v2009.01.01 - Convex Optimization

276 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.3.3.0.2 Exercise. Rank reduction of maximal complementarity.
Apply rank reduction Procedure 4.3.1.0.1 to the maximal complementarity
example (4.1.1.4.1). Demonstrate a rank-1 solution; which can certainly be
found (by Barvinok’s Proposition 2.9.3.0.1) because there is only one equality
constraint.
4.3.4 thoughts regarding rank reduction
Because the rank reduction procedure is guaranteed only to produce another
optimal solution conforming to Barvinok’s upper bound (245), the Procedure
will not necessarily produce solutions of arbitrarily low rank; but if they exist,
the Procedure can. Arbitrariness of search direction when matrix Z i becomes
indefinite, mentioned on page 272, and the enormity of choices for Z i (646)
are liabilities for this algorithm.
4.3.4.1 Inequality constraints
The question naturally arises: what to do when a semidefinite program (not
in prototypical form (584)) 4.20 has linear inequality constraints of the form
α T i svec X ≼ β i , i = 1... k (667)
where {β i } are given scalars and {α i } are given vectors. One expedient way
to handle this circumstance is to convert the inequality constraints to equality
constraints by introducing a slack variable γ ; id est,
α T i svec X + γ i = β i , i = 1... k , γ ≽ 0 (668)
thereby converting the problem to prototypical form.
4.20 Contemporary numerical packages for solving semidefinite programs can solve a range
of problems wider than prototype (584). Generally, they do so by transforming a given
problem into prototypical form by introducing new constraints and variables. [10] [336]
We are momentarily considering a departure from the primal prototype that augments the
constraint set with linear inequalities.