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An estimation of the slope of $ q(x)$ at the origin.

An estimation of the slope of $ q(x)$ at the origin is given by $ \lambda_1$. In the optimization program, we will only compute $ \lambda_1$ when we have interior convergence. The algorithm to find $ \lambda_1$ is the following:
  1. Set $ \lambda_L:=0$.
  2. Set $ \displaystyle \lambda_U := \min \bigg[ \max_i \Big[ [H]_{i,i} + \sum_{i\neq j } \vert [H]_{i,j} \vert \Big], \Vert H\Vert _F, \Vert H\Vert _{\infty} \bigg] $
  3. Set $ \displaystyle \lambda: = \frac{\lambda_L+\lambda_U}{2}$
  4. Try to factorize $ H(-\lambda)=L L^T$.
  5. If $ \lambda_L<0.99 \; \lambda_U$ go back to step 3.
  6. The required value of $ \lambda_1$ (=the approximation of the slope at the origin) is inside $ \lambda _L$

Frank Vanden Berghen 2004-04-19