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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 $F(x,y)\in \Cal H\subset \Bbb R^\nu,\ x\in K \subseteq H,\ y\in Y$ with a cone $\Cal H,\ H$ a real Hilbert space, $Y$ a parameter set. The impossibility of a generalized system means the condition $\Cal H\cap \Cal K_y=\emptyset$, where $\Cal K_y:=F(K,y)$ is the image of $K$ at $y.$ The main problem reduced to impossibility is: $\min f(x)$ subject to $x\in R:=\{x\in K:g(x)\in \Cal C\}$ with a convex and closed cone $\Cal C\subset \Bbb R^m.$ Applications to variational and quasivariational inequalities are shown. Results concerned with optimality conditions, saddle points, Lagrange multipliers, penalty method are presented, too. For the entire collection see [Zbl 0852.00018].

49K27Optimal control problems in abstract spaces (optimality conditions)
49J40Variational methods including variational inequalities
49N15Duality theory (optimization)