an:03854824
Zbl 0537.90087
Bard, Jonathan F.
Optimality conditions for the bilevel programming problem
EN
Nav. Res. Logist. Q. 31, 13-26 (1984).
00149009
1984
j
90C31 49K35 91A05 90B50 93A13 49J35 49K30 49J30
bilevel programming; competitive players; jointly dependent constraints; first-order necessary optimality conditions; Pareto optimality
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.