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The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs. (English) Zbl 1223.90061

The authors introduce a bilevel programming problem in terms of an upper level problem

minF(x,y)s.t.(x,y)X× m ,yΨ(x)( ULP )

where X is a closed subset of n and Ψ(x) is the solution set of the lower level problem defined as:


The functions F,f: n × m are continuous, X={xG(x)0,H(x)=0}, K(x)={yg(x,y)0,h(x,y)=0}, where the functions G: n k , H: n l , g: n × m p , h: n × m q are all continuous. Assuming feasibility of (ULP) the optimal value reformulation is

minF(x,y)s.t.xX,yK(x),f(x,y)ϕ(x)( OV )

where ϕ(x) is defined as: ϕ(x)=min{f(x,y)yK(x)}.

The authors contend that the problems (ULP) and (OV) are locally and globally equivalent. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of basic generalized differentiation constructions of Mordukhovich fails and a weakened form of the same is used to derive Karush-Kuhn-Tucker optimality conditions for (ULP). They also suggest new sufficient conditions for the partial calmness based on a more general notion of the weak sharp minimum.

90C30Nonlinear programming
90C46Optimality conditions, duality
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