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Optimality conditions for the bilevel programming problem. (English) Zbl 0537.90087
The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the first is determined implicitly by the solution to the second. Consider two decision-makers or competitive players who must find vectors x and y, respectively, to optimize their individual objective functions F(x,y) and f(x,y). It will be assumed that player 1 has the first choice and selects $$x\in X$$, followed by player 2 who selects $$y\in Y$$, where X and Y are nonempty subsets of $${\mathbb{R}}^{n^ 1}$$ and $${\mathbb{R}}^{n^ 2}$$. In addition, the choice made by player 1 may affect the set of feasible strategies, S, open to player 2, implying the existence of jointly dependent constraints.
Letting $$S=\{(x,y):$$ g(x,y)$$\geq 0\}$$ the above situation can be compactly stated as follows: $\max_{x\in X}F(x,y),\quad where\quad y\quad solves\quad \max_{y\in Y}f(x,y),\quad subject\quad to\quad G(x,y)\geq 0.$ It is first shown that the linear BLPP is equivalent to maximizing a linear function over a feasible region comprised of connected faces and edges of the original polyhedral constraint set. The solution is shown to occur at a vertex of that set. Next, under assumptions of differentiability, first-order necessary optimality conditions are developed for the more general BLPP, and a potentially equivalent mathematical program is formulated. Finally, the relationship between the solution to this problem and Pareto optimality is discussed and a number of examples given.

##### MSC:
 90C31 Sensitivity, stability, parametric optimization 49K35 Optimality conditions for minimax problems 91A05 2-person games 90B50 Management decision making, including multiple objectives 93A13 Hierarchical systems 49J35 Existence of solutions for minimax problems 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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