v2010.10.26 - Convex Optimization

224 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1.2.1.1 Exercise. Cone of convex functions.Prove that relation (498) implies: The set of all convex vector-valuedfunctions forms a convex cone in some space. Indeed, any nonnegativelyweighted sum of convex functions remains convex. So trivial function f = 0is convex. Relatively interior to each face of this cone are the strictly convexfunctions of corresponding dimension. 3.6 How do convex real functions fitinto this cone? Where are the affine functions?3.1.2.1.2 Example. Conic origins of Lagrangian.The cone of convex functions, implied by membership relation (498), providesfoundation for what is known as a Lagrangian function. [252, p.398] [280]Consider a conic optimization problem, for proper cone K and affine subset Aconvex w.r.t ⎣ RM +KAminimize f(x)xsubject to g(x) ≽ K 0h(x) ∈ A(504)A Cartesian product of convex functions remains convex, so we may write[ ] ⎡ ⎤[ ] ⎡ ⎤ ⎡fg fg w⎦ ⇔ [ w T λ T ν T ] convex ∀⎣λ ⎦∈h h ν⎤⎣ RM∗ +K ∗ ⎦A ⊥(505)where A ⊥ is a normal cone to A . A normal cone to an affine subset is theorthogonal complement of its parallel subspace (E.10.3.2.1).Membership relation (505) holds because of equality for h in convexitycriterion (497) and because normal-cone membership relation (451), givenpoint a∈A , becomesh ∈ A ⇔ 〈ν , h − a〉=0 for all ν ∈ A ⊥ (506)In other words: all affine functions are convex (with respect to any givenproper cone), all convex functions are translation invariant, whereas anyaffine function must satisfy (506).A real Lagrangian arises from the scalar term in (505); id est,[ ] fghL [w T λ T ν T ]= w T f + λ T g + ν T h (507)3.6 Strict case excludes cone’s point at origin and zero weighting.