v2007.09.13 - Convex Optimization

222 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSProve: f(X) is not quasiconcave except when N = 1, nor is it quasiconvexunless X 1 = X 2 .When a function is simultaneously quasiconvex and quasiconcave, itis called quasilinear. Quasilinear functions are completely determined byconvex level sets. One-dimensional function f(x) = x 3 and vector-valuedsignum function sgn(x) for example, are quasilinear. Any monotonicfunction is quasilinear. 3.143.4 Salient propertiesof convex and quasiconvex functions1.Aconvex (or concave) function is assumed continuous but notnecessarily differentiable on the relative interior of its domain.[228,10]A quasiconvex (or quasiconcave) function is not necessarily acontinuous function.2. convexity ⇒ quasiconvexity ⇔ convex sublevel setsconcavity ⇒ quasiconcavity ⇔ convex superlevel setsmonotonicity ⇒ quasilinearity ⇔ convex level sets3.(homogeneity) Convexity, concavity, quasiconvexity, andquasiconcavity are invariant to nonnegative scaling of function.g convex ⇔ −g concaveg quasiconvex ⇔ −g quasiconcave4. The line theorem (3.2.3.0.1) translates identically to quasiconvexity(quasiconcavity). [46,3.4.2]5. (affine transformation of argument) Composition g(h(X)) of aconvex (concave) function g with any affine function h : R m×n → R p×kremains convex (concave) in X ∈ R m×n , where h(R m×n ) ∩ dom g ≠ ∅ .[147,B.2.1] Likewise for the quasiconvex (quasiconcave) functions g .3.14 e.g., a monotonic concave function is therefore quasiconvex, but it is best to avoidthis confusion of terms.