The BOBYQA algorithm for bound constrained optimization without derivatives

BOBYQA is an iterative algorithm for finding a minimum of a function F(x), x2R n , subject to bounds axb on the variables, F being specified by a black box that returns the value F(x) for any feasible x. Each iteration employs a quadratic approximation Q to F that satisfies Q(y j )= F(y j ), j =1 ,2,...,m, the interpolation points y j being chosen and adjusted automatically, but m is a prescribed constant, the value m =2 n+1 being typical. These conditions leave much freedom in Q, taken up when the model is updated by the highly successful technique of minimizing the Frobenius norm of the change to the second derivative matrix of Q. Thus no first derivatives of F are required explicitly. Most changes to the…