Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Danial Davarnia, Hamed Rahimian

TL;DR

This paper tackles the polyhedral complexity of chance-constrained programs (CCPs) by focusing on the nonconvex mixing set with knapsack constraints. It introduces a unifying convexification framework that reformulates the mixing set as a bilinear set over a simplex in a lifted space and applies a step-by-step bilinear lift-and-project (BL&P) aggregation to derive facet-defining inequalities in the original variable space. The BL&P framework subsumes many existing valid-inequality families and generates new classes that capture a substantially larger portion of the convex hull, enabling stronger relaxations. Computational experiments show the proposed inequalities describe over 90% of the facets of benchmark convex hulls, indicating meaningful improvements in tightening CCP relaxations and advancing polyhedral understanding. The approach offers a cohesive path toward extending convexification to broader CCP variants, including intersections of multiple mixing sets and multi-row relaxations.

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.

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

TL;DR

This paper tackles the polyhedral complexity of chance-constrained programs (CCPs) by focusing on the nonconvex mixing set with knapsack constraints. It introduces a unifying convexification framework that reformulates the mixing set as a bilinear set over a simplex in a lifted space and applies a step-by-step bilinear lift-and-project (BL&P) aggregation to derive facet-defining inequalities in the original variable space. The BL&P framework subsumes many existing valid-inequality families and generates new classes that capture a substantially larger portion of the convex hull, enabling stronger relaxations. Computational experiments show the proposed inequalities describe over 90% of the facets of benchmark convex hulls, indicating meaningful improvements in tightening CCP relaxations and advancing polyhedral understanding. The approach offers a cohesive path toward extending convexification to broader CCP variants, including intersections of multiple mixing sets and multi-row relaxations.

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.
Paper Structure (11 sections, 18 theorems, 48 equations, 2 figures, 2 tables)

This paper contains 11 sections, 18 theorems, 48 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background on Chance-Constrained Programming
  3. MIP Reformulation of CCP
  4. Existing Families of Valid Inequalities in Closed Form
  5. A Convexification Procedure for Bilinear Sets over a Simplex
  6. Convexification of CCP through Bilinear Extended Reformulation
  7. Bilinear Reformulation
  8. Valid Inequalities for the Mixing Set with a Knapsack Constraint
  9. Valid Inequalities for the Mixing Set with a Cardinality Constraint
  10. Computational Experiments
  11. Conclusion and Future Research

Key Result

Proposition 1

gunluk2001mixingatamturk2000mixedguan2007sequential Consider $\{t_{1}, \cdots, t_{l}\} \subseteq M$ with $t_1 < \cdots < t_l$. Then, the star inequalities where $h_{t_{l+1}}:=0$, are valid for the mixing set eq:mixing.

Figures (2)

  • Figure 1: Relationship between existing inequalities (Propositions \ref{['prop:mixing']}--\ref{['prop:chance_lift_permutation']}) and BL&P inequalities (Theorems \ref{['thm:chance_lift_generic']}) in characterizing $\mathop{\rm conv}(\mathcal{F}_c)$ under a general probability distribution.
  • Figure 2: Relationship between existing inequalities (Propositions \ref{['prop:mixing']}--\ref{['prop:chance_lift_permutation']}) and BL&P inequalities (Theorems \ref{['thm:chance_lift_generic']}--\ref{['thm: chance_liftVIAgeneric_permutation']}) in characterizing $\mathop{\rm conv}(\mathcal{F}_c^=)$ under a uniform probability distribution.

Theorems & Definitions (42)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Example 1
  • Claim 1
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 32 more