2005
DOI: 10.1016/j.jmaa.2004.09.033
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Weak linear bilevel programming problems: existence of solutions via a penalty method

Abstract: We are concerned with a class of weak linear bilevel programs with nonunique lower level solutions. For such problems, we give via an exact penalty method an existence theorem of solutions. Then, we propose an algorithm.  2004 Elsevier Inc. All rights reserved.

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Cited by 44 publications
(44 citation statements)
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“…Even if all the functions are linear and the feasible sets are polyhedron, this problem remains non-convex. For this reason, this problem has received a large attention especially for the linear case, see for example Aboussoror and Mansouri (2005), Campelo et al (2000). For an extensive bibliography the reader can refer to Dempe (2003), Vicente and Calamai (1994).…”
Section: Introductionmentioning
confidence: 99%
“…Even if all the functions are linear and the feasible sets are polyhedron, this problem remains non-convex. For this reason, this problem has received a large attention especially for the linear case, see for example Aboussoror and Mansouri (2005), Campelo et al (2000). For an extensive bibliography the reader can refer to Dempe (2003), Vicente and Calamai (1994).…”
Section: Introductionmentioning
confidence: 99%
“…Pessimistic bilevel problem is generally formulated as a three level problem, as in (1)(2)(3). Apparently, it is more complicated than its bilevel optimistic counterpart OBL.…”
Section: A Tight Relaxation For Level Reductionmentioning
confidence: 99%
“…Under some sufficient conditions and a basic assumption (P), we show that the problem (S) admits solutions. As an application, we obtain the result given in [2] for a class of weak nonlinear bilevel problems. Indeed, this class corresponds to the case where M(x) is a set of solutions to another optimization problem parameterized by x.…”
Section: Introductionmentioning
confidence: 99%
“…Note that such a class of nonlinear bilevel problems presents a major difficulty in finding sufficient conditions that ensure the existence of solutions (comments and an exhaustive list are given in [9]). Sufficient condition for the existence of solutions to weak bilevel problems are given in [1][2][3]16]. Finally, we give some cases where the property (P) introduced for Min Sup problems is satisfied.…”
Section: Introductionmentioning
confidence: 99%