Removing Equality Constraints

We’re going to learn about good constraints. Those are constraints that make our lives easier.

Getting started with equality constraints

Remember the problem of finding the smaller surface of the soda cans? Here’s the problem formulation:

$$\min_{r, h} 2\pi rh + 2\pi r^2 \\ s.t.: \pi r^2h = 355$$

That constraint was helpful. We knew how to deal with the target functions of one variable and the constraint helped us to get rid of variable $h$.

We used the constraint to make $h$ a function of $r$:

$$h = \frac{355}{\pi r^2}$$

Then we transformed the target function accordingly:

$$\min_{r} 2\pi r\frac{355}{\pi r^2} + 2\pi r^2$$

We transformed a multivariate constrained problem into a univariate unconstrained one. A pretty good deal!