Impact of changing parameters on Lagrange Multipliers.

Question

I have the following problem:

$$\min f(x)\\ s.t. g_1(x)=w \\g_2(x) \geq 0$$.

I am setting up a Lagrange equation $L(x,\lambda,\mu) = f(x) + \alpha(g_1(x) - w) + \beta(g_2(x)$.

Importantly, $g_2(x)$ does not feature parameter $w$. My (very simple) questions are the following. Can I conclude that in the solution $\beta$ is positive or negative. If so, how can I prove it. Secondly, if I change parameter $w$, what will happen to $\beta$? My guess is that it won't change.

I would appreciate any help.

Thanks!

Answer 1

By definition of the dual, any feasible $\beta$ is negative.

Changing $w$ may or may not affect $\beta$. For example, when the last constraint is not binding for both values of $w$, $\beta=0$ in both cases.