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Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. (English) Zbl 0723.90070
The authors consider some algorithms for the bilevel programming problem of minimizing F(x,y) subject to $L\le y\le U$, $x\in K(y)$, and Z(x,y)(z- x)$\ge 0$ for all $z\in K(y)$ [cf. {\it P. Marcotte}, Math. Program. 34, 142-162 (1986; Zbl 0604.90053)]. For many applications the preceding variational inequality constraint defines x as a relatively smooth function of y. After presenting some results about sensitivity analysis of variational inequalities formal statements of three heuristic algorithms for the bilevel programming problem are given. The first two of them are descent algorithms and they differ from one another in the manner in which stepsizes are determined, either according to the Armijo rule or a priori. The third one is a generalization of the equilibrium decomposition optimization algorithm proposed by {\it C. Suwansirikul, T. L. Friesz} and {\it R. L. Tobin} [Transp. Sci. 21, 254-263 (1987; Zbl 0638.90097)]. Finally some numerical examples drawn from equilibrium network design, hierarchical mathematical programming, and game theory are discussed.

90C30Nonlinear programming
49J40Variational methods including variational inequalities
65K10Optimization techniques (numerical methods)
65K05Mathematical programming (numerical methods)
90-08Computational methods (optimization)
90C99Mathematical programming
49K40Sensitivity, stability, well-posedness of optimal solutions
91A65Hierarchical games
90C35Programming involving graphs or networks
Full Text: DOI
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