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Introduction to Bilevel Optimization: A perspective from Variational Analysis

David Salas

TL;DR

This collection of notes reframes bilevel optimization through variational and set-valued analysis, tracing the leader–follower structure from Stackelberg origins to modern formulations. It formalizes optimistic and pessimistic formulations, and specializes to linear bilevel problems, illustrating with classic applications such as toll pricing, security games, and electricity dispatch. Existence results are developed via set-valued continuity concepts and Berge’s Maximum Theorem, linking solvability to the semicontinuity of the follower’s reaction and the compactness of feasible sets. The text further extends to Mixed-Integer bilevel programming, bilevel games, and Bayesian approaches, outlining assumptions under which solutions exist and highlighting computational and theoretical challenges. Overall, the work provides a rigorous, variational-analysis perspective on the core structure, solvability, and extensions of bilevel optimization with practical relevance to economics, engineering, and operations research.

Abstract

Lecture Notes based on the course given at Toulouse School of Economics, on Fall 2024. It contains a quick introduction to the field of bilevel optimization, following a perspective from Variational Analysis.

Introduction to Bilevel Optimization: A perspective from Variational Analysis

TL;DR

This collection of notes reframes bilevel optimization through variational and set-valued analysis, tracing the leader–follower structure from Stackelberg origins to modern formulations. It formalizes optimistic and pessimistic formulations, and specializes to linear bilevel problems, illustrating with classic applications such as toll pricing, security games, and electricity dispatch. Existence results are developed via set-valued continuity concepts and Berge’s Maximum Theorem, linking solvability to the semicontinuity of the follower’s reaction and the compactness of feasible sets. The text further extends to Mixed-Integer bilevel programming, bilevel games, and Bayesian approaches, outlining assumptions under which solutions exist and highlighting computational and theoretical challenges. Overall, the work provides a rigorous, variational-analysis perspective on the core structure, solvability, and extensions of bilevel optimization with practical relevance to economics, engineering, and operations research.

Abstract

Lecture Notes based on the course given at Toulouse School of Economics, on Fall 2024. It contains a quick introduction to the field of bilevel optimization, following a perspective from Variational Analysis.

Paper Structure

This paper contains 36 sections, 17 theorems, 156 equations, 11 figures, 1 table.

Table of Contents

  1. The Model of Bilevel Optimization
  2. Stackelberg Duopoly
  3. The model: Optimistic and Pessimistic formulations
  4. Optimistic Approach.
  5. Pessimistic Approach.
  6. Linear bilevel problems
  7. Some Applications
  8. Toll installation problem
  9. Security games
  10. Dispatch problem in electricity markets
  11. Problems
  12. Existence of Solutions
  13. Preliminaries on Set-Valued Analysis
  14. Upper and Lower semicontinuity
  15. Characterizations of upper and lower semicontinuity
  16. ...and 21 more sections

Key Result

Proposition 2.1.7

Let $X,Y$ be two topological spaces and $M:X\mathop{\rightrightarrows}\nolimits Y$ be a set-valued map. Then,

Figures (11)

  • Figure 1: Graphs of $S(x)$ and $\theta(x)$, which are not functions.
  • Figure 2: Function $\varphi^o$ (right) and $\varphi^p$ (left) for the problem of Example \ref{['ch02:example:MultipleSolutionsFollower']}.
  • Figure 3: Illustration of Examples \ref{['ch03:ex:SetValuedMaps']}.1 (left) and \ref{['ch03:ex:SetValuedMaps']}.2 (right).
  • Figure 4: A set-valued map identified as the slices of its graph.
  • Figure 5: Illustration of gap between Upper and Lower semicontinuity.
  • ...and 6 more figures

Theorems & Definitions (50)

  • Definition 1.2.1: Bilevel Programming problem
  • Example 1.2.2
  • Definition 1.2.3: Optimistic Bilevel Programming problem
  • Definition 1.2.4: Pessimistic Bilevel Programming problem
  • Example 1.3.1
  • Definition 2.1.1: Set-Valued map
  • Example 2.1.2
  • Definition 2.1.3: Upper semicontinuity
  • Definition 2.1.4: Lower semicontinuity
  • Example 2.1.5: Gap between Upper and Lower semicontinuity
  • ...and 40 more