Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric and computational hardness of bilevel programming

Jérôme Bolte, Quoc-Tung Le, Edouard Pauwels, Samuel Vaiter

TL;DR

The paper investigates both geometric and computational hardness of bilevel programming. It shows that unconstrained $C^{\infty}$ bilevel problems can represent any extended-real-valued semi-algebraic function, making them as hard as general lower semicontinuous minimization, while polynomial bilevel problems exhibit expressivity depending on the lower-level constraints: unbounded boxes realize all SA functions, whereas convex lower-levels with bounded/compact sets yield the class $\mathcal{SA} \cap \mathcal{LSC} \cap \mathcal{CB}$ (with a broader equivalence $\mathcal{C}_{\text{cc}} = \mathcal{SA} \cap \mathcal{LSC} \cap \mathcal{CB}$ under convex-compact lower-level constraints). From a computational perspective, the decision version of polynomial bilevel optimization is shown to be $\Sigma^p_2$-hard through a reduction from the subset-sum-interval problem, indicating hardness beyond NP. These results collectively emphasize the irremediable difficulties of general bilevel optimization and motivate seeking regularity conditions or restricted sub-classes that admit tractable analysis and reliable algorithms.

Abstract

We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-algebraic function, possibly non-continuous, can be expressed as the value function of a polynomial bilevel program. Secondly, from a computational complexity perspective, the decision version of polynomial bilevel programming is one level above NP in the polynomial hierarchy ($Σ^p_2$-hard). Both types of difficulties are uncommon in non-linear programs for which objective functions are typically continuous and belong to the class NP. These results highlight the irremediable hardness attached to general bilevel optimization and the necessity of imposing some form of regularity on the lower level.

Geometric and computational hardness of bilevel programming

TL;DR

The paper investigates both geometric and computational hardness of bilevel programming. It shows that unconstrained bilevel problems can represent any extended-real-valued semi-algebraic function, making them as hard as general lower semicontinuous minimization, while polynomial bilevel problems exhibit expressivity depending on the lower-level constraints: unbounded boxes realize all SA functions, whereas convex lower-levels with bounded/compact sets yield the class (with a broader equivalence under convex-compact lower-level constraints). From a computational perspective, the decision version of polynomial bilevel optimization is shown to be -hard through a reduction from the subset-sum-interval problem, indicating hardness beyond NP. These results collectively emphasize the irremediable difficulties of general bilevel optimization and motivate seeking regularity conditions or restricted sub-classes that admit tractable analysis and reliable algorithms.

Abstract

We first show a simple but striking result in bilevel optimization: unconstrained smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-algebraic function, possibly non-continuous, can be expressed as the value function of a polynomial bilevel program. Secondly, from a computational complexity perspective, the decision version of polynomial bilevel programming is one level above NP in the polynomial hierarchy (-hard). Both types of difficulties are uncommon in non-linear programs for which objective functions are typically continuous and belong to the class NP. These results highlight the irremediable hardness attached to general bilevel optimization and the necessity of imposing some form of regularity on the lower level.
Paper Structure (17 sections, 18 theorems, 71 equations, 1 figure, 4 tables)

This paper contains 17 sections, 18 theorems, 71 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Geometric hardness of polynomial bilevel optimization
  3. Preliminaries
  4. Value functions of general polynomial bilevel programs
  5. Polynomial bilevel problems with convex lower-level
  6. Extension to arbitrary convex, compact, semi-algebraic lower-level constraints
  7. Summary of the results and pessimistic bilevel problems
  8. Computational hardness of polynomial bilevel optimization
  9. Preliminaries on the polynomial hierarchy
  10. Polynomial bilevel optimization is $\Sigma_2^p$-hard
  11. Observation 1
  12. Observation 2
  13. Conclusion
  14. The difficulty of smooth bilevel optimization
  15. Connection to existing work
  16. ...and 2 more sections

Key Result

Theorem 1.1

Any proper lower semicontinuous function $f:\mathbb{R}^n \to \mathbb{R}\cup \{+\infty\}$ whose domain $\mathrm{dom}(f):= \{x \in \mathbb{R}^n \mid f(x) \in \mathbb{R}\}$ is closed, is the value function of a bilevel problem with $C^\infty$ smooth upper and lower levels $P,Q$ and $\mathcal{Y} = \math

Figures (1)

  • Figure 1: Diagram describes the polynomial hierarchy. Arrows represent the inclusion relation.

Theorems & Definitions (38)

  • Theorem 1.1: Lower semicontinuous minimization problems are $C^\infty$ bilevel problems
  • Definition 2.1: Semi-algebraic sets and functions
  • Definition 2.2: Extended-real-valued semi-algebraic functions
  • Theorem 2.3: Tarski-Seidenberg theorem
  • Proposition 2.4: Characterization of closed semi-algebraic sets benedetti1990real
  • Proposition 2.5: Growth of semi-algebraic functions o-minimal-structures
  • Proposition 2.6: Semi-algebraicity of value functions
  • Definition 2.7: Outer semicontinuity
  • Theorem 2.8: Berge's Maximum Theorem maximum_theorem
  • Corollary 2.9: Semicontinuity of bilevel value functions with compact lower-level feasible sets
  • ...and 28 more