v2007.09.13 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 623Unique minimum-distance projection of H ∈ S n on the positivesemidefinite cone S n + in the Euclidean/Frobenius sense is accomplishedby eigen decomposition (diagonalization) followed by clipping allnegative eigenvalues to 0.Unique minimum-distance projection on the generally nonconvexsubset of all matrices belonging to S n + having rank not exceeding ρ(2.9.2.1) is accomplished by clipping all negative eigenvalues to 0 andzeroing the smallest nonnegative eigenvalues keeping only ρ largest.(7.1.2)Unique minimum-distance projection of H ∈ R m×n on the set of allm ×n matrices of rank no greater than k in the Euclidean/Frobeniussense is the singular value decomposition (A.6) of H having allsingular values beyond the k th zeroed. [244, p.208] This is also a solutionto the projection in the sense of spectral norm. [46,8.1]Projection on K of any point x∈−K ∗ , belonging to the polar cone, isequivalent to projection on the origin. (E.9.2)Projection on Lorentz cone: [46, exer.8.3(c)]P S N+ ∩ S N c= P S N+P S N c(1066)P RN×N+ ∩ S N h= P RN×NP + S Nh(7.0.1.1)P RN×N+ ∩ S= P N RN×N+P S N(E.9.5)E.9.4.0.1 Exercise. Largest singular value.Find the unique minimum-distance projection on the set of all m ×nmatrices whose largest singular value does not exceed 1. Deutsch [75] provides an algorithm for projection on polyhedral cones.Youla [297,2.5] lists eleven “useful projections”, of square-integrableuni- and bivariate real functions on various convex sets, in closed form.Unique minimum-distance projection on an ellipsoid. (Example 4.4.3.0.2,Figure 10)