v2009.01.01 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 529
7.3.1.4 Dual interpretation, projection on EDM cone
FromE.9.1.1 we learn that projection on a convex set has a dual form. In
the circumstance K is a convex cone and point x exists exterior to the cone
or on its boundary, distance to the nearest point Px in K is found as the
optimal value of the objective
‖x − Px‖ = maximize a T x
a
subject to ‖a‖ ≤ 1 (1892)
a ∈ K ◦
where K ◦ is the polar cone.
Applying this result to (1287), we get a convex optimization for any given
symmetric matrix H exterior to or on the EDM cone boundary:
maximize 〈A ◦ , H〉
minimize ‖D − H‖ 2 A ◦
F
D
subject to D ∈ EDM ≡ subject to ‖A ◦ ‖ N F ≤ 1 (1298)
A ◦ ∈ EDM N◦
Then from (1894) projection of H on cone EDM N is
D ⋆ = H − A ◦⋆ 〈A ◦⋆ , H〉 (1299)
Critchley proposed, instead, projection on the polar EDM cone in his 1980
thesis [78, p.113]: In that circumstance, by projection on the algebraic
complement (E.9.2.2.1),
which is equal to (1299) when A ⋆ solves
D ⋆ = A ⋆ 〈A ⋆ , H〉 (1300)
maximize 〈A , H〉
A
subject to ‖A‖ F = 1
(1301)
A ∈ EDM N
This projection of symmetric H on polar cone EDM N◦ can be made a convex
problem, of course, by relaxing the equality constraint (‖A‖ F ≤ 1).