Solving bilevel programs with the KKT-approach

Gemayqzel Bouza Allende,Georg Still

doi:10.1007/s10107-012-0535-x

Gemayqzel Bouza Allende, Georg Still

Open Access

https://doi.org/10.1007/s10107-012-0535-x

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Abstract

Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT- or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically.

Highlights

In the present article we consider bilevel problems (BL) of the form: PB L : min f (x, y) x,y s.t. (x, y) ∈ MBL (1.1)where x ∈ Rn, y ∈ Rm and the feasible set is given by G
An appealing way to deal with general BL’s is the so called Karush-Kuhn-Tucker (KKT) approach where the lower level constraint, that y is a global minimizer of the program Q(x), is firstly relaxed to the condition that y is a local minimizer of Q(x)
The whole investigations lead to an algorithmic approach for solving BL which is described in Sect. 5 along with some numerical experiments

Summary

Introduction

In the present article we consider bilevel problems (BL) of the form: PB L : min f (x, y) x,y s.t. (x, y) ∈ MBL (1.1). An appealing way to deal with general BL’s is the so called Karush-Kuhn-Tucker (KKT) approach where the lower level constraint, that y is a global minimizer of the program Q(x), is firstly relaxed to the condition that y is a local minimizer of Q(x). KKTBL Note that in [12] it has been shown (for n = 1) that the inclusion MBL ⊂ MKKTBL|Rn×Rm holds generically Both problems P , KKTBL PFJBL represent specially structured mathematical programs with complementarity constraints (MPCC). The consequences of these results in terms of the original BL problem are discussed in Sect. The whole investigations lead to an algorithmic approach for solving BL which is described in Sect. 5 along with some numerical experiments

Preliminaries

Genericity analysis of the KKT approach

Interpretation of the results in terms of PBL

A numerical approach for solving BL