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On the continuity of the minima for a family of constrained optimization problems. (English) Zbl 0574.49017

The work deals with a parametrized family Py of constrained minimum problems \(Py=\min \{f(x,y):\) \(x\in Ky\}\). Here the function \(f: X\times Y\to {\mathbb{R}}\) and the multifunction \(K: Y\Rightarrow X\) are the data and X,Y are fixed Banach spaces. Working mainly in convexity it is shown how the property of Tykhonov well posedness of the (limit) problems allows to prove stability results usually achievable under compactness hypotheses on the images of the multifuction.

MSC:

49K40 Sensitivity, stability, well-posedness
49J45 Methods involving semicontinuity and convergence; relaxation
90C48 Programming in abstract spaces
54C60 Set-valued maps in general topology
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