Is it true for the convex optimization problems?

Question

We know that if the inequality constraint is active then the optimal value of the Lagrange multiplier is positive ($>0$). And similarly if the constraint is inactive then the optimal Lagrange multiplier is $=0$.

Can we also say that if the optimal value of Lagrange multiplier $>0$ then the corresponding constraint is active? And similarly can we also say that if the optimal value of the Lagrange multiplier is $0$ then the corresponding constraint is inactive?

Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

The complementary slackness conditions say:

$$\lambda^*_i f_i(x^*) = 0 \quad \forall i,$$

where $\lambda^*_i$ is the $i$th optimal Lagrange multiplier (dual value) and $f_i(x) \le 0$ is the $i$th primal constraint.

In words: Either the $i$th optimal Lagrange multiplier is $0$, or the $i$th primal constraint has no slack (or both).

So, you can say:

• If the constraint is inactive (has slack) then the multiplier is $0$.
• If the multiplier is $>0$ then the constraint is active (has no slack).

You cannot say:

• If the constraint is active then the multiplier is $>0$.
• If the multiplier is $0$ then the constraint is inactive.

(Because it is possible for both the multiplier and the constraint slack to equal $0$.)