Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.6. EPIGRAPH, SUBLEVEL SET 233
q(x)
f(x)
quasiconvex
convex
x
x
Figure 73: Quasiconvex function q epigraph is not necessarily convex, but
convex function f epigraph is convex in any dimension. Sublevel sets are
necessarily convex for either function, but sufficient only for quasiconvexity.
3.6 Epigraph, Sublevel set
It is well established that a continuous real function is convex if and
only if its epigraph makes a convex set. [195] [301] [351] [364] [244]
Thereby, piecewise-continuous convex functions are admitted. Epigraph is
the connection between convex sets and convex functions. Its generalization
to a vector-valued function f(X) : R p×k →R M is straightforward: [289]
epi f {(X , t)∈ R p×k × R M | X ∈ domf , f(X) ≼
t } (534)
R M +
id est,
f convex ⇔ epif convex (535)
Necessity is proven: [59, exer.3.60] Given any (X, u), (Y , v)∈ epif , we
must show for all µ∈[0, 1] that µ(X, u) + (1−µ)(Y , v)∈ epif ; id est, we
must show
f(µX + (1−µ)Y ) ≼
µu + (1−µ)v (536)
R M +
Yet this holds by definition because f(µX+(1−µ)Y ) ≼ µf(X)+(1−µ)f(Y ).
The converse also holds.