v2010.10.26 - Convex Optimization

668 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSThen we have a symmetric decomposition from unitary matrices as in (1745)whereU A √ δ(ψ(δ(Υ))) , Q A √ δ(ψ(δ(Υ))) H , Σ A = |Υ| (1754)U B √ δ(ψ(δ(Λ))) , Q B √ δ(ψ(δ(Λ))) H , Σ B = |Λ| (1755)Procrustes solution (1739) again sees the transposition relationshipS ⋆ = U A U H B = R ⋆T ∈ C n×n (1748)but both optimal unitary matrices are now themselves diagonal. So,S ⋆ ΛR ⋆ = δ(ψ(δ(Υ)))Λδ(ψ(δ(Λ))) = δ(ψ(δ(Υ)))|Λ| (1756)C.5 Nonconvex quadratics[346,2] [321,2] Given A∈ S n , b∈ R n , and ρ>01minimizex 2 xT Ax + b T xsubject to ‖x‖ ≤ ρ(1757)is a nonconvex problem for symmetric A unless A≽0. But necessaryand sufficient global optimality conditions are known for any symmetric A :vector x ⋆ solves (1757) iff ∃ Lagrange multiplier λ ⋆ ≥ 0 i) (A + λ ⋆ I)x ⋆ = −bii) λ ⋆ (‖x ⋆ ‖ − ρ) = 0 ,iii) A + λ ⋆ I ≽ 0‖x ⋆ ‖ ≤ ρλ ⋆ is unique. C.5 x ⋆ is unique if A+λ ⋆ I ≻0. Equality constrained problemi) (A + λ ⋆ I)x ⋆ = −bii) ‖x ⋆ ‖ = ρiii) A + λ ⋆ I ≽ 0⇔1minimizex 2 xT Ax + b T xsubject to ‖x‖ = ρ(1758)is nonconvex for any symmetric A matrix. x ⋆ solves (1758) iff ∃ λ ⋆ ∈ Rsatisfying the associated conditions. λ ⋆ and x ⋆ are unique as before. In 1998,Hiriart-Urruty [197] disclosed global optimality conditions for maximizing aconvexly constrained convex quadratic; a nonconvex problem [307,32].C.5 The first two are necessary KKT conditions [61,5.5.3] while condition iii governspassage to nonconvex global optimality.