v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 213
q(x)
f(x)
quasiconvex
convex
x
x
Figure 65: 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.
3.1.6.0.4 Exercise. Level sets.
Given a function f and constant κ , its level sets are defined
L κ κf ∆ = {z | f(z)=κ} (492)
Give two distinct examples of convex function, that are not affine, having
convex level sets.
3.1.7 epigraph, sublevel set
It is well established that a continuous real function is convex if and
only if its epigraph makes a convex set. [173] [266] [311] [324] [215]
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: [255]
epif ∆ = {(X , t)∈ R p×k × R M | X ∈ domf , f(X) ≼
t } (493)
R M +
id est,
f convex ⇔ epi f convex (494)