v2007.09.13 - Convex Optimization

190 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1.4 invertedWe wish to implement objectives of the form x −1 . Suppose we have a 2×2matrix[ ]T =∆ x z∈ R 2 (442)z ywhich is positive semidefinite by (1310) whenT ≽ 0 ⇔ x > 0 and xy ≥ z 2 (443)This means we may formulate convex problems, having inverted variables,as semidefinite programs; e.g.,orminimize x −1x∈Rsubject to x > 0x ∈ C≡x > 0, y ≥ 1 x⇔minimizex , y ∈ Rsubject to[ x 11 y(inverted) For vector x=[x i , i=1... n]∈ R ny[ ] x 1≽ 01 yx ∈ C(444)]≽ 0 (445)orminimizex∈R nn∑i=1x −1isubject to x ≻ 0x ∈ C≡x ≻ 0, y ≥ tr ( δ(x) −1)minimizex∈R n , y∈Rsubject to⇔y[√ xin√ ] n≽ 0 , yi=1... nx ∈ C (446)[xi√ n√ n y]≽ 0 , i=1... n (447)3.1.5 fractional power[99] To implement an objective of the form x α for positive α , we quantizeα and work instead with that approximation. Choose nonnegative integer qfor adequate quantization of α like so:α ∆ = k 2 q (448)