Abstract
This paper deals with families of semi-infinite in general non-convex minimization problems where the restriction vector is considered as a parameter. To investigate those problems we introduce weakly pointwise convex minimization problems, which include convex and more general minimization problems. For these problems an always sufficient condition for a minimal point is also a necessary condition. Then we show: (1) If for every parameter this optimality condition is also necessary then the family of minimization problems is weakly pointwise convex. (2) The results are used to investigate the structure of the solvability set.
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