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Subsections

# Linear constraints

We want to find     which satisfies:

 Subject to: (8.3)

where is the dimension of the search space and is the number of linear constraints.

## Active set - Null space method

The material of this section is based on the following reference: [Fle87].

At each step computed from 8.1, we check if one of the linear constraints has been violated. On the figure 8.1, the linear constraint has been violated and it's thus "activated". Without loosing generality, let's define and the set of the active constraints.

Let and be and matrices respectively such that is non-singular, and in addition let and . The step must be feasible, so we must have to . A solution to this equation is given by: where can be any vector. Indeed, we have the following: . The situation is illustrated in figure 8.2. We will choose as the solution of

 subject to (8.4)

with . In other words, is the minimum of the quadratical approximation of the objective function limited to the reduced space of the active linear constraints and limited to the trust region boundaries. We have already developed an algorithm able to compute in chapter 4.
When using this method, there is no difference between " approach 1" and "approach 2".
This algorithm is very stable in regards to rounding error. It's very fast because we can make Newton step (quadratical convergence speed) in the reduced space. Beside, we can use software developed in chapter 4. For all these reasons, it has been chosen and implemented. It will be fully described in chapter 9.