v2010.10.26 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 581We can transform this problem to an equivalent Schur-form semidefiniteprogram; (3.5.2)minimizet∈R , Dsubject tot[]tI vec(D − H)vec(D − H) T ≽ 0 (1392)1D ∈ EDM Ncharacterized by great sparsity and structure. The advantage of this SDP islack of conditions on input H ; e.g., negative entries would invalidate anysolution provided by (1388). (7.0.1.2)7.3.1.3 Gram-form semidefinite program, Problem 3 convex caseFurther, this problem statement may be equivalently written in terms of aGram matrix via linear bijective (5.6.1) EDM operator D(G) (903);minimizeG∈S N c , t∈Rsubject tot[tI vec(D(G) − H)vec(D(G) − H) T 1]≽ 0(1393)G ≽ 0To include constraints on the list X ∈ R n×N , we would rewrite this:minimizetG∈S N c , t∈R , X∈ Rn×Nsubject to[tI vec(D(G) − H)vec(D(G) − H) T 1[ ] I XX T ≽ 0G]≽ 0(1394)X ∈ Cwhere C is some abstract convex set. This technique is discussed in5.4.2.3.5.7.3.1.4 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 cone