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A Sensitivity-Based Method for Bilevel Optimization Problems: Theoretical Analysis and Computational Performance

Eduardo Nolasco, Ross D. King, Vassilios S. Vassiliadis

TL;DR

This work tackles deterministic, optimistic bilevel optimization by treating the lower-level solution as an implicit function $ar{y}(x)$ and solving the resulting problem within an augmented Lagrangian framework. A sensitivity analysis of the lower level yields the total gradient of the upper-level objective with respect to $x$, enabling a gradient-based outer-loop update without single-level reformulations. The method combines a robust inner solver ($L ext{-}BFGS ext{-}B$) with an ALM-based outer loop and proves convergence to a KKT point of the implicit problem, with equivalence to S-stationarity under standard MPCC regularity. Computational tests on benchmark problems, including Clark–Westerberg and BOLIB instances, demonstrate efficiency, robustness, and the practical value of a dual-criterion stopping condition to address asymmetric primal–dual convergence rates.

Abstract

Bilevel optimization provides a powerful framework for modeling hierarchical decision-making systems. This work presents a novel sensitivity-based algorithm that directly addresses the bilevel structure by treating the lower-level optimal solution as an implicit function of the upper-level variables, thus avoiding classical single-level reformulations. This implicit problem is solved within a robust Augmented Lagrangian framework, where the inner subproblems are managed by a quasi-Newton (L-BFGS-B) solver to handle the ill-conditioned and non-smooth landscapes that can arise. The validity of the proposed method is established through both theoretical convergence guarantees and extensive computational experiments. These experiments demonstrate the algorithm's efficiency and robustness and validate the use of a pragmatic dual-criterion stopping condition to address the practical challenge of asymmetric primal-dual convergence rates.

A Sensitivity-Based Method for Bilevel Optimization Problems: Theoretical Analysis and Computational Performance

TL;DR

This work tackles deterministic, optimistic bilevel optimization by treating the lower-level solution as an implicit function and solving the resulting problem within an augmented Lagrangian framework. A sensitivity analysis of the lower level yields the total gradient of the upper-level objective with respect to , enabling a gradient-based outer-loop update without single-level reformulations. The method combines a robust inner solver () with an ALM-based outer loop and proves convergence to a KKT point of the implicit problem, with equivalence to S-stationarity under standard MPCC regularity. Computational tests on benchmark problems, including Clark–Westerberg and BOLIB instances, demonstrate efficiency, robustness, and the practical value of a dual-criterion stopping condition to address asymmetric primal–dual convergence rates.

Abstract

Bilevel optimization provides a powerful framework for modeling hierarchical decision-making systems. This work presents a novel sensitivity-based algorithm that directly addresses the bilevel structure by treating the lower-level optimal solution as an implicit function of the upper-level variables, thus avoiding classical single-level reformulations. This implicit problem is solved within a robust Augmented Lagrangian framework, where the inner subproblems are managed by a quasi-Newton (L-BFGS-B) solver to handle the ill-conditioned and non-smooth landscapes that can arise. The validity of the proposed method is established through both theoretical convergence guarantees and extensive computational experiments. These experiments demonstrate the algorithm's efficiency and robustness and validate the use of a pragmatic dual-criterion stopping condition to address the practical challenge of asymmetric primal-dual convergence rates.

Paper Structure

This paper contains 15 sections, 2 theorems, 38 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. BLP formulation and classical solution approaches
  3. Bilevel optimization problems formulation
  4. Classical solution methods
  5. The Sensitivity-Based Solution Method
  6. The Implicit Upper-Level Problem
  7. Sensitivity Analysis of the Lower Level
  8. Total Gradient Computation for the Upper Level
  9. The Proposed Algorithm
  10. Convergence Analysis
  11. Equivalence to S-Stationarity
  12. Computational Tests
  13. Implementation Details
  14. Test Problems
  15. Conclusions

Key Result

Theorem 3.1

Let $\{x_k, \mu_k \}$ be a sequence of iterates generated by Algorithm alg:MainAlg, with the upper-level update step given by Algorithm alg:ALM_revised. Assume that: Then any limit point of the sequence $\{x_k, \mu_k \}$ is a KKT point of eq:Implicit_BLP.

Figures (5)

  • Figure 1: High-level schematic of the overall sensitivity-based Augmented Lagrangian framework, corresponding to Algorithm \ref{['alg:MainAlg']}.
  • Figure 2: Workflow for the ALM subproblem solution (Algorithm \ref{['alg:ALM_revised']}). At each outer iteration, a NLP solver is used to find an approximate minimizer of the implicit Augmented Lagrangian function.
  • Figure 3: Detailed workflow for the implicit objective and gradient evaluation. This multi-step process, which includes solving the lower-level NLP and the linear sensitivity system, is called at each iteration of the inner-loop.
  • Figure 4: The implicit upper-level objective $F(x,\bar{y}(x))$ and the lower-level optimal response $\bar{y}(x)$ as a function of the upper-level variable $x$ for the ClarkWesterberg1990 problem. The local and global optima are highlighted.
  • Figure 5: Convergence behavior for the Outrata_Cervinka_2009 problem. The algorithm converges in a few iterations as the KKT residual drops below the tolerance $\epsilon = 10^{-5}$.

Theorems & Definitions (4)

  • Theorem 3.1: Convergence to a KKT point
  • proof
  • Theorem 3.2: Equivalence to S-stationarity
  • proof