v2007.09.13 - Convex Optimization

234 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.2 Framework4.2.1 Feasible setsDenote by C and C ∗ the convex sets of primal and dual points respectivelysatisfying the primal and dual constraints in (546), each assumed nonempty;⎧ ⎡ ⎤ ⎫⎨〈A 1 , X〉 ⎬C =⎩ X ∈ Sn + | ⎣ . ⎦= b⎭ = A ∩ Sn +〈A m , X〉(558){}m∑C ∗ = S ∈ S n + , y = [y i ]∈ R m | y i A i + S = CThese 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 . 4.5 Both sets are closed and convex andthe objective functions on a Euclidean vector space are linear, hence (546P)and (546D) are convex optimization problems.4.2.1.1 A ∩ S n + emptiness determination via Farkas’ lemma4.2.1.1.1 Lemma. Semidefinite Farkas’ lemma.Given an arbitrary set {A i ∈ S n , i=1... m} and a vector b = [b i ]∈ R m ,define the affine subseti=1A = {X ∈ S n | 〈A i , X〉 = b i , i=1... m} (549)Primal feasible set A ∩ S n + is nonempty if and only if y T b ≥ 0 holds foreach and every vector y = [y i ]∈ R m ∑such that m y i A i ≽ 0.4.5 The inequality C − ∑ y i A i ≽0 follows directly from (546D) (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 [64]) since its inclusion raisesthis inequality to equality.i=1