Optimization—Theory and Applications pp 474-502 | Cite as

# Duality and Upper Semicontinuity of Set Valued Functions

## 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 *u*_{1}, *u*_{2} ∈ *U*, 0 ≤ *α* ≤ 1, implies *F*(*αu*_{1} + (1 − *α*)*u*_{2}) ≤ *αF*(*u*_{1}) + (1 − *α*)*F*(*u*_{2}). 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## Preview

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