v2010.10.26 - Convex Optimization

3.3. INVERTED FUNCTIONS AND ROOTS 2353.3 Inverted functions and rootsA given function f is convex iff −f is concave. Both functions are looselyreferred to as convex since −f is simply f inverted about the abscissa axis,and minimization of f is equivalent to maximization of −f .A given positive function f is convex iff 1/f is concave; f inverted aboutordinate 1 is concave. Minimization of f is maximization of 1/f .We wish to implement objectives of the form x −1 . Suppose we have a2×2 matrix[ ] x zT ∈ R 2 (537)z ywhich is positive semidefinite by (1514) whenT ≽ 0 ⇔ x > 0 and xy ≥ z 2 (538)A polynomial constraint such as this is therefore called a conic constraint. 3.13This means we may formulate convex problems, having inverted variables,as semidefinite programs in Schur-form; e.g.,ratherminimize 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 nminimizex∈R nn∑i=1x −1isubject to x ≻ 0ratherx ∈ C≡x ≻ 0, y ≥ tr ( δ(x) −1)minimizex∈R n , y∈Rsubject to⇔y[ ] x 1≽ 01 yx ∈ C(539)]≽ 0 (540)y[√ xin√ ] n≽ 0 , yi=1... nx ∈ C (541)[xi√ n√ n y]≽ 0 , i=1... n (542)3.13 In this dimension, the convex cone formed from the set of all values {x , y , z} satisfyingconstraint (538) is called a rotated quadratic or circular cone or positive semidefinite cone.