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On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

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Gaoxi Li at Chongqing Technology and Business University
Gaoxi Li
Zhongping Wan at Wuhan University
Zhongping Wan
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Abstract

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.
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Journal of Optimization Theory and Applications (2018) 179:820–837
https://doi.org/10.1007/s10957-018-1392-4
On Bilevel Programs with a Convex Lower-Level Problem
Violating Slater’s Constraint Qualification
Gaoxi Li1,2 ·Zhongping Wan2
Received: 27 October 2017 / Accepted: 11 September 2018 / Published online: 21 September 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract
This paper focuses on bilevel programs with a convex lower-level problem violating
Slater’s constraint qualification. We relax the constrained domain of the lower-level
problem. Then, an approximate solution of the original bilevel program can be obtained
by solving this perturbed bilevel program. As the lower-level problem of the perturbed
bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a
mathematical program with complementarity constraints which can be solved by stan-
dard algorithms. The lower convergence properties of the constraint set mapping and
the solution set mapping of the lower-level problem of the perturbed bilevel program
are expanded. We show that the solutions of a sequence of the perturbed bilevel pro-
grams are convergent to that of the original bilevel program under some appropriate
conditions. And this convergence result is applied to simple trilevel programs.
Keywords Nonlinear programs ·Bilevel programs ·Slater’s constraint qualification ·
Complementarity constraints ·Lower convergence
Mathematics Subject Classification 90C33 ·90C30
1 Introduction
Bilevel program (BP) is an active research area in mathematical programs at present.
This model has a framework to deal with decision processes involving two decision
makers with a hierarchical nested structure. The upper-level decision maker (leader)
BGaoxi Li
ligaoxicn@126.com
Zhongping Wan
mathwanzhp@whu.edu.cn
1School of Mathematics and Statistics, Chongqing Technology and Business University,
Chongqing, China
2School of Mathematics and Statistics, Wuhan University, Wuhan, China
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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