New PDF release: Bilevel Programming Problems: Theory, Algorithms and

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

ISBN-10: 3662458276

ISBN-13: 9783662458273

This e-book describes contemporary theoretical findings proper to bilevel programming more often than not, and in mixed-integer bilevel programming particularly. It describes fresh purposes in power difficulties, comparable to the stochastic bilevel optimization methods utilized in the normal gasoline undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are offered and explained.

From the again Cover

This booklet describes fresh theoretical findings correct to bilevel programming ordinarily, and in mixed-integer bilevel programming specifically. It describes fresh purposes in power difficulties, reminiscent of the stochastic bilevel optimization methods utilized in the usual gasoline undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are awarded and explained.

About the Author

Stephan Dempe studied arithmetic on the Technische Hochschule Karl-Marx-Stadt and bought a PhD from a similar collage. this day he's professor for mathematical optimization on the TU Bergakademie Freiberg, Germany. concentration of his paintings is on parametric and nonconvex optimization.

Vyacheslav Kalashnikov studied arithmetic at Novosibirsk nation collage, he received his PhD in Operations study from the Siberian department of the Academy of Sciences of the USSR and his Dr.Sc. (Habilitation measure) from the imperative Economics and arithmetic Institute (CEMI), Moscow, Russia. this day he's Professor at Tecnológico de Monterrey, Mexico, on the CEMI, and at Sumy country college, Ukraine. the most components of his paintings are bilevel programming, hierarchical video games and their purposes in engineering and economics.

Gerardo Alfredo Perez Valdes studied arithmetic on the Universidad Autónoma de Nuevo León and acquired his PhDs in Engineering from Tecnológico de Monterrey, Mexico, and from Texas Tech college, Lubbock, united states. this present day he's Professor at collage of technology and expertise in Trondheim (NTNU), Norway. the point of interest of his paintings is on resolution algorithms in mathematical optimization.

Nataliya Kalashnykova studied arithmetic at Novosibirsk country college and obtained her PhD in Operations study from the Siberian department of the Academy of Sciences of the USSR. at the present time she is Professor on the Universidad Autónoma de Nuevo León, Mexico, and at Sumy kingdom collage, Ukraine. Her services lies in stochastic optimum keep an eye on and mathematical versions of optimization.

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

Example text

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.

G. 2). 4) are no longer fully equivalent. 4). 4) is related to a local optimal solution (x, y, λ) for each λ ∈ Λ(x, y) := {λ ≥ 0 : λ g(x, y) = 0, 0 ∈ ∂ y f (x, y) + λ ∂ y g(x, y)}, provided that Slater’s condition is satisfied. 1) be a convex optimization problem and assume that Slater’s condition is satisfied for all x ∈ X with Ψ (x) = ∅. 2) for each λ ∈ Λ(x, y). 4) for all λ ∈ Λ(x, y). 4). 4) converging to (x, y) such that k k F(x , y ) < F(x, y) for all k. Since the KKT conditions are necessary optimalk k k ity conditions there exists a sequence {λk }∞ k=1 with λ ∈ Λ(x , y ).

23). 8. 23). Then, M B ⊆ M R and C M R is a Bouligand cone to a convex set. Hence, for (x, y) ∈ M B sufficiently close to (x k , y k ) we have d k := ((x, y) − (x k , y k ))/ (x, y) − (x k , y k ) ∈ C M R (x k , y k ) and (a b )d k ≥ γ for sufficiently large k. The Bouligand cone to M B is defined analogously to the Bouligand cone to M R . Let (x, y) be an arbitrary accumulation point of the sequence {(x k , y k )}∞ k=1 computed by the local algorithm. Assume that (x, y) is not a local optimal 38 2 Linear Bilevel Optimization Problem solution.

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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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