v2007.09.13 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 4717.3.1.4 Dual interpretation, projection on EDM coneFromE.9.1.1 we learn that projection on a convex set has a dual form. Inthe circumstance K is a convex cone and point x exists exterior to the coneor on its boundary, distance to the nearest point Px in K is found as theoptimal value of the objective‖x − Px‖ = maximize a T xasubject to ‖a‖ ≤ 1 (1779)a ∈ K ◦where K ◦ is the polar cone.Applying this result to (1185), we get a convex optimization for any givensymmetric matrix H exterior to or on the EDM cone boundary:maximize 〈A ◦ , H〉minimize ‖D − H‖ 2 A ◦FDsubject to D ∈ EDM ≡ subject to ‖A ◦ ‖ N F ≤ 1 (1196)A ◦ ∈ EDM N◦Then from (1781) projection of H on cone EDM N isD ⋆ = H − A ◦⋆ 〈A ◦⋆ , H〉 (1197)Critchley proposed, instead, projection on the polar EDM cone in his 1980thesis [61, p.113]: In that circumstance, by projection on the algebraiccomplement (E.9.2.2.1),which is equal to (1197) when A ⋆ solvesD ⋆ = A ⋆ 〈A ⋆ , H〉 (1198)maximize 〈A , H〉Asubject to ‖A‖ F = 1(1199)A ∈ EDM NThis projection of symmetric H on polar cone EDM N◦ can be made a convexproblem, of course, by relaxing the equality constraint (‖A‖ F ≤ 1).