Get Bilevel Programming Problems: Theory, Algorithms and PDF

By Stephan Dempe, Vyacheslav Kalashnikov, Gerardo A. Pérez-Valdés, Nataliya Kalashnykova

ISBN-10: 3662458268

ISBN-13: 9783662458266

ISBN-10: 3662458276

ISBN-13: 9783662458273

This booklet describes contemporary theoretical findings appropriate to bilevel programming usually, and in mixed-integer bilevel programming particularly. It describes contemporary functions in power difficulties, corresponding to the stochastic bilevel optimization methods utilized in the common fuel undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are awarded and explained.

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Additional resources for Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks

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3) is a linear optimization problem. Thus, its optimal solution can be found at a vertex of the set {(x, y) : C y ≤ x, Ax ≤ c}. 3) has an optimal solution, at least one global optimal solution occurs at a vertex of the set {(x, y) : C y ≤ x, Ax ≤ c}. g. Bard [10]). 1) with upper level connecting constraints is considered. As it can be seen in Fig. 2, the bilevel optimization problem is a nonconvex optimization problem, it has a feasible set which is not given by explicit constraints. As a result, besides a global optimal solution bilevel optimization problems can have local extrema and stationary solutions which are not local optimal solutions.

17), it is only necessary to consider possible index sets I ∈ I (y). 17) if and only if y is a (global) optimal solution for all problems (A I ): F(y) → min y,b Ay = b y ≥ 0 yi = 0 ∀i ∈ I Bb = b with I ∈ I (y). 17) and assume that there is a set I ∈ I (y) with y being not optimal for (A I ). Then there exists a sequence {y k }∞ k=1 of feasible solutions of (A I ) with limk→∞ y k = y and F(y k ) < F(y) for all k. 18). 17) with lim k→∞ y = y and k F(y ) < F(y) for all k. 17) there the condition yik > 0 for all i ∈ are sets I ∈ I (y) such that y k is feasible for problem (A I ).

Take an arbitrary vertex y of the set {y : Ay = b, y ≥ 0}. Then, by parametric linear optimization, there exists c such that Ψ (b, c) = {y} for all c sufficiently close to c, formally ∀ c ∈ U (c) for some open neighborhood U (c) of c. Hence, if U (c) ∩ C = ∅, there exists z satisfying A z ≤ c, y (A z − c) = 0 for some c ∈ U (c) ∩ C such that (y, z, b, c) is a local optimal solution of the problem F(y) → min y,z,b,c Ay = b, y ≥ 0, A z ≤ c, y (A z − c) = 0 Bb = b, Cc = c. 2 Optimality Conditions 29 _T c x = const.

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Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks by Stephan Dempe, Vyacheslav Kalashnikov, Gerardo A. Pérez-Valdés, Nataliya Kalashnykova


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