v2006.03.09 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 149KK ∗0Figure 42: 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.)2.13.1.0.1 Example. Dual problem. (confer6.1)Essentially, duality theory concerns the representation of a given optimizationproblem as half a minimax problem (minimize x maximize y ˆf(x,y)) [194,36][38,5.4] whose saddle-value [83] exists. [191, p.3] Consider primal conicproblem (p) and its corresponding dual problem (d): [184,3.3.1] [148,2.1]given vectors α,β and matrix constant C(p)minimize α T xxsubject to x ∈ KCx = βmaximize β T zy,zsubject to y ∈ K ∗C T z + y = α(d) (249)Observe the dual problem is also conic, and its objective 2.45 never exceedsthat of the primal;α T x ≥ β T zx T (C T z + y) ≥ (Cx) T zx T y ≥ 02.45 The objective is the function that is argument to minimization or maximization.(250)