Download Bilevel Programming Problems: Theory, Algorithms and by Stephan Dempe, Vyacheslav Kalashnikov, Gerardo A. PDF
By Stephan Dempe, Vyacheslav Kalashnikov, Gerardo A. Pérez-Valdés, Nataliya Kalashnykova
This ebook describes fresh theoretical findings suitable to bilevel programming as a rule, and in mixed-integer bilevel programming specifically. It describes fresh functions in strength difficulties, akin to the stochastic bilevel optimization ways utilized in the common fuel undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are awarded and explained.
From the again Cover
This e-book describes contemporary theoretical findings proper to bilevel programming quite often, and in mixed-integer bilevel programming specifically. It describes fresh purposes in power difficulties, resembling the stochastic bilevel optimization methods utilized in the average fuel undefined. New algorithms for fixing linear and mixed-integer bilevel programming difficulties are offered and explained.
About the Author
Stephan Dempe studied arithmetic on the Technische Hochschule Karl-Marx-Stadt and acquired a PhD from a similar collage. 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 nation college, he bought his PhD in Operations examine from the Siberian department of the Academy of Sciences of the USSR and his Dr.Sc. (Habilitation measure) from the critical Economics and arithmetic Institute (CEMI), Moscow, Russia. at the present time he's Professor at Tecnológico de Monterrey, Mexico, on the CEMI, and at Sumy nation collage, Ukraine. the most parts 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 bought 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 know-how in Trondheim (NTNU), Norway. the focal point of his paintings is on answer algorithms in mathematical optimization.
Nataliya Kalashnykova studied arithmetic at Novosibirsk kingdom college and bought her PhD in Operations learn 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 nation collage, Ukraine. Her services lies in stochastic optimum regulate and mathematical types of optimization.
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Extra info for Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks
Sample text
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.
5x ≥ 2 if x ≤ 0. Hence, (x, y, z) = (0, 1, 1) is the global (and unique local) optimal solution of the problem. At this point, three constraints in the lower level problem are active and the linear independence constraint qualification is violated. Note that small smooth perturbations of the data of both the lower and the upper level problems will have no impact on this property. g. the case when the right-hand side of the first constraint x + y ≤ 1 in the follower’s problem is perturbed and the new lower level problem reads as follows: Ψ L (x) := Argmin{−y − z : x + y ≤ 1 + α, −x + y ≤ 1, 0 ≤ z ≤ 1}.
Let, without loss of generality (x ∗ , y ∗ ) be a limit point of the sequence {x k , y k }∞ k=1 . Continuity of the function ϕ(·) leads to lim ϕ(x k ) = lim x k k→∞ k→∞ yk = x ∗ y∗, where y ∗ is again without loss of generality a limit point of the sequence {y k }∞ k=1 . Then, we have x k y k ≤ x k y k−1 by the formulae in the algorithm. Hence, by convergence of the sequences, we derive x ∗ y ∗ ≤ ϕ(x ∗ ) = x ∗ y ∗ . Consequently, the point (x ∗ , y ∗ ) is feasible and, thus, also globally optimal.