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Continuous version of the Choquet integral representation theorem

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Puchała P.

Studia Mathematica

2005

tom 168

nr 1

15-24

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if $sup_{||f|| ≤ 1} |f(x) - ∫_{K} fdμ| < γ$ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family $(μ_{t})$ of regular Borel probability measures on X γ-representing points in P(t). Two cases are considered: in the first case the values of P are compact, while in the second they are closed. For this purpose it is shown (using geometrical tools) that the mapping t ↦ ext P(t) is lower semicontinuous. Continuous versions of the Krein-Milman theorem are obtained as corollaries.

Minimization of functional with integrand expressed as minimum of quasiconvex functions - general and special cases

100%

Puchała P.

Banach Center Publications

2014

tom 101

nr 1

169-186

We present Z. Naniewicz method of optimization a coercive integral functional 𝒥 with integrand being a minimum of quasiconvex functions. This method is applied to the minimization of functional with integrand expressed as a minimum of two quadratic functions. This is done by approximating the original nonconvex problem by appropriate convex ones.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Puchała P.

1 2014

1 2005

Wyniki wyszukiwania

Continuous version of the Choquet integral representation theorem

Minimization of functional with integrand expressed as minimum of quasiconvex functions - general and special cases