In the Lagrangian, do we add or subtract terms from the objective function? Are there two accepted definitions of the Lagrangian?
Most of the time, I have seen the Lagrangian of an optimization problem defined as the objective function minus a bunch of terms related to the constraints. However, the version below from Convex Optimization by Boyd/Vandenberghe seems to define a totally different function. Surely the addition of the terms (rather than their subtraction) is not a typo by the authors?

 A: For the optimization itself, the sign doesn’t matter – as noted in a comment, $\lambda_i$ can have either sign anyway, and also the constraint $f_i(x)=0$ could be multiplied by $-1$ without changing the problem.
There’s one reason, however, that I learned about only recently (after decades of applying Lagrange multipliers) to use a particular sign. If you have a constraint of the form $f(x,y)=b$, then if you introduce the Lagrange multiplier in the form $L(x,y)=u(x,y)-\lambda f(x,y)$ you can interpret $\lambda$ as $\frac{\mathrm du}{\mathrm db}$ (where $u(b)$ is the optimal value of $u$ given the constant $b$ in the constraint). The first-order stationarity conditions are
$$
\frac{\partial u}{\partial x}-\lambda\frac{\partial f}{\partial x}=0\;,\\
\frac{\partial u}{\partial y}-\lambda\frac{\partial f}{\partial y}=0\;,
$$
and this yields
\begin{eqnarray}
\frac{\mathrm du}{\mathrm db}
&=&
\frac{\partial u}{\partial x}\frac{\mathrm dx}{\mathrm db}+\frac{\partial u}{\partial y}\frac{\mathrm dy}{\mathrm db}
\\
&=&
\lambda\frac{\partial f}{\partial x}\frac{\mathrm dx}{\mathrm db}+\lambda\frac{\partial f}{\partial y}\frac{\mathrm dy}{\mathrm db}
\\
&=&
\lambda\left(\frac{\partial f}{\partial x}\frac{\mathrm dx}{\mathrm db}+\frac{\partial f}{\partial y}\frac{\mathrm dy}{\mathrm db}\right)
\\
&=&
\lambda\frac{\mathrm df}{\mathrm db}
\\
&=&
\lambda\;.
\end{eqnarray}
So if $b$ is some sort of budget you can spend on $x$ and $y$ (for instance with $f(x,y)=xp_y+yp_y$, where $p_x$ and $p_y$ are the prices per unit of $x$ and $y$), then $\lambda$ tells you how much your utility $u$ increases when your budget increases. This is used a lot in economics. Obviously it doesn’t work if you use the other sign.
