixed-integer convex case, showing that for a given β, solutions to Problem 1 can be obtained by solving P -SP[K] with appropriate K = (K 1 , . .…”

“…The computational effort of solving P -SP[K] depends mainly on the constraints of P -SP[K]. Different metrics can be used to characterize this computational effort, e.g., by considering the number of constraints of P -SP[K], as proposed in [14]. In this work, we argue that the cost of evaluating the constraints should be considered explicitly, e.g., by considering the number of floating point operations (FLOPs) [11, p. 12] required to evaluate the constraints, or if the constraints are linear in x, the number of non-zero elements (NNZs) of the matrix encoding the constraints.…”

“…In order to reduce the computational cost of solving P -SP[K], similarly to [14], we make two simplifications:…”

“…A multiple chance constrained program (MCP), defined in [25], was constructed in [14] to minimize the number of constraints of the resulting scenario program. In this work, we focus on mixed-integer convex CCPs and further develop the perspective of [14]. We construct multiple chance constrained formulations of the original CCP such that the resulting scenario program has reduced computational cost.…”

“…The contributions are as follows: 1) We propose to optimize the partitioning of the constraints into multiple chance constraints, with the objective of minimizing the computational cost of the resulting scenario program. Differently from [14], where a fixed partition was considered, we also optimize over the partitioning, allowing to more effectively exploit the problem structure. Moreover, we explicitly consider minimizing the computational cost of the resulting scenario program, according to a given cost metric, e.g., the number of floating point operations required for evaluating the constraint.…”