v2006.03.09 - Convex Optimization

482 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSFor A∈ S N having eigenvalues λ(A)∈ R N [148,2.1] [28,I.6.15]min{λ(A) i } = inf x T Ax = minimizei‖x‖=1X∈ S N +subject to trX = 1max{λ(A) i } = sup x T Ax = maximizei‖x‖=1X∈ S N +subject to trX = 1tr(XA) = maximize tt∈Rsubject to A ≽ tI(1276)tr(XA) = minimize tt∈Rsubject to A ≼ tI(1277)The minimum eigenvalue of any symmetric matrix is always a concavefunction of its entries, while the maximum eigenvalue is always convex.[38, exmp.3.10] For v 1 a normalized eigenvector of A corresponding tothe largest eigenvalue, and v N a normalized eigenvector correspondingto the smallest eigenvalue,v N = arg inf‖x‖=1 xT Ax (1278)v 1 = arg sup x T Ax (1279)‖x‖=1For A∈ S N having eigenvalues λ(A)∈ R N , consider the unconstrainednonconvex optimization that is a projection (7.1.2) on the rank 1subset (2.9.2.1) of the boundary of positive semidefinite cone S N + :defining λ 1 ∆ = maxi{λ(A) i } and corresponding eigenvector v 1minimizex‖xx T − A‖ 2 F = minimize tr(xx T (x T x) − 2Axx T + A T A)x{‖λ(A)‖ 2 , λ 1 ≤ 0=‖λ(A)‖ 2 − λ 2 1 , λ 1 > 0 (1280)