next up previous contents
Next: Remarks about the constrained Up: Detailed description of the Previous: The SQP algorithm   Contents

The constrained step in detail

The step $ s_k$ of the constrained algorithm are the solution of:

\begin{displaymath}\begin{array}{l} \min_{s \in \Re^n} q(x_k+s) \equiv f(x_k)
 +...
...\ldots,l \\  \Vert s \Vert _2 < \Delta \end{cases}
 \end{array}\end{displaymath}    

We will use a null-space, active set approach. We will follow the notations of section 9.1.1.
  1. Let $ \lambda $ be a vector of Lagrange Multiplier associated with all the linear constraints. This vector is recovered from the previous calculation of the constrained step. Set $ k=1$, $ {\cal
A}=$The constraints which are active are determined by a non-null $ \lambda $ , $ s_1=0$. If a $ \lambda $ associated with a non-linear constraint is not null, set NLActive$ =true$, otherwise set NLActive$ =false$.
  2. Compute the matrix $ Y$ and $ Z$ associated with the reduced space of the active box and linear constraints. The active set is determined by $ \cal A$.
  3. We will now compute the step in the reduced-space of the active box and linear constraints. Check NLActive:
  4. Compute the Lagrange multipliers $ \lambda_k$. If $ \lambda_k
\geq 0$ for all constraints then terminate. Remove from $ \cal A$ the constraints which have negative $ \lambda_k$.
  5. Check if a non-linear constraint has been violated. If the test is true, set NLActive$ =true$, set $ k:=k+1$ and go to (2).
  6. Solve 9.13 and add if necessary a new box or linear constraint inside $ \cal A$. Set $ k:=k+1$ and go to (2).
This is really a small, simple sketch of the implemented algorithm. The real algorithm has some primitive techniques to avoid cycling. As you can see, the algorithm is also able to "warm start", using the previous $ \lambda $ computed at the previous step.
next up previous contents
Next: Remarks about the constrained Up: Detailed description of the Previous: The SQP algorithm   Contents
Frank Vanden Berghen 2004-04-19