v2007.09.13 - Convex Optimization

216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS→Y −Xwhere dg(X) is the directional derivative (D.1.4) of function g at X indirection Y −X . By discretized dual generalized inequalities, (2.13.5)g(Y ) − g(X) −→Y −Xdg(X) ≽0 ⇔〈g(Y ) − g(X) −→Y −X〉dg(X) , ww T ≥ 0 ∀ww T (≽ 0)S M +For each and every X,Y ∈ domg (confer (516))(525)S M +g(Y ) ≽g(X) +→Y −Xdg(X) (526)S M +must therefore be necessary and sufficient for convexity of a matrix-valuedfunction of matrix variable on open convex domain.3.2.2 epigraph of matrix-valued function, sublevel setsWe generalize the epigraph to a continuous matrix-valued functiong(X) : R p×k →S M :epig ∆ = {(X , T )∈ R p×k × S M | X ∈ domg , g(X) ≼T } (527)S M +from which it followsg convex ⇔ epi g convex (528)Proof of necessity is similar to that in3.1.7 on page 196.Sublevel sets of a matrix-valued convex function corresponding to eachand every S ∈ S M (confer (462))L Sg ∆ = {X ∈ dom g | g(X) ≼S } ⊆ R p×k (529)S M +are convex. There is no converse.