A short review of the available techniques.

There are two possible approaches. The first approach is now described.

The steps of the unconstrained algorithm are the solution of:

(8.1) | ||

subject to |

In the first approach (=``

The problem of solving 8.2 is not trivial at all. It's in fact as difficult as the original problem. The only advantage in solving this subproblem at each step is that the objective function (which is ) evaluations are cheap and thus we can have very precise steps leading (hopefully) to a fast convergence to the solution of the original problem.

The same methods used for solving the subproblem 8.2 can be directly applied to the original non-linear objective function. This is our second approach (=``

There are specific methods for box or linear constraints and for non-linear constraints. We will describe them in two separate chapters.

- Linear constraints

- Non-Linear constraints

- Primal-dual interior point
- Duality
- A primal-dual Algorithm
- Central path
- Link between Barrier method and Interior point method
- A final note on primal-dual algorithms.
- SQP Methods
- A small note about the H matrix in the SQP algorithm.
- A final note about the SQP algorithm.

- The final choice of a constrained algorithm