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

This e-book describes contemporary theoretical findings suitable to bilevel programming quite often, and in mixed-integer bilevel programming specifically. It describes contemporary functions in power difficulties, corresponding to the stochastic bilevel optimization techniques utilized in the typical gasoline undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are awarded and explained.

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

**Sample text**

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 ).

The set of all induced region directions is T (x, y = y(x)). 16) for the index set of active constraints I . This set is a convex cone. 15) and all λ ∈ Λ(x, y). The number of cones used in this union is finite but can be arbitrarily large. This result can now be used to derive necessary and sufficient optimality conditions.

The matrix Q is assumed to symmetric and positive definite. The lower level problem is in this case a strictly convex quadratic optimization problem, its optimal solution exists and is unique whenever the feasible set of this problem is not empty. Denote Ψ Q (x) = Argminz {qx (z) : Ax + By ≤ c}. Let x, d ∈ Rn be given such that Y (x + td) := {y : A(x + td) + By ≤ c} = ∅ for t > 0 sufficiently small. Then, since the objective function qx (·) is strictly convex and quadratic, Ψ Q (x + td) = ∅ and the optimal solution is unique.