v2010.10.26 - Convex Optimization

3.9. SALIENT PROPERTIES 2713.8.0.0.3 Exercise. Nonconvexity of matrix product.Consider real function f on a positive definite domain[ ] [X1rel int SN+f(X) = tr(X 1 X 2 ) , X ∈ domf X 2 rel int S N +](645)with superlevel setsL s f = {X ∈ dom f | 〈X 1 , X 2 〉 ≥ s } (646)Prove: f(X) is not quasiconcave except when N = 1, nor is it quasiconvexunless X 1 = X 2 .3.8.1 bilinearBilinear function 3.27 x T y of vectors x and y is quasiconcave (monotonic) onthe entirety of the nonnegative orthants only when vectors are of dimension 1.3.8.2 quasilinearWhen a function is simultaneously quasiconvex and quasiconcave, it is calledquasilinear. Quasilinear functions are completely determined by convexlevel sets. One-dimensional function f(x) = x 3 and vector-valued signumfunction sgn(x) , for example, are quasilinear. Any monotonic function isquasilinear 3.28 (3.8.0.0.1,3.9 no.4, but not vice versa; Exercise 3.6.4.0.2).3.9 Salient propertiesof convex and quasiconvex functions1.A convex function is assumed continuous but not necessarilydifferentiable on the relative interior of its domain. [307,10]A quasiconvex 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.3.27 Convex envelope of bilinear functions is well known. [3]3.28 e.g., a monotonic concave function is quasiconvex, but dare not confuse these terms.