v2007.09.13 - Convex Optimization

140 CHAPTER 2. CONVEX GEOMETRYthe primal program. Third, the maximum value achieved by the dual problemis often equal to the minimum of the primal. [221,2.1.3] Essentially, dualitytheory concerns representation of a given optimization problem as half aminimax problem. [228,36] [46,5.4] Given any real function f(x,z)minimizexalways holds. Whenminimizexmaximizezmaximizezf(x,z) ≥ maximizezf(x,z) = maximizezminimize f(x,z) (261)xminimize f(x,z) (262)xwe have strong duality and then a saddle value [103] exists. (Figure 45)[225, p.3] Consider primal conic problem (p) and its corresponding dualproblem (d): [216,3.3.1] [174,2.1] given vectors α , β and matrixconstant C(p)minimize α T xxsubject to x ∈ KCx = βmaximize β T zy , zsubject to y ∈ K ∗C T z + y = α(d) (263)Observe the dual problem is also conic, and its objective function value neverexceeds that of the primal;α T x ≥ β T zx T (C T z + y) ≥ (Cx) T zx T y ≥ 0(264)which holds by definition (258). Under the sufficient condition: (263p) isa convex problem and satisfies Slater’s condition, 2.46 then each problem (p)and (d) attains the same optimal value of its objective and each problemis called a strong dual to the other because the duality gap (primal−dualobjective difference) is 0. Then (p) and (d) are together equivalent to theminimax problemminimize α T x − β T zx,y,zsubject to x ∈ K ,y ∈ K ∗Cx = β , C T z + y = α(p)−(d) (265)2.46 A convex problem, essentially, has convex objective function optimized over a convexset. (4) In this context, (p) is convex if K is a convex cone. Slater’s condition is satisfiedwhenever any primal strictly feasible point exists. (p.235)