v2010.10.26 - Convex Optimization

3.5. EPIGRAPH, SUBLEVEL SET 241{a T z 1 + b 1 | a∈ R}supa T pz i + b ii{a T z 2 + b 2 | a∈ R}{a T z 3 + b 3 | a∈ R}a{a T z 4 + b 4 | a∈ R}{a T z 5 + b 5 | a∈ R}Figure 74: Pointwise supremum of convex functions remains convex; byepigraph intersection. Supremum of affine functions in variable a evaluatedat argument a p is illustrated. Topmost affine function per a is supremum.Give two distinct examples of convex function, that are not affine, havingconvex level sets.3.4.0.0.5 Exercise. Epigraph intersection. (confer Figure 74)Draw three hyperplanes in R 3 representing max(0,x) sup{0,x i | x∈ R n }in R 2 ×R to see why maximum of nonnegative vector entries is a convexfunction of variable x . What is the normal to each hyperplane? 3.15 Why ismax(x) convex?3.5 Epigraph, Sublevel setIt is well established that a continuous real function is convex if andonly if its epigraph makes a convex set; [199] [307] [358] [371] [250]epigraph is the connection between convex sets and convex functions(p.219). Piecewise-continuous convex functions are admitted, thereby, and3.15 Hint: page 258.