v2010.10.26 - Convex Optimization

160 CHAPTER 2. CONVEX GEOMETRYKK ∗0Figure 57: Polyhedral cone K is a halfspace about origin in R 2 . Dual cone K ∗is a ray base 0, hence not full-dimensional in R 2 ; so K cannot be pointed,hence has no extreme directions. (Both convex cones appear truncated.)2.13.1.0.3 Example. Dual problem. (confer4.1)Duality is a powerful and widely employed tool in applied mathematics fora number of reasons. First, the dual program is always convex even if theprimal is not. Second, the number of variables in the dual is equal to thenumber of constraints in the primal which is often less than the number ofvariables in the primal program. Third, the maximum value achieved bythe dual problem is often equal to the minimum of the primal. [299,2.1.3]When not equal, the dual always provides a bound on the primal optimalobjective. For convex problems, the dual variables provide necessary andsufficient optimality conditions:Essentially, Lagrange duality theory concerns representation of a givenoptimization problem as half of a minimax problem. [307,36] [61,5.4] Givenany real function f(x,z)minimizexalways holds. Whenminimizexmaximizezmaximizezf(x,z) ≥ maximizezf(x,z) = maximizezminimize f(x,z) (300)xminimize f(x,z) (301)x