v2007.09.17 - Convex Optimization

3.1. CONVEX FUNCTION 197When an epigraph (459) is artificially bounded above, t ≼ ν , then thecorresponding sublevel set can be regarded as an orthogonal projection ofthe epigraph on the function domain.Sense of the inequality is reversed in (459), for concave functions, and weuse instead the nomenclature hypograph. Sense of the inequality in (462) isreversed, similarly, with each convex set then called superlevel set.3.1.7.0.2 Example. Matrix pseudofractional function.Consider a real function of two variables on domf = S n +× R(A)f(A, x) : S n × R n → R = x T A † x (464)This function is convex simultaneously in both variables when variablematrix A belongs to the entire positive semidefinite cone S n + and variablevector x is confined to range R(A) of matrix A .To explain this, we need only demonstrate that the function epigraph isconvex. Consider Schur-form (1311) fromA.4: for t ∈ R[ ] A zG(A, z , t) =z T ≽ 0t⇔z T (I − AA † ) = 0t − z T A † z ≥ 0A ≽ 0(465)Inverse image of the positive semidefinite cone S n+1+ under affine mappingG(A, z , t) is convex by Theorem 2.1.9.0.1. Of the equivalent conditions forpositive semidefiniteness of G , the first is an equality demanding vector zbelong to R(A). Function f(A, z)=z T A † z is convex on S+× n R(A) becausethe Cartesian product constituting its epigraphepif(A, z) = { (A, z , t) | A ≽ 0, z ∈ R(A), z T A † z ≤ t } = G −1( S n+1+is convex.)(466)3.1.7.0.3 Exercise. Matrix product function.Continue Example 3.1.7.0.2 by introducing vector variable x and makingthe substitution z ←Ax . Because of matrix symmetry (E), for all x∈ R nf(A, z(x)) = z T A † z = x T A T A † Ax = x T Ax = f(A, x) (467)