Minimization of a convex functional on a class of sets in a measure space
Let X be a Banach space with the sigma-field \(\Sigma\) of Borel sets and with an atomless finite measure \(\mu\). Let z(x) be a convex \(\mu\)- integrable functional and \(\Omega\) a class of sets \(\omega\) from \(\Sigma\) with property \(\mu (\omega)\geq \mu_ 0\) for some \(\mu_ 0<\mu (X)\). In this paper properties of the extremum problem: minimize the functional \[ \phi_ z(\omega)=\int_{\omega}z(x)\mu (d\omega)/\int_{\omega}\mu (d\omega) \] over \(\omega\in \Sigma\), is considered. Conditions for the existence of an optimal solution \(\omega^*\) are presented. For the special case of X a Hilbert space and z an affine functional, the problem is reduced to a quadratic programming problem in Hilbert space.
scientific article; zbMATH DE number 4160515
scientific article