If it lies on the interior, the trust region may as well not have been there and therefore is the unconstrained minimizer of . We have seen in Equation 2.4 () how to find it. We have seen in Equation 2.6 that

in order to be able to apply 2.4.

If we found a value of using 2.4 which lies outside the trust region, it means that lies on the trust region boundary. Let's take a closer look to this case:

First, we rewrite the constraints as . Now, we introduce a Lagrange multiplier for the constraint and use first-order optimality conditions (see Annexe, section 13.3 ). This gives:

(4.3) |

Using first part of Equation 13.22,we have

which is 4.3.

We will now proof that must be positive (semi)definite.

Suppose is a feasible point ( ), we obtain:

Using the secant equation (see Annexes, Section 13.4), , we can rewrite 4.5 into . This and the restriction that implies that:

Combining 4.6 and 4.7

Let's define a line as a function of the scalar . This line intersect the constraints for two values of : and at which . So , and therefore, using 4.8, we have that

If is positive definite, then for any , and therefore 4.8 shows that whenever is feasible. Thus is the unique global minimizer.

Using 4.2 (which is concerned about an interior minimizer) and the previous paragraph (which is concerned about a minimizer on the boundary of the trust region), we can state:

The parameter is said to ``regularized'' or ``modify'' the model such that the modified model is convex and so that its minimizer lies on or within the trust region boundary.