The Basic Properties And Algorithm Of A Linear Bilevel Programming Problem With Multiple Followers

A model of bilevel programming was introduced by Von Stackelberg in 1952 , and the bilevel linear programming can handle very availably actual decentralized management system . In this paper , a model of a linear bilevel programming problem with multiple followers(MFBLP) is proposed . Under the assumption that the constrained set is bounded and nonempty , we discussed the geometric properties of the feasible set for MFBLP model and Optimality conditions of MFBLP . And we proved that the feasible set is composed of the union of the faces of the constrained set and is a weak quasi-convex set , and we obtained two sufficient and necessary conditions about the Optimality condition of MFBLP problem by duality theory . Finally , we design a finer algorithms for the MFBLP problem by using these geometric properties and Optimality conditions of MFBLP in this paper . By analyzing example , we know that the theory and algorithm is legitimacy .This paper is divided into 5 sections . The first chapter contains the history , development and actuality of multilevel programming . In section 2 , we introduce some concepts and results which we'll use in the sequel. In section 3 , we introduce a model of MFBLP and some backgrounds . With the assumption that constraint sets are nonempty and bounded , we studied the geometric properties of feasible sets and the Optimality conditions of MFBLP . The results we obtained lay a foundation for solving this kind of problem . In section 4 , we propose an algorithm and some numerical experiments . Finally , we present an outline of the article and some prospects in the field .