v2007.09.17 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1352.13.1 Dual coneFor any set K (convex or not), the dual cone [46,2.6.1] [73,4.2]K ∗ ∆ = { y ∈ R n | 〈y , x〉 ≥ 0 for all x ∈ K } (258)is a unique cone 2.44 that is always closed and convex because it is anintersection of halfspaces (halfspaces theorem (2.4.1.1.1)) whose partialboundaries each contain the origin, each halfspace having inward-normal xbelonging to K ; e.g., Figure 42(a).When cone K is convex, there is a second and equivalent construction:Dual cone K ∗ is the union of each and every vector y inward-normal toa hyperplane supporting K or bounding a halfspace containing K ; e.g.,Figure 42(b). When K is represented by a halfspace-description such as(246), for example, where⎡a T1A =∆ ⎣ .⎤⎡c T1⎦∈ R m×n , C =∆ ⎣ .⎤⎦∈ R p×n (259)a T mc T pthen the dual cone can be represented as the conic hullK ∗ = cone{a 1 ,..., a m , ±c 1 ,..., ±c p } (260)a vertex-description, because each and every conic combination of normalsfrom the halfspace-description of K yields another inward-normal to ahyperplane supporting or bounding a halfspace containing 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} (1793)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 43 by usingdual cone equation (258).2.44 The dual cone is the negative polar cone defined by many authors; K ∗ = −K ◦ .[148,A.3.2] [230,14] [29] [20] [247,2.7]