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Optimality conditions for bilevel programming problems. (English) Zbl 0820.65032

Summary: 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 in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minimax problem satisfy the calmness condition automatically.
A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementarity lower level problem are given. First-order necessary optimality conditions are given using nonsmooth analysis. Second-order sufficient optimality conditions are also given for the case where the lower level problem is unconstrained.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
49J52 Nonsmooth analysis
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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References:

[1] DOI: 10.1016/0305-0548(82)90007-7
[2] DOI: 10.1137/0331063 · Zbl 0791.90040
[3] DOI: 10.1287/moor.11.4.632 · Zbl 0623.90072
[4] Bazaraa M.S., Nonlinear Programming (1979)
[5] Clarke F.H., Optimization and Nonsmooth Analysis (1983) · Zbl 0582.49001
[6] DOI: 10.1080/02331939208843831 · Zbl 0817.90104
[7] Ferris M. C. Weak sharp minima and penalty functions in mathematical programming Computer Sciences Department, University of Wisconsin June 1988 Technical Report 779
[8] DOI: 10.1016/0022-247X(67)90163-1 · Zbl 0149.16701
[9] DOI: 10.1007/BF01442544 · Zbl 0389.90088
[10] DOI: 10.1007/BF00940846 · Zbl 0642.90107
[11] DOI: 10.1016/0167-6377(88)90047-8 · Zbl 0653.90055
[12] Outrata J.V., Zeitschrift für Operations Research 4 pp 255– (1990)
[13] DOI: 10.1007/BF00939610 · Zbl 0802.49007
[14] Rockafellar R.T., Convex Analysis (1970) · Zbl 0193.18401
[15] DOI: 10.1016/0022-247X(77)90255-4 · Zbl 0355.90066
[16] Von Stackelberg H., The Theory of the Market Economy (1952)
[17] DOI: 10.1109/TAC.1981.1102607 · Zbl 0472.93004
[18] Simaan M., Journal of Optimization Theory and Applications 11 pp 535– (1973) · Zbl 0261.65023
[19] Zhang R., SIAM J. Optimization
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