Optimality conditions for bilevel programming problems




Ye, J. J.
Zhu, D. L.

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The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that for certain problems such as the linear bilevel programming problem and the minmax problem, the calmness condition is satisfied. First order necessary optimality conditions are given by using the nonsmooth analysis technique. Second-order sufficient optimality conditions are also given for the case where the lower level problem is unconstrained.


Originally published September 1992. Revised July 1993.


Optimality conditions, Bilevel programming problem, Nonsmooth analysis, Constraint qualification