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A Surrogate Value Function Formulation for Bilevel Optimization

Mengwei Xu, Yu-Hong Dai, Xin-Wei Liu, Meiqi Ma

TL;DR

This paper tackles bilevel optimization in regimes where lower‑level degeneracy undermines traditional VP and KP reformulations. It proposes a Surrogate Value Function (SVF) that replaces the implicit lower‑level value function with an explicit surrogate anchored at a lower‑level stationary point and enforces hierarchy via a dominance constraint, preserving the hierarchical structure while remaining computationally tractable. The SVF is shown to be equivalent to the original BP under pseudoconvexity and related regularity assumptions, with strong connections to existing stationarity concepts; a smoothing barrier augmented Lagrangian (SBAL) method is developed to handle complementarity and prove convergence to solutions and Clarke stationary points. Empirical results on 109 benchmarks, including the Mirrlees problem, demonstrate that SVF solved with SBAL outperforms KP and other reformulations in robustness and precision, particularly in nonconvex settings where KKT‑based models fail.

Abstract

The value function formulation captures the hierarchical nature of bilevel optimization through the optimal value function of the lower level problem, yet its implicit and nonsmooth characteristics pose significant analytical and computational difficulties. We introduce a surrogate value function formulation that replaces the intractable value function with an explicit surrogate derived from lower level stationarity conditions. This surrogate formulation preserves the essential idea of the classical value function model but fundamentally departs from Karush Kuhn Tucker (KKT) formulations, which embed lower level stationary points into the upper level feasible region and obscure the hierarchical dependence. Instead, it enforces the hierarchy through a dominance constraint that remains valid even when lower level constraint qualifications fail at the solution. We establish equivalence with the original bilevel problem, reveal the failure of standard constraint qualifications, and show that its strong stationarity implies that of KKT models. To handle the complementarity constraints in the surrogate formulation, we apply a smoothing barrier augmented Lagrangian method and prove its convergence to solutions and Clarke stationary points. Extensive experiments demonstrate the robustness and high numerical precision of this formulation, especially in nonconvex settings, including the classical Mirrlees problem where KKT models fail.

A Surrogate Value Function Formulation for Bilevel Optimization

TL;DR

This paper tackles bilevel optimization in regimes where lower‑level degeneracy undermines traditional VP and KP reformulations. It proposes a Surrogate Value Function (SVF) that replaces the implicit lower‑level value function with an explicit surrogate anchored at a lower‑level stationary point and enforces hierarchy via a dominance constraint, preserving the hierarchical structure while remaining computationally tractable. The SVF is shown to be equivalent to the original BP under pseudoconvexity and related regularity assumptions, with strong connections to existing stationarity concepts; a smoothing barrier augmented Lagrangian (SBAL) method is developed to handle complementarity and prove convergence to solutions and Clarke stationary points. Empirical results on 109 benchmarks, including the Mirrlees problem, demonstrate that SVF solved with SBAL outperforms KP and other reformulations in robustness and precision, particularly in nonconvex settings where KKT‑based models fail.

Abstract

The value function formulation captures the hierarchical nature of bilevel optimization through the optimal value function of the lower level problem, yet its implicit and nonsmooth characteristics pose significant analytical and computational difficulties. We introduce a surrogate value function formulation that replaces the intractable value function with an explicit surrogate derived from lower level stationarity conditions. This surrogate formulation preserves the essential idea of the classical value function model but fundamentally departs from Karush Kuhn Tucker (KKT) formulations, which embed lower level stationary points into the upper level feasible region and obscure the hierarchical dependence. Instead, it enforces the hierarchy through a dominance constraint that remains valid even when lower level constraint qualifications fail at the solution. We establish equivalence with the original bilevel problem, reveal the failure of standard constraint qualifications, and show that its strong stationarity implies that of KKT models. To handle the complementarity constraints in the surrogate formulation, we apply a smoothing barrier augmented Lagrangian method and prove its convergence to solutions and Clarke stationary points. Extensive experiments demonstrate the robustness and high numerical precision of this formulation, especially in nonconvex settings, including the classical Mirrlees problem where KKT models fail.
Paper Structure (13 sections, 11 theorems, 60 equations, 3 figures, 1 table)

This paper contains 13 sections, 11 theorems, 60 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and An Illustrative Example
  3. Stationarity Concepts
  4. An illustrative Counterexample
  5. Analysis of the SVF Formulation
  6. Optimality and Stationarity Analysis
  7. Stationarity Relationships with Existing Formulations
  8. Smoothing‑Barrier Augmented Lagrangian Approximation
  9. Numerical experiments
  10. Algorithmic Framework
  11. Experimental Setup
  12. Results and Analysis
  13. Conclusion

Key Result

Proposition 2.1

m1yw (i) If $S$ is inner semicontinuous at $x^*$ for some $y^*\in S(x^*)$, i.e., for every $x_k\to x^*$, there is a sequence $y_k\in S(x_k)$ converging to $y^*$ and the MFCQ holds at $y^*$, then the value function $V(x)$ is Lipschitz continuous near $x^*$ and where $W_{y^*}(x^*):= \left \{\nabla_{1} f(x^*, y^*)+s^T \nabla_{1} g(x^*,y^*): s\in M(x^*, y^*) \right \}$ and (ii) Assume that $Y(x)$ i

Figures (3)

  • Figure 1: Feasible region of the lower level problem in Example \ref{['Counterexample']} at $x=0$
  • Figure 2: Performance profiles
  • Figure :

Theorems & Definitions (14)

  • Proposition 2.1
  • Definition 2.1
  • Example 2.1
  • Theorem 3.1
  • Definition 3.1: MPEC-NNAMCQ
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 4 more