Model Predictive Control

14 2 Optimality Conditions
From the last equation we can conclude that u ∗′ g(z ∗ ) = m
i=1 u∗ i gi(z ∗ ) = 0 and
since u ∗ i ≥ 0 and gi(z ∗ ) ≤ 0, we have
u ∗ i gi(z ∗ ) = 0, i = 1, . . .,m (2.16)
Conditions (2.16) are called complementary slackness conditions. Complementary
slackness conditions can be interpreted as follows. If the i-th constraint of
the primal problem is inactive at optimum (gi(z∗ ) < 0) then the i-th dual optimizer
has to be zero (u ∗ i
= 0). Vice versa, if the i-th dual optimizer is different from zero
(u ∗ i > 0), then the i-th constraint is active at the optimum (gi(z ∗ ) = 0).
Relation (2.16) implies that the inequality in (2.14) holds as equality
f(z ∗ ) +
u ∗ i gi(z ∗ ) +
i
j
v ∗ jhj(z ∗ ) = min
z∈Z
⎛
⎝f(z) +
u ∗ i gi(z) +
v ∗ ⎞
jhj(z) ⎠.
(2.17)
Therefore, complementary slackness implies that z ∗ is a minimizer of L(z, u ∗ , v ∗ ).
2.4 Karush-Kuhn-Tucker Conditions
Consider the (primal) optimization problem (2.1) and its dual (2.10). Assume that
strong duality holds. Assume that the cost functions and constraint functions f,
gi, hi are differentiable. Let z ∗ and (u ∗ , v ∗ ) be primal and dual optimal points,
respectively. Complementary slackness implies that z ∗ minimizes L(z, u ∗ , v ∗ ) under
no constraints (equation (2.17)). Since f, gi, hi are differentiable, the gradient of
L(z, u ∗ , v ∗ ) must be zero at z ∗
∇f(z ∗ ) +
u ∗ i ∇gi(z ∗ ) +
v ∗ j ∇hj(z ∗ ) = 0.
i
In summary, the primal and dual optimal pair z ∗ , (u ∗ , v ∗ ) of an optimization
problem with differentiable cost and constraints and zero duality gap, have to
satisfy the following conditions:
∇f(z ∗ ) +
m
u ∗ i ∇gi(z ∗ ) +
i=1
j
i
p
v ∗ j ∇hi(z ∗ ) = 0, (2.18a)
j=1
u ∗ i gi(z ∗ ) = 0, i = 1, . . .,m (2.18b)
u ∗ i ≥ 0, i = 1, . . .,m (2.18c)
gi(z ∗ ) ≤ 0, i = 1, . . .,m (2.18d)
hj(z ∗ ) = 0 j = 1, . . .,p (2.18e)
where equations (2.18d)-(2.18e) are the primal feasibility conditions, equation (2.18c)
is the dual feasibility condition and equations (2.18b) are the complementary slackness
conditions.
j