# Duality and Upper Semicontinuity of Set Valued Functions

• Lamberto Cesari
Part of the Applications of Mathematics book series (SMAP, volume 17)

## Abstract

We shall consider here a real valued function F(u) defined and everywhere finite on a set U of R n . If U is convex (Section 8.4), then F is said to be convex in U if u1, u2U, 0 ≤ α ≤ 1, implies F(αu1 + (1 − α)u2) ≤ αF(u1) + (1 − α)F(u2). The function F is said to be extended (Section 8.5) if we take F = + ∞ in R n U. With obvious conventions the convexity of F in R n is equivalent to the statement that U is convex and F is convex in U. As mentioned in Section 8.5, the set $$\tilde{Q} = {\text{ }}\left[ {({{z}^{0}},u) + \infty > {{z}^{0}} > F\left( u \right),u \in U} \right]$$ is said to be the epigraph of F, or epi F.

## Keywords

Convex Function Convex Subset Half Space Convex Combination Lower Semicontinuous
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.