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

This ebook describes fresh theoretical findings appropriate to bilevel programming in most cases, and in mixed-integer bilevel programming specifically. It describes fresh purposes in power difficulties, corresponding to the stochastic bilevel optimization methods utilized in the usual fuel undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are offered and explained.

From the again Cover

This publication describes fresh theoretical findings suitable to bilevel programming in most cases, and in mixed-integer bilevel programming specifically. It describes fresh functions in strength difficulties, comparable to the stochastic bilevel optimization ways utilized in the average fuel undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are provided and explained.

About the Author

Stephan Dempe studied arithmetic on the Technische Hochschule Karl-Marx-Stadt and received a PhD from a similar college. this present 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 kingdom college, 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 important 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 functions 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 collage, Lubbock, united states. this day he's Professor at collage of technology and expertise in Trondheim (NTNU), Norway. the point of interest of his paintings is on answer algorithms in mathematical optimization.

Nataliya Kalashnykova studied arithmetic at Novosibirsk nation college and bought her PhD in Operations study from the Siberian department of the Academy of Sciences of the USSR. this present day she is Professor on the Universidad Autónoma de Nuevo León, Mexico, and at Sumy nation collage, Ukraine. Her services lies in stochastic optimum regulate and mathematical versions of optimization.

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

**Example text**

4) for x = x 1 . Setk := 1. Step 1 Select a basic matrix D for y k , compute the region of stability R D and solve the problem min{a x + b y : x = (x D x N ) , y = D −1 x D , x ∈ R D }. x Let (x, D −1 x D ) be an optimal solution. Step 2 Set x k+1 = x and compute an optimal basic solution y k+1 of the problem min{b y : y ∈ Ψ (x k+1 )} y Stop if the optimal solution has not changed: (x k+1 , y k+1 ) = (x k , y k ). Otherwise goto Step 1. 3 Solution Algorithms 35 This algorithm computes a local optimal solution since either one of the problems in Steps 1 or 2 of the algorithm would lead to a better solution.

4 Let the functions F, G be continuous and the functions f, g be continuously differentiable. Assume that the set {(x, y) : G(x) ≤ 0, g(x, y) ≤ 0} is not empty and bounded, X = Rn , and let (MFCQ) be satisfied at all points (x, y) ∈ gph Y . 4) has an optimal solution. 4) is a nonconvex optimization problem. 4 it has a global optimal solution. But, in general, it can also have local optima. 3) has an optimal solution. 6) we need lower semicontinuity (or at least inner semicontinuity at an optimal solution) of the point-to-set mapping Ψ (see Lucchetti et al.

This can only be done if the lower level problem is (for a fixed value of the parameter x) convex. Otherwise, the feasible set of the bilevel optimization problem is enlarged by local optimal solutions and stationary points of the lower level problem. In this case, the global optimal solution of the bilevel problem is in general not a stationary solution for the resulting problem (Mirrlees [232]). Let Y (x) := {y : g(x, y) ≤ 0} denote the feasible set of the lower level problem and assume that Y (x) is a convex set, y → f (x, y) is a convex function, and T ⊆ Rm is a convex set.