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Images, separation of sets and extremum problems. (English) Zbl 0888.49018
Agarwal, R. P. (ed.), Recent trends in optimization theory and applications. Singapore: World Scientific. World Sci. Ser. Appl. Anal. 5, 79-106 (1995).
The authors present a method of solving the constrained extremum problem through converting it into separating of the sets in the image space. The problem is formulated as the generalized system F(x,y) ν ,xKH,yY with a cone ,H a real Hilbert space, Y a parameter set. The impossibility of a generalized system means the condition 𝒦 y =, where 𝒦 y :=F(K,y) is the image of K at y· The main problem reduced to impossibility is: minf(x) subject to xR:={xK:g(x)𝒞} with a convex and closed cone 𝒞 m · Applications to variational and quasivariational inequalities are shown. Results concerned with optimality conditions, saddle points, Lagrange multipliers, penalty method are presented, too.
MSC:
49K27Optimal control problems in abstract spaces (optimality conditions)
49J40Variational methods including variational inequalities
49N15Duality theory (optimization)