v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 20321.510.5Y 20−0.5−1−1.5−2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2Figure 58: Gradient in R 2 evaluated on grid over some open disc in domainof convex quadratic bowl f(Y )= Y T Y : R 2 → R illustrated in Figure 59.Circular contours are level sets; each defined by a constant function-value.Y 1‖X‖ 2 F − ‖A − X‖ 2 F = 2 tr(XA) − ‖A‖ 2 F (490)So problem (489) is equivalent to the convex optimizationminimize tr(XA)X∈ S Msubject to X ∈ C(491)But this problem (489) does not have a Schur formminimize t − αX∈ S M , α , tsubject to X ∈ C‖X‖ 2 F ≤ t(492)‖A − X‖ 2 F ≥ αbecause the constraint in α is nonconvex. (2.9.1.0.1)