Existence of solutions of minimization problems with an increasing cost function and porosity

Abstract

We consider the minimization problem $f\left(x\right)\to min$, $x\in K$, where $K$ is a closed subset of an ordered Banach space $X$ and $f$ belongs to a space of increasing lower semicontinuous functions on $K$. In our previous work, we showed that the complement of the set of all functions $f$, for which the corresponding minimization problem has a solution, is of the first category. In the present paper we show that this complement is also a $\sigma$-porous set.