v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 175From this and Cone Table 1 (p.167) we get a vertex-description, (1637)K ∗ = {[Z †T (ÂZ)T C T −C T ]c | c ≽ 0} (387)Yet becauseK = {x | Ax ≽ 0} ∩ {x | Cx = 0} (388)then, by (272), we get an equivalent vertex-description for the dual coneK ∗ = {x | Ax ≽ 0} ∗ + {x | Cx = 0} ∗= {[A T C T −C T ]b | b ≽ 0}(389)from which the conically dependent columns may, of course, be removed.2.13.10 Dual cone-translateFirst-order optimality condition (308) inspires a dual-cone variant: For anyset K , the negative dual of its translation by any a∈ R n is−(K − a) ∗ ∆ = { y ∈ R n | 〈y , x − a〉≤0 for all x ∈ K }= { y ∈ R n | 〈y , x〉≤0 for all x ∈ K − a } (390)a closed convex cone called the normal cone to K at point a . (E.10.3.2.1)From this, a new membership relation like (276) for closed convex cone K :y ∈ −(K − a) ∗ ⇔ 〈y , x − a〉≤0 for all x ∈ K (391)2.13.10.1 first-order optimality condition - restatementThe general first-order necessary and sufficient condition for optimalityof solution x ⋆ to a minimization problem with real differentiable convexobjective function f(x) : R n →R over convex feasible set C is [227,3](confer (308))−∇f(x ⋆ ) ∈ −(C − x ⋆ ) ∗ , x ⋆ ∈ C (392)id est, the negative gradient (3.1.8) belongs to the normal cone at x ⋆ as inFigure 53.