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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 $F\left(x,y\right)\in ℋ\subset {ℝ}^{\nu },\phantom{\rule{4pt}{0ex}}x\in K\subseteq H,\phantom{\rule{4pt}{0ex}}y\in Y$ with a cone $ℋ,\phantom{\rule{4pt}{0ex}}H$ a real Hilbert space, $Y$ a parameter set. The impossibility of a generalized system means the condition $ℋ\cap {𝒦}_{y}=\varnothing$, where ${𝒦}_{y}:=F\left(K,y\right)$ is the image of $K$ at $y·$ The main problem reduced to impossibility is: $minf\left(x\right)$ subject to $x\in R:=\left\{x\in K:g\left(x\right)\in 𝒞\right\}$ with a convex and closed cone $𝒞\subset {ℝ}^{m}·$ Applications to variational and quasivariational inequalities are shown. Results concerned with optimality conditions, saddle points, Lagrange multipliers, penalty method are presented, too.
##### MSC:
 49K27 Optimal control problems in abstract spaces (optimality conditions) 49J40 Variational methods including variational inequalities 49N15 Duality theory (optimization)