v2007.11.26 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 2173.2.1 first-order convexity condition, matrix functionFrom the scalar-definition we have, for differentiable matrix-valuedfunction g and for each and every real vector w of unit norm ‖w‖= 1,w T g(Y )w ≥ w T →Y −XTg(X)w + w dg(X) w (524)that follows immediately from the first-order condition (510) for convexity ofa real function because→Y −XTw dg(X) w = 〈 ∇ X w T g(X)w , Y − X 〉 (525)→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 (517))(526)S M +g(Y ) ≽g(X) +→Y −Xdg(X) (527)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 } (528)S M +from which it follows