v2010.10.26 - Convex Optimization

284 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.2 Framework4.2.1 Feasible setsDenote by D and D ∗ the convex sets of primal and dual points respectivelysatisfying the primal and dual constraints in (649), each assumed nonempty;⎧ ⎡ ⎤ ⎫⎨〈A 1 , X〉 ⎬D =⎩ X ∈ Sn + | ⎣ . ⎦= b⎭ = A ∩ Sn +〈A m , X〉(661){}m∑D ∗ = S ∈ S n + , y = [y i ]∈ R m | y i A i + S = CThese are the primal feasible set and dual feasible set. Geometrically,primal feasible A ∩ S n + represents an intersection of the positive semidefinitecone S n + with an affine subset A of the subspace of symmetric matrices S nin isometrically isomorphic R n(n+1)/2 . A has dimension n(n+1)/2 −mwhen the vectorized A i are linearly independent. Dual feasible set D ∗ isa Cartesian product of the positive semidefinite cone with its inverse image(2.1.9.0.1) under affine transformation 4.9 C − ∑ y i A i . Both feasible sets areconvex, and the objective functions are linear on a Euclidean vector space.Hence, (649P) and (649D) 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 set {A i ∈ S n , i=1... m} , vector b = [b i ]∈ R m , and affine subset(652) A = {X ∈ S n |〈A i , X〉=b i , i=1... m} {A svec X |X ≽ 0} (380) is closed,then 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.Equivalently, primal feasible set A ∩ S n + is nonempty if and only if∑y T b ≥ 0 holds for each and every vector ‖y‖= 1 such that m y i A i ≽ 0. ⋄4.9 Inequality C − ∑ y i A i ≽0 follows directly from (649D) (2.9.0.1.1) and is known as alinear matrix inequality. (2.13.5.1.1) Because ∑ y i A i ≼C , matrix S is known as a slackvariable (a term borrowed from linear programming [94]) since its inclusion raises thisinequality to equality.i=1i=1i=1