v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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354 CHAPTER 4. SEMIDEFINITE PROGRAMMINGThis is transformed to an equivalent convex problem by moving the rankconstraint to the objective: We iterate solution ofwithminimize 〈G , Y 〉X∈S N , x∈R Nsubject to Ax = b[X CxG =x T C T 1trX = 1minimizeY ∈ S N+1 〈G ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = N]≽ 0(782)(783)until convergence. Direction matrix Y ∈ S N+1 , initially 0, regulates rank.(1700a) Singular value decomposition G ⋆ = UΣQ T ∈ S N+1+ (A.6) provides anew direction matrix Y = U(:, 2:N+1)U(:, 2:N+1) T that optimally solves(783) at each iteration. An optimal solution to (779) is thereby found in afew iterations, making convex problem (782) 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 randomized element:Y = −U(:,2:N+1) ∗ (U(:,2:N+1) ′ + randn(N,1) ∗U(:,1) ′ ) (784)in Matlab notation. This heuristic is quite effective for problem (779) whichis exceptionally easy to solve by convex iteration.When b /∈R(A) then problem (779) must be restated as a projection:minimize ‖Ax − b‖x∈R Nsubject to ‖Cx‖ = 1(785)This is a projection of point b on an ellipsoid boundary because any affinetransformation of an ellipsoid remains an ellipsoid. Problem (782) in turnbecomesminimize 〈G , Y 〉 + ‖Ax − b‖X∈S N , x∈R N[ ]X Cxsubject to G =x T C T ≽ 0 (786)1trX = 1

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 355We iterate this with calculation (783) of direction matrix Y as before untila rank-1 G matrix is found.4.6.0.0.2 Example. Procrustes problem. [52]Example 4.6.0.0.1 is extensible. An orthonormal matrix Q∈ R n×p ischaracterized Q T Q = I . Consider the particular case Q = [x y ]∈ R n×2 asvariable to a Procrustes problem (C.3): given A∈ R m×n and B ∈ R m×2minimize ‖AQ − B‖ FQ∈R n×2subject to Q T Q = I(787)which is nonconvex. By vectorizing matrix Q we can make the assignment:⎡ ⎤ ⎡ ⎤ ⎡ ⎤x [ x T y T 1] X Z x xx T xy T xG = ⎣ y ⎦ = ⎣ Z T Y y ⎦⎣yx T yy T y ⎦∈ S 2n+1 (788)1x T y T 1 x T y T 1Now orthonormal Procrustes problem (787) can be equivalently restated:minimizeX , Y ∈ S , Z , x , y‖A[x y ] − B‖ F⎡X Z⎤xsubject to G = ⎣ Z T Y y ⎦(≽ 0)x T y T 1rankG = 1trX = 1trY = 1trZ = 0(789)To solve this, we form the convex problem sequence:minimizeX , Y , Z , x , y‖A[x y ] −B‖ F + 〈G , W 〉⎡X Z⎤xsubject to G = ⎣ Z T Y y ⎦ ≽ 0x T y T 1trX = 1trY = 1trZ = 0(790)

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 355We iterate this with calculation (783) of direction matrix Y as before untila rank-1 G matrix is found.4.6.0.0.2 Example. Procrustes problem. [52]Example 4.6.0.0.1 is extensible. An orthonormal matrix Q∈ R n×p ischaracterized Q T Q = I . Consider the particular case Q = [x y ]∈ R n×2 asvariable to a Procrustes problem (C.3): given A∈ R m×n and B ∈ R m×2minimize ‖AQ − B‖ FQ∈R n×2subject to Q T Q = I(787)which is nonconvex. By vectorizing matrix Q we can make the assignment:⎡ ⎤ ⎡ ⎤ ⎡ ⎤x [ x T y T 1] X Z x xx T xy T xG = ⎣ y ⎦ = ⎣ Z T Y y ⎦⎣yx T yy T y ⎦∈ S 2n+1 (788)1x T y T 1 x T y T 1Now orthonormal Procrustes problem (787) can be equivalently restated:minimizeX , Y ∈ S , Z , x , y‖A[x y ] − B‖ F⎡X Z⎤xsubject to G = ⎣ Z T Y y ⎦(≽ 0)x T y T 1rankG = 1trX = 1trY = 1trZ = 0(789)To solve this, we form the convex problem sequence:minimizeX , Y , Z , x , y‖A[x y ] −B‖ F + 〈G , W 〉⎡X Z⎤xsubject to G = ⎣ Z T Y y ⎦ ≽ 0x T y T 1trX = 1trY = 1trZ = 0(790)

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