Barrier method

Suppose we want to minimize a Convex Function subject to .
We introduce barrier functions in order to enforce the constraint:

Now stays away from because would blow up.

Theorem

Let minimize subject to and let minimize .
Then:

where is the number of constraints.

The method:

1. Pick some and a feasible
2. Find the minimum of using favourite Convex optimisation method
3. make smaller and repeat until satisfied.