v2007.11.26 - Convex Optimization

196 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS{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 58: Pointwise supremum of convex functions remains a convexfunction. Illustrated is a supremum of affine functions in variable a evaluatedfor a particular argument a p . Topmost affine function is supremum for eachvalue of a .q(x)f(x)quasiconvexconvexxxFigure 59: Quasiconvex function q epigraph is not necessarily convex, butconvex function f epigraph is convex in any dimension. Sublevel sets arenecessarily convex for either.