v2007.09.13 - Convex Optimization

C.2. DIAGONAL, TRACE, SINGULAR AND EIGEN VALUES 537For X ∈ S m , Y ∈ S n , A∈ C ⊆ R m×n for set C convex, and σ(A)denoting the singular values of A [90,3]minimizeA∑σ(A) isubject to A ∈ Ci≡1minimize 2A , X , Ysubject totrX + trY[ ] X AA T ≽ 0YA ∈ C(1463)For A∈ S N + and β ∈ Rβ trA = maximize tr(XA)X∈ S Nsubject to X ≼ βI(1464)But the following statement is numerically stable, preventing anunbounded solution in direction of a 0 eigenvalue:maximize sgn(β) tr(XA)X∈ S Nsubject to X ≼ |β|IX ≽ −|β|I(1465)where β trA = tr(X ⋆ A). If β ≥ 0 , then X ≽−|β|I ← X ≽ 0.For A∈ S N having eigenvalues λ(A)∈ R N , its smallest and largesteigenvalue is respectively [9,4.1] [31,I.6.15] [149,4.2] [174,2.1]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(1466)tr(XA) = minimize tt∈Rsubject to A ≼ tI(1467)The smallest eigenvalue of any symmetric matrix is always a concavefunction of its entries, while the largest eigenvalue is always convex.[46, exmp.3.10] For v 1 a normalized eigenvector of A corresponding to