Deterministic Equivalents for Optimizing and Satisficing under Chance Constraints

TLDR

Chance‑constrained programming permits random data variations and limited constraint violations, and under certain conditions can be reformulated as a deterministic problem by selecting appropriate decision rules and objectives. The authors prove that deterministic convex equivalents exist for linear decision rules under maximum‑expected‑value, minimum‑variance, and maximum‑probability objectives, and discuss how the P model enables comparison of satisficing strategies with traditional E and V objectives.

Abstract

Chance constrained programming admits random data variations and permits constraint violations up to specified probability limits. Different kinds of decision rules and optimizing objectives may be used so that, under certain conditions, a programming problem (not necessarily linear) can be achieved that is deterministic—in that all random elements have been eliminated. Existence of such “deterministic equivalents” in the form of specified convex programming problems is here established for a general class of linear decision rules under the following 3 classes of objectives (1) maximum expected value (“E model”), (2) minimum variance (“V model”), and (3) maximum probability (“P model”). Various explanations and interpretations of these results are supplied along with other aspects of chance constrained programming. For example, the “P model” is interpreted so that H. A. Simon's suggestions for “satisficing” can be studied relative to more traditional optimizing objectives associated with “E” and “V model” variants.