v2010.10.26 - Convex Optimization

264 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSthen we could define a taxicab distance matrixD 1 (X) (I ⊗ 1 T n) | vec(X)1 T − 1 ⊗X | ∈ S N h ∩ R N×N+⎡⎤0 ‖x 1 − x 2 ‖ 1 ‖x 1 − x 3 ‖ 1 · · · ‖x 1 − x N ‖ 1‖x 1 − x 2 ‖ 1 0 ‖x 2 − x 3 ‖ 1 · · · ‖x 2 − x N ‖ 1=‖x⎢ 1 − x 3 ‖ 1 ‖x 2 − x 3 ‖ 1 0 ‖x 3 − x N ‖ 1⎥⎣ . .... . ⎦‖x 1 − x N ‖ 1 ‖x 2 − x N ‖ 1 ‖x 3 − x N ‖ 1 · · · 0(623)where 1 n is a vector of ones having dim1 n = n and where ⊗ representsKronecker product. This matrix-valued function is convex with respect to thenonnegative orthant since, for each and every Y,Z ∈ R n×N and all 0≤µ≤1D 1 (µY + (1 − µ)Z)≼R N×N+µD 1 (Y ) + (1 − µ)D 1 (Z) (624)3.7.0.0.3 Exercise. 1-norm distance matrix.The 1-norm is called taxicab distance because to go from one point to anotherin a city by car, road distance is a sum of grid lengths. Prove (624). 3.7.1 first-order convexity condition, matrix-valued fFrom the scalar-definition (3.7.0.0.1) of a convex matrix-valued function,for differentiable function g and for each and every real vector w of unitnorm ‖w‖= 1, we havew T g(Y )w ≥ w T →Y −XTg(X)w + w dg(X) w (625)that follows immediately from the first-order condition (607) for convexity ofa real function because→Y −XTw dg(X) w = 〈 ∇ X w T g(X)w , Y − X 〉 (626)→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) ≽S M +0 ⇔〈g(Y ) − g(X) −→Y −X〉dg(X) , ww T ≥ 0 ∀ww T (≽ 0)(627)S M∗+