Stochastic Programming - Index of

84 STOCHASTIC PROGRAMMING
as the necessary condition
∃û ≥ 0 such that ∇f(ˆx)+
m∑
û i ∇g i (ˆx) =0,
i=1
m∑
û i g i (ˆx) =0.
(b) if we have a primal feasible and a dual feasible solution ˜x and ũ
respectively, such that the difference of the respective objectives is zero
then, by the weak duality theorem (Proposition 1.17), ˜x solves the primal
problem; in other words, given a feasible ˜x, the condition
∃ũ ≥ 0 such that ∇f(˜x)+
i=1
m∑
ũ i ∇g i (˜x) =0,
i=1
m∑
ũ i g i (˜x) =0
i=1
is sufficient for ˜x to be a solution of the program (8.4).
✷
Remark 1.10 The optimality condition derived in Remark 1.9 for the linear
case could be formulated as follows:
(1) For the feasible ˆx the negative gradient of the objective f—i.e. the
direction of the greatest (local) descent of f—is equal (with the multipliers
û i ≥ 0) to a nonnegative linear combination of the gradients of those
constraint functions g i that are active at ˆx, i.e. that satisfy g i (ˆx) =0.
(2) This corresponds to the fact that the multipliers satisfy the complementarity
conditions û i g i (ˆx) =0, i =1, ···,m, stating that the multipliers
û i are zero for those constraints that are not active at ˆx, i.e. that satisfy
g i (ˆx) < 0.
In conclusion, this optimality condition says that −∇f(ˆx) mustbecontained
in the convex polyhedral cone generated by the gradients ∇g i (ˆx) of the constraints
being active in ˆx. This is one possible formulation of the Kuhn–Tucker
conditions illustrated in Figure 28.
✷
Let us now return to the more general nonlinear case and consider the
following question. Given that ˆx is a (local) solution, under what assumption