What does it mean to say Lagrange multiplier $\lambda$ denotes "tightness" of constraint? The problem is to $\max_x f(x)$ given the constraint that $x \le C$. 
I have two questions.


*

*When we have inequality constraints, how do we form the Lagrange function? Do we add $-\lambda$ or $+\lambda$ to our objective function, and do we multiply this $\lambda$ with $C - x$ or $x - C$?

*I am reading a book which says that at the optimum, the Lagrange multiplier is a measure of the "tightness" of the constraint. What exactly is meant by this, and how can I convince myself that it is true?
 A: With inequality constraints, one uses the Karush-Kuhn-Tucker (KKT) conditions which generalize the Lagrange conditions. There are a set of constraint qualifications which allow us to use the KKT conditions. In this case, your problem has linear constraints so the KKT conditions are accurate.
If the problem is to maximize $f(x)$ subject to $g_i(x) \le 0$ and $h_j(x) = 0$ then we form a multiplier system
$$ \nabla f(x) = \sum_i \mu_i \nabla g_i(x) + \sum_j \lambda_j \nabla h_j(x). $$
This is exactly the same as the Lagrange condition. The new feature is that for each inequality constraint, we demand that the multiplier be positive. That is that $\mu_i \ge 0$ for all $i$. And, of course, we also have the original inequalities and equalities: $g_i(x) \le 0$ and $h_j(x) = 0$
The KKT conditions have a complementary slackness feature. This says that at an optimum $x^*$, either the inequality $g_i(x^*) \le 0$ is tight, meaning that $g_i(x^*) = 0$ or $\mu_i = 0$. This is sometimes written as $\mu_i g_i(x^*) = 0$.
Complementary slackness means that if we are not on the part of the feasible region where the inequality $g_i(x) \le 0$ is preventing us from moving in one direction or another, then we can ignore the inequality all together.
In your problem, we have $g_1(x) = x - C \le 0$. So the KKT conditions give us the following system:
\begin{align*}
x - C &\le 0 \\
f'(x) &= \mu_1 \\
\mu_1 &\ge 0.
\end{align*}
(Since $g_1'(x) = 1$.) Complementary slackness says that either $x = C$ or $\mu_1 = 0$. That is, either the optimum is at $x = C$ with $f'(C) \ge 0$ or the optimum is with $x < C$ and $f'(x) = 0$ (the usual condition for optimality).
I believe, and I could be wrong, that "measure of tightness" refers to complementary slackness. The actual size of the multiplier doesn't say to much about how close $g_i(x) \le C$ is to being tight (i.e. to being equal).
