v2010.10.26 - Convex Optimization

162 CHAPTER 2. CONVEX GEOMETRYwe have strong duality and then a saddle value [152] exists. (Figure 58)[304, p.3] Consider primal conic problem (p) (over cone K) and itscorresponding dual problem (d): [293,3.3.1] [239,2.1] [240] givenvectors 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) (302)Observe: the dual problem is also conic, and its objective function valuenever exceeds that of the primal;α T x ≥ β T zx T (C T z + y) ≥ (Cx) T zx T y ≥ 0(303)which holds by definition (297). Under the sufficient condition: (302p) is aconvex problem 2.61 satisfying Slater’s condition (p.285), then equalityx ⋆T y ⋆ = 0 (304)is achieved; which is necessary and sufficient for optimality (2.13.10.1.5);each problem (p) and (d) attains the same optimal value of its objective andeach problem is called a strong dual to the other because the duality gap(optimal primal−dual objective difference) becomes 0. Then (p) and (d) aretogether equivalent to the minimax problemminimize α T x − β T zx,y,zsubject to x ∈ K ,y ∈ K ∗Cx = β , C T z + y = α(p)−(d) (305)whose optimal objective always has the saddle value 0 (regardless of theparticular convex cone K and other problem parameters). [359,3.2] Thusdetermination of convergence for either primal or dual problem is facilitated.Were convex cone K polyhedral (2.12.1), then problems (p) and (d)would be linear programs. Selfdual nonnegative orthant K yields the primal2.61 A convex problem, essentially, has convex objective function optimized over a convexset. (4) In this context, problem (p) is convex if K is a convex cone.