v2006.03.09 - Convex Optimization

362 CHAPTER 6. SEMIDEFINITE PROGRAMMING6.1.1.4 Later developmentsThis rational example (855) indicates the need for a more generally applicableand simple algorithm to identify an optimal solution X ⋆ satisfying Barvinok’sProposition 2.9.3.0.1. We will review such an algorithm in6.3, but first weprovide more background.6.2 Framework6.2.1 Feasible setsDenote by C and C ∗ the convex sets of primal and dual points respectivelysatisfying the primal and dual constraints in (845), each assumed nonempty;⎧ ⎡⎨C =⎩ X ∈ Sn + | ⎣C ∗ =〈A 1 , X〉.〈A m , X〉{S ∈ S n + , y = [y i ]∈ R m |⎤ ⎫⎬⎦= b⎭ = A ∩ Sn +m∑i=1}y i A i + S = C(857)These are the primal feasible set and dual feasible set in domain intersectionof the respective constraint functions. Geometrically, primal feasible A ∩ S n +represents an intersection of the positive semidefinite cone S n + with anaffine subset A of the subspace of symmetric matrices S n in isometricallyisomorphic R n(n+1)/2 . The affine subset has dimension n(n+1)/2 −m whenthe A i are linearly independent. Dual feasible set C ∗ is the Cartesian productof the positive semidefinite cone with its inverse image (2.1.9.0.1) underaffine transformation C − ∑ y i A i . 6.5 Both sets are closed and convex andthe objective functions on a Euclidean vector space are linear, hence (845)(P)and (845)(D) are convex optimization problems.6.5 The inequality C − ∑ y i A i ≽0 follows directly from (845)(D) (2.9.0.1.1) and is knownas a linear matrix inequality. (2.13.5.1.1) Because ∑ y i A i ≼C , matrix S is known as aslack variable (a term borrowed from linear programming [53]) since its inclusion raisesthis inequality to equality.