v2007.11.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 177(confer (309))−∇f(x ⋆ ) ∈ −(C − x ⋆ ) ∗ , x ⋆ ∈ C (393)id est, the negative gradient (3.1.8) belongs to the normal cone at x ⋆ as inFigure 55.2.13.10.1.1 Example. Normal cone to orthant.Consider proper cone K = R n + , the self-dual nonnegative orthant in R n .The normal cone to R n + at a∈ K is (1886)K ⊥ R n + (a∈ Rn +) = −(R n + − a) ∗ = −R n + ∩ a ⊥ , a∈ R n + (394)where −R n += −K ∗ is the algebraic complement of R n + , and a ⊥ is theorthogonal complement of point a . This means: When point a is interior toR n + , the normal cone is the origin. If n p represents the number of nonzeroentries in point a ∈ ∂R n + , then dim(−R n + ∩ a ⊥ )= n − n p and there is acomplementary relationship between the nonzero entries in point a and thenonzero entries in any vector x∈−R n + ∩ a ⊥ .2.13.10.1.2 Example. Optimality conditions for conic problem.Consider a convex optimization problem having real differentiable convexobjective function f(x) : R n →R defined on domain R n ;minimize f(x)xsubject to x ∈ K(395)The feasible set is a pointed polyhedral cone K possessing a linearlyindependent set of generators and whose subspace membership is madeexplicit by fat full-rank matrix C ∈ R p×n ; id est, we are given thehalfspace-descriptionK = {x | Ax ≽ 0, Cx = 0} ⊆ R n(247a)where A∈ R m×n . The vertex-description of this cone, assuming (ÂZ)†skinny-or-square full-rank, isK = {Z(ÂZ)† b | b ≽ 0} (386)where Â∈ Rm−l×n , l is the number of conically dependent rows in AZ(2.10) that must be removed, and Z ∈ R n×n−rank C holds basis N(C)columnar.