UOBYQA: unconstrained optimization by quadratic approximation.

*(English)*Zbl 1014.65050A new algorithm for general unconstrained optimization calculations is described. It takes account of the curvature of the objective function by forming quadratic models by interpolation. Obviously, no first derivatives are required. A typical iteration of the algorithm generates a new vector of variables either by minimizing the quadratic model subject to a trust region bound, or by a procedure that should improve the accuracy of the model. The paper addresses the initial positions of the interpolation points and the adjustment of trust region radii.

The algorithm works with the Lagrange functions of the interpolation equations explicitly; therefore their coefficients are updated when an interpolation point is moved. The Lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation of the function being minimized. It is pointed out that results are very promising for functions with less than twenty variables.

The algorithm works with the Lagrange functions of the interpolation equations explicitly; therefore their coefficients are updated when an interpolation point is moved. The Lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation of the function being minimized. It is pointed out that results are very promising for functions with less than twenty variables.

Reviewer: R.P.Tewarson (Stony Brook)

##### MSC:

65K05 | Numerical mathematical programming methods |

90C30 | Nonlinear programming |

90C56 | Derivative-free methods and methods using generalized derivatives |