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

This e-book describes contemporary theoretical findings proper to bilevel programming commonly, and in mixed-integer bilevel programming specifically. It describes contemporary functions in strength difficulties, akin to the stochastic bilevel optimization techniques utilized in the ordinary gasoline undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are offered and explained.

**Read or Download Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks PDF**

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

**Sample text**

16) has a unique solution (r, γ ) and, hence, r equals the directional derivative y (x; d) of the solution function y(·) at x into direction d, see Bigelow and Shapiro [22]. 16) has a solution, this does not need to be true since this system is not suitable to the direction. 10). 16) equals (y (x; d), γ ) for some direction d is Ai d + Bi r ≤ 0 for i with Ai x + Bi y − ci = 0 and γi ≥ 0 for i with λi0 = 0. 12) is piecewise affine-linear. This implies that it is a Lipschitz continuous function, its directional derivative 1 d → y (x; d) = lim [y(x + td) − y(x)] t↓0 t is also Lipschitz continuous.

1 (Mersha and Dempe [227]). Consider the problem min −x − 2y x,y 2x − 3y ≥ −12 x + y ≤ 14 subject to and y ∈ Argmin {−y : −3x + y ≤ −3, 3x + y ≤ 30}. y © Springer-Verlag Berlin Heidelberg 2015 S. 1007/978-3-662-45827-3_2 21 22 2 Linear Bilevel Optimization Problem Fig. 1 The problem with upper level connecting constraints. The feasible set is depicted with bold lines. The point C is global optimal solution, point A is a local optimal solution Fig. 2 The problem when the upper level connecting constraints are shifted into the lower level problem.

20) yi = 0 ∀i ∈ I (y) b ∈ B 0 and the tangent cone TR (y) = {d : Ad = r, Br = 0, di ≥ 0, i ∈ I (y)\ I 0 (y), di = 0, i ∈ I 0 (y)} to the feasible set of this problem at the point y again relative to y only. e. the set of all linear combinations of elements in S with nonnegative coefficients. Let spanS denote the set of all linear combinations of elements in S. 22) where Ai denotes the ith column of the matrix A. 4 All non degenerated vertices of Ay = b, y ≥ 0 satisfy the full rank condition. This condition allows us now to establish equality between the cones above.