5.1. Example of the solution of the constrained optimization.
We continue to study constrained optimization.
And we'll begin with one more example.
We set the problem of maximization, minimization subject to constraint.
We have learned that we need to start with checking NDCQ condition.
or rather its gradient will turn zero at some point.
So our constraint is the function.
gradient of g is a vector.
doesn't have this point, so it doesn't belong.
That means the origin, this point, the origin.
And this is the set.
We call it the constraint set.
So we claim NDCQ holds.
After that we form Lagrangian function using the formula.
the second equations, move these terms to the right.
After that, having in mind that lam is not zero, it's clear that it's not zero.
right hand sides by right hand sides and finally we get a smaller system.
And it's very easy to solve it.
The first point has coordinates.
And the second point, let it be B.
Is symmetric to A point with respect to the origin.
minimum of the given function 3x + 4y.
they are points of extremum.
such a function takes the greatest and the least values.
So let me make the statement, precise statement.
So Weierstrass is German mathematician of the 19th century.