Procedures

This subroutine seeks the least value of a function of many variables, by applying a trust region method that forms quadratic models by interpolation. There is usually some freedom in the interpolation conditions, which is taken up by minimizing the Frobenius norm of the change to the second derivative of the model, beginning with the zero matrix. The values of the variables are constrained by upper and lower bounds.

Test problem for bobyqa, the objective function being the sum of the reciprocals of all pairwise distances between the points P_I, I=1,2,...,M in two dimensions, where M=N/2 and where the components of P_I are X(2I-1) and X(2I). Thus each vector X of N variables defines the M points P_I. The initial X gives equally spaced points on a circle. Four different choices of the pairs (N,NPT) are tried, namely (10,16), (10,21), (20,26) and (20,41). Convergence to a local minimum that is not global occurs in both the N=10 cases. The details of the results are highly sensitive to computer rounding errors. The choice IPRINT=2 provides the current X and optimal F so far whenever RHO is reduced. The bound constraints of the problem require every component of X to be in the interval [-1,1].

This subroutine minimizes an objective function F(X) subject to M inequality constraints on X, where X is a vector of variables that has N components. The algorithm employs linear approximations to the objective and constraint functions, the approximations being formed by linear interpolation at N+1 points in the space of the variables. We regard these interpolation points as vertices of a simplex. The parameter RHO controls the size of the simplex and it is reduced automatically from RHOBEG to RHOEND. For each RHO the subroutine tries to achieve a good vector of variables for the current size, and then RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and RHOEND should be set to reasonable initial changes to and the required accuracy in the variables respectively, but this accuracy should be viewed as a subject for experimentation because it is not guaranteed.

Test routine for cobyla.

This subroutine seeks the least value of a function of many variables,  subject to general linear inequality constraints, by a trust region  method that forms quadratic models by interpolation.

Test problem for lincoa.

This subroutine seeks the least value of a function of many variables, by a trust region method that forms quadratic models by interpolation. There can be some freedom in the interpolation conditions, which is taken up by minimizing the Frobenius norm of the change to the second derivative of the quadratic model, beginning with a zero matrix.

The Chebyquad test problem (Fletcher, 1965) for N = 2,4,6 and 8, with NPT = 2N+1.

This subroutine seeks the least value of a function of many variables, by a trust region method that forms quadratic models by interpolation.

The Chebyquad test problem (Fletcher, 1965) for N = 2,4,6,8.