v2007.09.17 - Convex Optimization

138 CHAPTER 2. CONVEX GEOMETRYKK ∗0Figure 44: K is a halfspace about the origin in R 2 . K ∗ is a ray base 0,hence has empty interior in R 2 ; so K cannot be pointed. (Both convexcones appear truncated.)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 has emptyinterior. Rockafellar formulates the dimension of K and K ∗ . [230,14] 2.45To further motivate our understanding of the dual cone, consider theease with which convergence can be observed in the following optimizationproblem (p):2.13.1.0.3 Example. Dual problem. (confer4.1)Duality is a powerful and widely employed tool in applied mathematics for anumber of reasons. First, the dual program is always convex even if the primalis not. Second, the number of variables in the dual is equal to the number ofconstraints in the primal which is often less than the number of variables in2.45 His monumental work Convex Analysis has not one figure or illustration. See[20,II.16] for a good illustration of Rockafellar’s recession cone [30].