v2010.10.26 - Convex Optimization

656 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSFor A∈ S N + and β ∈ Rβ trA = maximize tr(XA)X∈ S Nsubject to X ≼ βI(1689)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(1690)where β trA = tr(X ⋆ A). If β ≥ 0, then X ≽−|β|I ← X ≽ 0.For symmetric A∈ S N , its smallest and largest eigenvalue in λ(A)∈ R Nis respectively [11,4.1] [43,I.6.15] [202,4.2] [239,2.1] [240]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(1691)tr(XA) = minimize tt∈Rsubject to A ≼ tI(1692)The largest eigenvalue λ 1 is always convex in A∈ S N because, givenparticular x , x T Ax is linear in matrix A ; supremum of a family oflinear functions is convex, as illustrated in Figure 74. So for A,B∈ S N ,λ 1 (A + B) ≤ λ 1 (A) + λ 1 (B). (1504) Similarly, the smallest eigenvalueλ N of any symmetric matrix is a concave function of its entries;λ N (A + B) ≥ λ N (A) + λ N (B). (1504) For v 1 a normalized eigenvectorof A corresponding to the largest eigenvalue, and v N a normalizedeigenvector corresponding to the smallest eigenvalue,v N = arg inf x T Ax (1693)‖x‖=1v 1 = arg sup x T Ax (1694)‖x‖=1