v2004.06.19 - Convex Optimization

78 CHAPTER 2. CONVEX GEOMETRY2.8.1 Dual coneFor any cone K (convex or not), the dual cone [1, §2.6.1]K ∗ = {y ∈ R n | 〈y, x〉 ≥ 0 for all x ∈ K} (170)is a unique cone 2.30 that is always closed and convex because it is an intersectionof halfspaces (halfspaces theorem, §2.3.1.1.1) whose partial boundarieseach contain the origin, each having inward-normal x belonging to K ; e.g.,Figure 2.19(a).When cone K is convex, the dual cone K ∗ is the union of each and everyvector y inward-normal to a hyperplane supporting or containing K ; e.g.,Figure 2.19(b). When K is represented by a halfspace description such as(159), for example, where⎡a T1A =∆ ⎣ .⎤⎡c T1⎦∈ R m×n , C =∆ ⎣ .⎤⎦∈ R k×n (171)a T mc T kthen the dual cone can be represented as the conic hullK ∗ = cone{a 1 ,..., a m , ±c 1 ,..., ±c k } (172)a vertex-description, because each and every conic combination of normalsfrom the halfspace description of K yields another inward-normal to a hyperplanesupporting or containing K .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 has emptyinterior. Rockafellar formulates the dimension of K and K ∗ . [30, §14] 2.31To further motivate our understanding of the dual cone, consider theease with which convergence can be observed in the following optimizationproblem (p):Example. Dual problem. Essentially, duality theory concerns therepresentation of a given optimization problem as half a minimax problem2.30 The dual cone is the negative of the polar cone defined by some authors; K ∗ = −K ◦ .[29] [30] [65] [23] [31]2.31 His monumental work Convex Analysis has not one figure or illustration. See[23, §II.16] for a good illustration of Rockafellar’s recession cone [33].