v2006.03.09 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 159And from this, alternative systems with respect to the nonnegativeorthant attributed to Gordan in 1873: [89] [36,2.2] substituting A ←A Tand setting b = 0A T y ≺ 0or in the alternativeAx = 0, x ≽ 0, ‖x‖ 1 = 1(284)2.13.3 Optimality conditionThe general first-order necessary and sufficient condition for optimalityof solution x ⋆ to a convex problem ((249)(p) for example) with realdifferentiable objective function ˆf(x) : R n →R [193,3] is∇ ˆf(x ⋆ ) T (x − x ⋆ ) ≥ 0 ∀x ∈ C , x ⋆ ∈ C (285)where C is the feasible set, the convex set of all variable values satisfying theproblem constraints.2.13.3.0.1 Example. Equality constrained problem.Given a real differentiable convex function ˆf(x) : R n →R defined ondomain R n , a fat full-rank matrix C ∈ R p×n , and vector d∈ R p , the convexoptimization problemminimize ˆf(x)x(286)subject to Cx = dis characterized by the well-known necessary and sufficient optimalitycondition [38,4.2.3]∇ ˆf(x ⋆ ) + C T ν = 0 (287)where ν ∈ R p is the eminent Lagrange multiplier. [192] Feasible solution x ⋆is optimal, in other words, if and only if ∇ ˆf(x ⋆ ) belongs to R(C T ). Viamembership relation, we now derive this particular condition from the generalfirst-order condition for optimality (285):In this case, the feasible set isC ∆ = {x∈ R n | Cx = d} = {Zξ + x p | ξ ∈ R n−rank C } (288)