v2009.01.01 - Convex Optimization

312 CHAPTER 4. SEMIDEFINITE PROGRAMMING
{X(:, i) | X(:, i) T X(:, i) = 1}
{X(:, i) | 1 T X(:, i) = 1}
Figure 84: Permutation matrix i th column-sum and column-norm constraint
in two dimensions when rank constraint is satisfied. Optimal solutions reside
at intersection of hyperplane with unit circle.
1) norm of each row and column is 1 , 4.32
‖Ξ(:, i)‖ = 1, ‖Ξ(i , :)‖ = 1, i=1... n (709)
2) sum of each nonnegative row and column is 1, (2.3.2.0.4)
Ξ T 1=1, Ξ1=1, Ξ ≥ 0 (710)
solution via rank constraint
The idea is to individually constrain each column of variable matrix X to
have unity norm. Matrix X must also belong to that polyhedron, (91) in
the nonnegative orthant, implied by constraints (710); so each row-sum and
4.32 This fact would be superfluous were the objective of minimization linear, because the
permutation matrices reside at the extreme points of a polyhedron (91) implied by (710).
But as posed, only either rows or columns need be constrained to unit norm because
matrix orthogonality implies transpose orthogonality. (B.5.1) Absence of vanishing inner
product constraints that help define orthogonality, like trZ = 0 from Example 4.6.0.0.2,
is a consequence of nonnegativity; id est, the only orthogonal matrices having exclusively
nonnegative entries are permutations of the identity.