v2006.03.09 - Convex Optimization

6.3. RANK REDUCTION 385Alternatively, we say an inequality constraint is active when it is met withequality; id est, when for particular i in (927), αi T psvec X ⋆ = β ip . An optimalhigh-rank solution X ⋆ is, of course, feasible satisfying all the constraints.But for the purpose of rank reduction, the inactive inequality constraints areignored while the active inequality constraints are interpreted as equalityconstraints. In other words, we take the union of the active inequalityconstraints (as equalities) with the equality constraints A svec X = b to forma composite affine subset  for the primal problem (845)(P). Then weproceed with rank reduction of X ⋆ as though the semidefinite program werein prototypical form (845)(P).