# Trust region method

### From Glossary

The iteration is defined as , where is the (complete) change determined by a subproblem of the form:

where F depends on the iterate and is an approximation of the change in objective function value. The particular norm and the magnitude of D determine the set of admissible change values (p), and this is called the *trust region*. A common choice of F is the quadratic form using the Taylor expansion about the current iterate, x, as:

Using the Euclidean norm and applying the Lagrange Multiplier Rule to the subproblem yields p from the equation:

Note that for u=0, the iteration is Newton's method, and for
very large u, the iteration is nearly Cauchy's steepest ascent.