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Theoretical and numerical comparison of seven single-level reformulations for bilevel programs

Yu-Wei Li, Gui-Hua Lin, Xide Zhu

TL;DR

This work systematically compares seven single-level reformulations for bilevel programs, including three new tight-feasibility formulations TWDP, TMDP, and eTMDP, against the traditional MPCC and Wolfe/Mond-Weir reformulations. It develops two projection-based algorithms (direct and relaxation) to solve these reformulations and tests them on 450 randomly generated bilevel problems spanning LP, QP, and QCQP lower-level constraints. Contrary to the conjecture that tighter feasible regions yield better performance, the results show that WDP/MDP/TWDP/TMDP generally outperform MPCC, while eMDP/eTMDP underperform among the duality-based reformulations, with relaxation methods delivering substantial gains (roughly 3–5x over MPCC). The study also demonstrates that TWDP/TMDP/eTMDP may satisfy MFCQ at feasible points, provides duality-theoretic guarantees, and outlines concrete directions for extending the approach to more general bilevel settings and related minimax problems.

Abstract

This paper considers a bilevel program. To solve this bilevel program, it is generally necessary to transform it into some single-level optimization problem. One approach is to replace the lower-level program by its KKT conditions to transform the bilevel program as a mathematical program with complementarity constraints (MPCC). Another approach is to apply the lower-level Wolfe/Mond-Weir/extended Mond-Weir duality to transform the bilevel program into some duality-based single-level reformulations, called WDP, MDP, and eMDP respectively in the literature. In this paper, inspired by a conjecture from a recent publication that the tighter feasible region of a reformulation, the better its numerical performance, we present three new duality-based single-level reformulations, called TWDP/TMDP/eTMDP, with tighter feasible regions. Our main goal is to compare all above-mentioned reformulations by designing some direct and relaxation algorithms with projection and implementing these algorithms on 450 test examples generated randomly. Our numerical experiments show that, whether overall comparison or pairwise comparison, at least in our tests, the WDP/MDP/TWDP/TMDP reformulations were always better than the MPCC reformulation, while the eMDP/eTMDP reformulations were always the worst ones among six duality-based reformulations, which indicates that the above conjecture is incorrect. In particular, for the relaxation algorithms, the WDP/MDP/TWDP/TMDP reformulations performed 3-5 times better than the MPCC reformulation, while the eMDP/eTMDP reformulations performed 2 times better than the MPCC reformulation.

Theoretical and numerical comparison of seven single-level reformulations for bilevel programs

TL;DR

This work systematically compares seven single-level reformulations for bilevel programs, including three new tight-feasibility formulations TWDP, TMDP, and eTMDP, against the traditional MPCC and Wolfe/Mond-Weir reformulations. It develops two projection-based algorithms (direct and relaxation) to solve these reformulations and tests them on 450 randomly generated bilevel problems spanning LP, QP, and QCQP lower-level constraints. Contrary to the conjecture that tighter feasible regions yield better performance, the results show that WDP/MDP/TWDP/TMDP generally outperform MPCC, while eMDP/eTMDP underperform among the duality-based reformulations, with relaxation methods delivering substantial gains (roughly 3–5x over MPCC). The study also demonstrates that TWDP/TMDP/eTMDP may satisfy MFCQ at feasible points, provides duality-theoretic guarantees, and outlines concrete directions for extending the approach to more general bilevel settings and related minimax problems.

Abstract

This paper considers a bilevel program. To solve this bilevel program, it is generally necessary to transform it into some single-level optimization problem. One approach is to replace the lower-level program by its KKT conditions to transform the bilevel program as a mathematical program with complementarity constraints (MPCC). Another approach is to apply the lower-level Wolfe/Mond-Weir/extended Mond-Weir duality to transform the bilevel program into some duality-based single-level reformulations, called WDP, MDP, and eMDP respectively in the literature. In this paper, inspired by a conjecture from a recent publication that the tighter feasible region of a reformulation, the better its numerical performance, we present three new duality-based single-level reformulations, called TWDP/TMDP/eTMDP, with tighter feasible regions. Our main goal is to compare all above-mentioned reformulations by designing some direct and relaxation algorithms with projection and implementing these algorithms on 450 test examples generated randomly. Our numerical experiments show that, whether overall comparison or pairwise comparison, at least in our tests, the WDP/MDP/TWDP/TMDP reformulations were always better than the MPCC reformulation, while the eMDP/eTMDP reformulations were always the worst ones among six duality-based reformulations, which indicates that the above conjecture is incorrect. In particular, for the relaxation algorithms, the WDP/MDP/TWDP/TMDP reformulations performed 3-5 times better than the MPCC reformulation, while the eMDP/eTMDP reformulations performed 2 times better than the MPCC reformulation.

Paper Structure

This paper contains 19 sections, 11 theorems, 68 equations, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. New reformulations with tighter feasible regions
  3. Duality theorems
  4. Equivalence between TWDP/TMDP/eTMDP and BP
  5. TWDP/TMDP/eTMDP may satisfy the MFCQ at feasible points
  6. Relationship among seven single-level reformulations
  7. Direct and relaxation algorithms with projection
  8. Direct algorithm with projection
  9. Relaxation algorithm with projection
  10. Numerical comparison of seven single-level reformulations
  11. Test problems
  12. Group I: bilevel programs with LP constraints
  13. Group II: bilevel programs with QP constraints
  14. Group III: bilevel programs with QCQP constraints
  15. Numerical comparison
  16. ...and 4 more sections

Key Result

Theorem 2.1

(Wolfe case) Assume that, for any given $x\in X$, $u\in\mathbb{R}^p_{+}$, and $v\in\mathbb{R}^q$, $L(x, \cdot, u, v)$ is pseudoconvex on $\mathcal{Z}_1(x,u,v)$. Then, the weak duality holds, i.e., Moreover, if $\mathrm{P}_x$ satisfies the Guignard constraint qualification (GCQ) at some solution $y_x\in S(x)$, there exists $(z_x,u_x,v_x)\in \mathcal{F}_1(x)$ such that the strong duality holds, i.e

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Example 2.1
  • Example 2.2
  • Theorem 3.1
  • Theorem 3.2
  • ...and 3 more