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Cardinality Constraints in Single-Leader-Multi-Follower games

Didier Aussel, Daniel Lasluisa, David Salas

Abstract

This work explores bilevel problems in the context of cardinality constraints. More specifically Single-Leader-Multi-Follower games (SLMFG) involving cardinality constraints are considered in two different configurations: one with the cardinality constraint at the leader's level and a mixed structure in which the cardinality constraint is split between leader and followers problem. We prove existence results in both cases and provided equivalent reformulations allowing the numerical treatment of these complex problems. The obtained results are illustrated thanks to an application to a facility location problem.

Cardinality Constraints in Single-Leader-Multi-Follower games

Abstract

This work explores bilevel problems in the context of cardinality constraints. More specifically Single-Leader-Multi-Follower games (SLMFG) involving cardinality constraints are considered in two different configurations: one with the cardinality constraint at the leader's level and a mixed structure in which the cardinality constraint is split between leader and followers problem. We prove existence results in both cases and provided equivalent reformulations allowing the numerical treatment of these complex problems. The obtained results are illustrated thanks to an application to a facility location problem.
Paper Structure (17 sections, 6 theorems, 56 equations, 3 figures, 4 tables)

This paper contains 17 sections, 6 theorems, 56 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaties and notation
  3. Constraint qualification for convex optimization problems:
  4. Single-Leader-Multi-Follower games:
  5. SLMF games with upper-level cardinality constraints
  6. Existence of solutions
  7. Reformulations
  8. SLMF games with mixed cardinality constraints
  9. Existence of solutions and MPCC reformulation
  10. Single-level reformulations for the linear lower-level problems
  11. $(b)\implies (c)$:
  12. $(c)\implies (b)$:
  13. Application to Facility location problems with cardinality constraints
  14. Upper-level and mixed formulations
  15. Calibration of numerical experiments
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $X\subset \mathbb{R}^{n}$ be a closed set, and consider a set-valued map $F:X\mathop{\rightrightarrows}\nolimits \mathbb{R}^m$ given by where, for each $k\in[m]$, $\varphi_k:\mathbb{R}^n\to\mathbb{R}$ are continuous functions and the functions $\{g_k\}_{k\in[m]}$ are continuous convex and weakly analytic, then the set-valued map $F$ is lower semicontinuous.

Figures (3)

  • Figure 1: Optimality gap for each configuration in each case were at least one feasible solution was found. For the cases where no feasible solution was found, they are not displayed since the gap is $+\infty$.
  • Figure 2: Each bar corresponds to the number of built facilities of type $a_1$, $a_2$ or $a_3$, for cases 1 and 5. The left bar corresponds to Upper configuration while the right bar corresponds to Mixed configuration.
  • Figure 3: Boxplot of the values for the different objective functions of the leader, for each configuration.

Theorems & Definitions (17)

  • Proposition 2.1: bank1983nonlinear
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Example 3.3: Infeasibility at the upper level problem by the cardinality constraint
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • Remark 3.6
  • Example 3.7
  • ...and 7 more