v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 357{X(:, i) | X(:, i) T X(:, i) = 1}{X(:, i) | 1 T X(:, i) = 1}Figure 105: Permutation matrix i th column-sum and column-norm constraintin two dimensions when rank constraint is satisfied. Optimal solutions resideat intersection of hyperplane with unit circle.1) norm of each row and column is 1 , 4.38‖Ξ(:, i)‖ = 1, ‖Ξ(i , :)‖ = 1, i=1... n (792)2) sum of each nonnegative row and column is 1, (2.3.2.0.4)Ξ T 1=1, Ξ1=1, Ξ ≥ 0 (793)solution via rank constraintThe idea is to individually constrain each column of variable matrix X tohave unity norm. Matrix X must also belong to that polyhedron, (100) inthe nonnegative orthant, implied by constraints (793); so each row-sum and4.38 This fact would be superfluous were the objective of minimization linear, because thepermutation matrices reside at the extreme points of a polyhedron (100) implied by (793).But as posed, only either rows or columns need be constrained to unit norm becausematrix orthogonality implies transpose orthogonality. (B.5.1) Absence of vanishing innerproduct 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 exclusivelynonnegative entries are permutations of the identity.