v2009.01.01 - Convex Optimization

242 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.3.2 quasilinear
When a function is simultaneously quasiconvex and quasiconcave, it is called
quasilinear. Quasilinear functions are completely determined by convex
level sets. One-dimensional function f(x) = x 3 and vector-valued signum
function sgn(x) , for example, are quasilinear. Any monotonic function is
quasilinear 3.21 (but not vice versa, Exercise 3.1.11.0.1).
3.4 Salient properties
of convex and quasiconvex functions
1.
Aconvex (or concave) function is assumed continuous but not
necessarily differentiable on the relative interior of its domain.
[266,10]
A quasiconvex (or quasiconcave) function is not necessarily a
continuous function.
2. convex epigraph ⇔ convexity ⇒ quasiconvexity ⇔ convex sublevel sets.
convex hypograph ⇔ concavity ⇒ quasiconcavity ⇔ convex superlevel.
monotonicity ⇒ quasilinearity ⇔ convex level sets.
3.
4.
log-convex ⇒ convex ⇒ quasiconvex.
concave ⇒ quasiconcave ⇐ log-concave ⇐ positive concave. 3.22
(homogeneity) Convexity, concavity, quasiconvexity, and
quasiconcavity are invariant to nonnegative scaling of function.
g convex ⇔ −g concave
g quasiconvex ⇔ −g quasiconcave
g log-convex ⇔ 1/g log-concave
5. The line theorem (3.2.3.0.1) translates identically to quasiconvexity
(quasiconcavity). [53,3.4.2]
3.21 e.g., a monotonic concave function is quasiconvex, but dare not confuse these terms.
3.22 Log-convex means: logarithm of function f is convex on dom f .