On regular minimax optimization

We consider functions of maximum type (max functions for short), subject to (in)equality constraints. The space dimension is finite, and the maximum is taken over a compact manifold with boundary. Effective local minimization algorithms based on Newton's method can be derived in the case where a local minimum is nondegenerate (in a two-level sense). In fact, nondegeneracy refers on the one hand to a local (implicit) reduction of the original max function to another one, where the maximum is taken over a finite set. On the other hand, it refers to strict complementarity and nondegeneracy of the underlying quadratic form with respect to the reduced stationary situation.As the main goal, we show that the set ofn-parameter families of functions, for which the stationary points of the corresponding max function are nondegenerate, constitutes an open and dense subset in the space of alln-parameter families (the topology used takes derivatives up to second order into account). An application to approximation problems of Chebyshev type is presented.

[1]  H. Th. Jongen,et al.  On the Local Structure of the Feasible Set in Semi-Infinite Optimization , 1984 .

[2]  W. Wetterling,et al.  Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben , 1970 .

[3]  H. Jongen,et al.  On the Local Continuity of the Chebyshev Operator , 1981 .

[4]  R. P. Hettich,et al.  Semi-infinite programming: Conditions of optimality and applications , 1978 .

[5]  K. R. Schneider.  Ljusternik, L. A./ Sobolew, W. I., Elemente der Funktionalanalysis, 5. verbess. Auflage, Berlin. Akademie-Verlag. 1976. 375 S., 7 Abb., M 34,— , 1978 .

[6]  V. F. Demʹi︠a︡nov,et al.  Introduction to minimax , 1976 .

[7]  C. Gibson,et al.  Topological Stability of Smooth Mappings , 1976 .

[8]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[9]  Singularities of the maximum of a parametrically dependent function , 1977 .

[10]  L. A. Li︠u︡sternik,et al.  Elemente der Funktionalanalysis , 1955 .

[11]  M. Kojima,et al.  Continuous deformation of nonlinear programs , 1984 .

[12]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[13]  Okitsugu Fujiwara.  A Note on Differentiability of Global Optimal Values , 1985, Math. Oper. Res..

[14]  H. Jongen,et al.  On Regular Semi-Infinite Optimization , 1985 .

[15]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[16]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[17]  Rainer Hettich,et al.  Numerische Methoden der Approximation und semi-infiniten Optimierung , 1982 .

[18]  V. F. Demʹi︠a︡nov,et al.  Introduction to minimax , 1976 .