47.1 Introduction to cobyla

fmin_cobyla is a Common Lisp translation (via f2cl) of the Fortran constrained optimization routine COBYLA by Powell[1][2][3].

COBYLA minimizes an objective function \(F(X)\) subject to \(M\) inequality constraints of the form \(g(X) \ge 0\) on \(X\), where \(X\) is a vector of variables that has \(N\) components.

Equality constraints \(g(X) = 0\) can often be implemented by a pair of inequality constraints \(g(X) \ge 0\) and \(-g(X) \ge 0.\) Maxima’s interface to COBYLA allows equality constraints and internally converts the equality constraints to a pair of inequality constraints.

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. The interpolation points are regarded 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. The routine treats each constraint individually when calculating a change to the variables, rather than lumping the constraints together into a single penalty function. The name of the subroutine is derived from the phrase Constrained Optimization BY Linear Approximations.

References:

[1] Fortran Code is from http://plato.asu.edu/sub/nlores.html#general

[2] M. J. D. Powell, "A direct search optimization method that models the objective and constraint functions by linear interpolation," in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.

[3] M. J. D. Powell, "Direct search algorithms for optimization calculations," Acta Numerica 7, 287-336 (1998). Also available as University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from http://www.damtp.cam.ac.uk/user/na/reports.html