A Unifying Convexification Framework for Chance-Constrained Programs via Bilinear Extended Formulations over a Simplex

Abstract: Chance-constrained programming is a widely used framework for decision-making under uncertainty, yet its mixed-integer reformulations involve nonconvex mixing sets with a knapsack constraint, leading to weak relaxations and computational challenges. Most existing approaches for strengthening the relaxations of these sets rely primarily on extensions of a specific class of valid inequalities, limiting both convex hull coverage and the discovery of fundamentally new structures. In this paper, we develop a novel convexification framework that reformulates chance-constrained sets as bilinear sets over a simplex in a lifted space and employs a step-by-step aggregation procedure to derive facet-defining inequalities in the original space of variables. Our approach generalizes and unifies established families of valid inequalities in the literature while introducing new ones that capture substantially larger portions of the convex hull. Main contributions include: (i) the development of a new aggregation-based convexification technique for bilinear sets over a simplex in a lower-dimensional space; (ii) the introduction of a novel bilinear reformulation of mixing sets with a knapsack constraint -- arising from single-row relaxations of chance constraints -- over a simplex, which enables the systematic derivation of strong inequalities in the original variable space; and (iii) the characterization of facet-defining inequalities within a unified framework that contains both existing and new families. Preliminary computational experiments demonstrate that our inequalities describe over 90\% of the facet-defining inequalities of the convex hull of benchmark instances, significantly strengthening existing relaxations and advancing the polyhedral understanding of chance-constrained programs.