v2007.09.13 - Convex Optimization

4.3. RANK REDUCTION 2554.3.3.0.2 Exercise. Rank reduction of maximal complementarity.Apply rank reduction Procedure 4.3.1.0.1 to the maximal complementarityexample (4.1.1.3.1). 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 the rank reduction procedure is guaranteed only to produce anotheroptimal solution conforming to Barvinok’s upper bound (232), the Procedurewill not necessarily produce solutions of arbitrarily low rank; but if they exist,the Procedure can. Arbitrariness of search direction when matrix Z i becomesindefinite, mentioned on page 251, and the enormity of choices for Z i (608)are liabilities for this algorithm.4.3.4.1 Inequality constraintsThe question naturally arises: what to do when a semidefinite program (notin prototypical form (546)) 4.19 has inequality constraints of the formα T i svec X ≼ β i , i = 1... k (629)where the β i are scalars. One expedient way to handle this circumstance isto convert the inequality constraints to equality constraints by introducing aslack variable γ ; id est,α T i svec X + γ i = β i , i = 1... k , γ ≽ 0 (630)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 (629), αi T 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 as4.19 Contemporary numerical packages for solving semidefinite programs can solve a widerrange of problem than our conic prototype (546). Generally, they do so by transforming agiven problem into some prototypical form by introducing new constraints and variables.[9] [290] We are momentarily considering a departure from the primal prototype thataugments the constraint set with affine inequalities.