v2010.10.26 - Convex Optimization

4.3. RANK REDUCTION 3074.3.3.0.2 Exercise. Rank reduction of maximal complementarity.Apply rank reduction Procedure 4.3.1.0.1 to the maximal complementarityexample (4.1.2.3.1). Demonstrate a rank-1 solution; which can certainly befound (by Barvinok’s Proposition 2.9.3.0.1) because there is only one equalityconstraint.4.3.4 thoughts regarding rank reductionBecause rank reduction Procedure 4.3.1.0.1 is guaranteed only to produceanother optimal solution conforming to Barvinok’s upper bound (272), theProcedure will not necessarily produce solutions of arbitrarily low rank; but ifthey exist, the Procedure can. Arbitrariness of search direction when matrixZ i becomes indefinite, mentioned on page 303, and the enormity of choicesfor Z i (712) are liabilities for this algorithm.4.3.4.1 inequality constraintsThe question naturally arises: what to do when a semidefinite program (notin prototypical form (649)) 4.22 has linear inequality constraints of the formα T i svec X ≼ β i , i = 1... k (733)where {β i } are given scalars and {α i } are given vectors. One expedient wayto handle this circumstance is to convert the inequality constraints to equalityconstraints by introducing a slack variable γ ; id est,α T i svec X + γ i = β i , i = 1... k , γ ≽ 0 (734)thereby converting the problem to prototypical form.Alternatively, we say the i th inequality constraint is active when it ismet with equality; id est, when for particular i in (733), α T i svec X ⋆ = β i .An optimal high-rank solution X ⋆ is, of course, feasible (satisfying all theconstraints). But for the purpose of rank reduction, inactive inequalityconstraints are ignored while active inequality constraints are interpreted as4.22 Contemporary numerical packages for solving semidefinite programs can solve a rangeof problems wider than prototype (649). Generally, they do so by transforming a givenproblem into prototypical form by introducing new constraints and variables. [11] [385]We are momentarily considering a departure from the primal prototype that augments theconstraint set with linear inequalities.