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Exploiting Overlap Information in Chance-constrained Program with Random Right-hand Side

Wei Lv, Wei-Kun Chen, Yu-Hong Dai, Xiao-Jiao Tong

TL;DR

The paper tackles CCPs with discrete random RHS by identifying and mitigating overlap in the MILP-based branch-and-cut search tree that hampers solver performance. It introduces nonlinear if-then constraints to remove overlap and develops a practical dominance-based branching strategy, augmented by overlap-oriented node pruning and variable fixing, to create smaller, more tractable subproblems. Preprocessing to derive additional dominance relations further enhances branching effectiveness, and the authors embed these techniques in SCIP, demonstrating substantial reductions in tree size and considerable speedups on CCRP, CCMPP, and CCLS instances. The results indicate that overlap-aware strategies can significantly improve the practicality of MILP reformulations for CCPs, with scalable performance improvements and clear guidance for extensions to more general CCPs.

Abstract

We consider the chance-constrained program (CCP) with random right-hand side under a finite discrete distribution. It is known that the standard mixed integer linear programming (MILP) reformulation of the CCP is generally difficult to solve by general-purpose solvers as the branch-and-cut search trees are enormously large, partly due to the weak linear programming relaxation. In this paper, we identify another reason for this phenomenon: the intersection of the feasible regions of the subproblems in the search tree could be nonempty, leading to a wasteful duplication of effort in exploring the uninteresting overlap in the search tree. To address the newly identified challenge and enhance the capability of the MILP-based approach in solving CCPs, we first show that the overlap in the search tree can be completely removed by a family of valid nonlinear if-then constraints, and then propose two practical approaches to tackle the highly nonlinear if-then constraints. In particular, we use the concept of dominance relations between different scenarios of the random variables, and propose a novel branching, called dominance-based branching, which is able to create a valid partition of the problem with a much smaller overlap than the classic variable branching. Moreover, we develop overlap-oriented node pruning and variable fixing techniques, applied at each node of the search tree, to remove more overlaps in the search tree. Computational results demonstrate the effectiveness of the proposed dominance-based branching with the overlap-oriented node pruning and variable fixing techniques in reducing the search tree size and improving the overall solution efficiency.

Exploiting Overlap Information in Chance-constrained Program with Random Right-hand Side

TL;DR

The paper tackles CCPs with discrete random RHS by identifying and mitigating overlap in the MILP-based branch-and-cut search tree that hampers solver performance. It introduces nonlinear if-then constraints to remove overlap and develops a practical dominance-based branching strategy, augmented by overlap-oriented node pruning and variable fixing, to create smaller, more tractable subproblems. Preprocessing to derive additional dominance relations further enhances branching effectiveness, and the authors embed these techniques in SCIP, demonstrating substantial reductions in tree size and considerable speedups on CCRP, CCMPP, and CCLS instances. The results indicate that overlap-aware strategies can significantly improve the practicality of MILP reformulations for CCPs, with scalable performance improvements and clear guidance for extensions to more general CCPs.

Abstract

We consider the chance-constrained program (CCP) with random right-hand side under a finite discrete distribution. It is known that the standard mixed integer linear programming (MILP) reformulation of the CCP is generally difficult to solve by general-purpose solvers as the branch-and-cut search trees are enormously large, partly due to the weak linear programming relaxation. In this paper, we identify another reason for this phenomenon: the intersection of the feasible regions of the subproblems in the search tree could be nonempty, leading to a wasteful duplication of effort in exploring the uninteresting overlap in the search tree. To address the newly identified challenge and enhance the capability of the MILP-based approach in solving CCPs, we first show that the overlap in the search tree can be completely removed by a family of valid nonlinear if-then constraints, and then propose two practical approaches to tackle the highly nonlinear if-then constraints. In particular, we use the concept of dominance relations between different scenarios of the random variables, and propose a novel branching, called dominance-based branching, which is able to create a valid partition of the problem with a much smaller overlap than the classic variable branching. Moreover, we develop overlap-oriented node pruning and variable fixing techniques, applied at each node of the search tree, to remove more overlaps in the search tree. Computational results demonstrate the effectiveness of the proposed dominance-based branching with the overlap-oriented node pruning and variable fixing techniques in reducing the search tree size and improving the overall solution efficiency.
Paper Structure (28 sections, 10 theorems, 79 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 28 sections, 10 theorems, 79 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions and outline
  3. Notations and assumptions
  4. Overlaps arising in solving formulation \ref{['milp']}
  5. Overlaps in the search tree
  6. Removing the overlaps
  7. Dominance-based branching
  8. Description of the dominance-based branching
  9. Relation to the result in Ruszczynski2002
  10. Deriving more dominance relations
  11. An illustrative example
  12. Overlap-oriented node pruning and variable fixing
  13. The exact approach
  14. The polynomial-time approximation approach
  15. Deriving lower bounds for variables $v$
  16. ...and 13 more sections

Key Result

Theorem 2.4

Let $a=(\mathcal{N}_0^a, \mathcal{N}_1^a)$ and $b=(\mathcal{N}_0^b, \mathcal{N}_1^b)$ be any two nodes in the search tree of formulation minlp and $c$ be their first common ancestor. If $c$ differs from $a$ and $b$, then where $\mathcal{F}^{\text{\rm {MINLP}\xspace}}(\mathcal{N}_0^a, \mathcal{N}_1^a)$ and $\mathcal{F}^{\text{\rm {MINLP}\xspace}}(\mathcal{N}_0^b, \mathcal{N}_1^b)$ are the feasible

Figures (3)

  • Figure 1: The B&C search tree of the problem in \ref{['example1']} with the classic variable branching applied.
  • Figure 2: The B&C search tree of the problem in \ref{['example1']} with the proposed dominance-based branching applied.
  • Figure 3: The B&C search tree of the problem in \ref{['example1']} with the proposed dominance-based branching with the overlap-oriented variable fixing applied.

Theorems & Definitions (26)

  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1: Ruszczynski2002
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 16 more