v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 157represented by a halfspace-description such as (287), for example, where⎡a T1 ⎤⎡c T1A ⎣ . ⎦∈ R m×n , C ⎣ .⎤⎦∈ R p×n (298)a T mc T pthen the dual cone can be represented as the conic hullK ∗ = cone{a 1 ,..., a m , ±c 1 ,..., ±c p } (299)a vertex-description, because each and every conic combination of normalsfrom the halfspace-description of K yields another inward-normal to ahyperplane supporting K .K ∗ can also be constructed pointwise using projection theory fromE.9.2:for P K x the Euclidean projection of point x on closed convex cone K−K ∗ = {x − P K x | x∈ R n } = {x∈ R n | P K x = 0} (2024)2.13.1.0.1 Exercise. Manual dual cone construction.Perhaps the most instructive graphical method of dual cone construction iscut-and-try. Find the dual of each polyhedral cone from Figure 56 by usingdual cone equation (297).2.13.1.0.2 Exercise. Dual cone definitions.What is {x∈ R n | x T z ≥0 ∀z∈R n } ?What is {x∈ R n | x T z ≥1 ∀z∈R n } ?What is {x∈ R n | x T z ≥1 ∀z∈R n +} ?As defined, dual cone K ∗ exists even when the affine hull of the originalcone is a proper subspace; id est, even when the original cone is notfull-dimensional. 2.60To further motivate our understanding of the dual cone, consider theease with which convergence can be ascertained in the following optimizationproblem (302p):2.60 Rockafellar formulates dimension of K and K ∗ . [307,14.6.1] His monumental workConvex Analysis has not one figure or illustration. See [26,II.16] for a good illustrationof Rockafellar’s recession cone [42].