v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 275Any rank-1 solution must have this form. (B.1.0.2) Ellipsoidally constrainedfeasibility problem (648) is equivalent to:findX∈S Nx ∈ R Nsubject to Ax = b[X CxG =x T C T 1(G ≽ 0)rankG = 1trX = 1](650)This is transformed to an equivalent convex problem by moving the rankconstraint to the objective: We iterate solution ofminimize 〈G , Y 〉X∈S N , x∈R Nsubject to Ax = b[X CxG =x T C T 1trX = 1]≽ 0(651)withminimizeY ∈ S N+1 〈G ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = N(652)Direction matrix Y ∈ S N+1 , initially 0, controls rank. (1475a) Takingsingular value decomposition G ⋆ = UΣQ T ∈ R N+1 , (A.6) then a newdirection matrix Y = U(:, 2:N+1)U(:, 2:N+1) T optimally solves (652) ateach iteration. An optimal solution to (648) is thereby found in a fewiterations, making convex problem (651) its equivalent.It remains possible for the iteration to stall; were a rank-1 G matrix notfound. In that case, the current search direction is momentarily reversedwith an added random element:Y = −U(:, 2:N+1) ( U(:, 2:N+1) T +randn(N,1)U(:, 1) T) (653)This heuristic is quite effective for this problem.