M. J. D. Powell

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The problem of constructing a function 4,(x, y) of two variables on a triangle, such that ~(x, y) and its first derivatives take given values at the vertices, where ¢(x, y) is composed of quadratic pieces, is considered. Two methods of constructing piecewise quadratic approximations are described which have the property that, if they are applied on each(More)
UOBYQA is a new algorithm for general unconstrained optimization calculations, that takes account of the curvature of the objective function, F say, by forming quadratic models by interpolation. Therefore, because no rst derivatives are required, each model is deened by 1 2 (n+1)(n+2) values of F, where n is the number of variables, and the interpolation(More)
Lagrangian functions are the basis of many of the more successful methods for nonlinear constraints in optimization calculations. Sometimes they are used in conjunction with linear approximations to the constraints and sometimes penalty terms are included to allow the use of algorithms for unconstrained optimization. Much has been discovered about these(More)
Many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large [2]. We consider how large these approximations have to be, if they prevent convergence when the objective function is bounded below and continuously differentiable. Thus we obtain a useful(More)
Quadratic models of objective functions are highly useful in many optimization algorithms. They are updated regularly to include new information about the objective function, such as the diierence between two gradient vectors. We consider the case, however, when each model interpolates some function values, so an update is required when a new function value(More)
Many diierent procedures have been proposed for optimization calculations when rst derivatives are not available. Further, several researchers have contributed to the subject, including some who wish to prove convergence theorems , and some who wish to make any reduction in the least calculated value of the objective function. There is not even a key idea(More)