v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1632.13.7.1.2 Example. Expansion respecting nonpositive orthant.Suppose x ∈ K any orthant in R n . 2.52 Then coordinates for biorthogonalexpansion of x must be nonnegative; in fact, absolute value of the Cartesiancoordinates.Suppose, in particular, x belongs to the nonpositive orthant K = R n − .Then the biorthogonal expansion becomes an orthogonal expansionn∑n∑x = XX T x = −e i (−e T i x) = −e i |e T i x| ∈ R n − (343)i=1and the coordinates of expansion are nonnegative. For this orthant K we haveorthonormality condition X T X = I where X = −I , e i ∈ R n is a standardbasis vector, and −e i is an extreme direction (2.8.1) of K .Of course, this expansion x=XX T x applies more broadly to domain R n ,but then the coordinates each belong to all of R .2.13.8 Biorthogonal expansion, derivationBiorthogonal expansion is a means for determining coordinates in a pointedconic coordinate system characterized by a nonorthogonal basis. Studyof nonorthogonal bases invokes pointed polyhedral cones and their duals;extreme directions of a cone K are assumed to constitute the basis whilethose of the dual cone K ∗ determine coordinates.Unique biorthogonal expansion with respect to K depends upon existenceof its linearly independent extreme directions: Polyhedral cone K must bepointed; then it possesses extreme directions. Those extreme directions mustbe linearly independent to uniquely represent any point in their span.We consider nonempty pointed polyhedral cone K having possibly emptyinterior; id est, we consider a basis spanning a subspace. Then we needonly observe that section of dual cone K ∗ in the affine hull of K because, byexpansion of x , membership x∈aff K is implicit and because any breachof the ordinary dual cone into ambient space becomes irrelevant (2.13.9.3).Biorthogonal expansioni=1x = XX † x ∈ aff K = aff cone(X) (344)is expressed in the extreme directions {Γ i } of K arranged columnar in2.52 An orthant is simplicial and self-dual.X = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (240)