<P> the Lagrangian dual problem is </P> <Dl> <Dd> maximize u inf x (f (x) + ∑ j = 1 m u j g j (x)) s u b j e c t t o u i ≥ 0, i = 1,..., m (\ displaystyle (\ begin (aligned) & (\ underset (u) (\ operatorname (maximize))) && \ inf _ (x) \ left (f (x) + \ sum _ (j = 1) ^ (m) u_ (j) g_ (j) (x) \ right) \ \ & \ operatorname (subject \; to) &&u_ (i) \ geq 0, \ quad i = 1, \ dots, m \ end (aligned))) </Dd> </Dl> <Dd> maximize u inf x (f (x) + ∑ j = 1 m u j g j (x)) s u b j e c t t o u i ≥ 0, i = 1,..., m (\ displaystyle (\ begin (aligned) & (\ underset (u) (\ operatorname (maximize))) && \ inf _ (x) \ left (f (x) + \ sum _ (j = 1) ^ (m) u_ (j) g_ (j) (x) \ right) \ \ & \ operatorname (subject \; to) &&u_ (i) \ geq 0, \ quad i = 1, \ dots, m \ end (aligned))) </Dd> <P> where the objective function is the Lagrange dual function . Provided that the functions f (\ displaystyle f) and g 1, ⋯, g m (\ displaystyle g_ (1), \ cdots, g_ (m)) are continuously differentiable, the infimum occurs where the gradient is equal to zero . The problem </P>

Using duality theory show that the optimal values of the primal and dual variables are given by