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Existence of solutions in parametric optimization and porosity

Zaslavski A. J.

Journal of Mathematics and Applications

2006

Vol. 28

161-184

In this work we study the parametric family of the minimization problems f (b, x) → min, x ∈ X on a complete metric space X with a parameter b which belongs to a Hausdorff compact space Β. Here f(•,•) belongs to a space of functions on Β x X, say Μ, endowed with an appropriate metric. We study the set of all functions f(•,•) ∈ Μ for which the corresponding parametric family of the minimization problems has solutions for all parameters b ∈ Β. We show that the complement of this set is not only of the first category but also a σ-porous set. This result and its extensions are obtained as realizations of a variational principle.