On nonlinear optimization since 1959

This view of the development of algorithms for nonlinear optimization is based on the research that has been of particular interest to the author since 1959, including several of his own contributions. After a brief survey of classical methods, which may require good starting points in order to converge successfully, the huge impact of variable metric and conjugate gradient methods is addressed. It encouraged the use of penalty and barrier functions for expressing constrained calculations in unconstrained forms, which are introduced briefly including the augmented Lagrangian method. Direct methods that make linear approximations to constraints became more popular in the late 1970s, especially sequential quadratic programming, which receives attention too. Sometimes the linear approximations are satisfied only if the changes to the variables are so large that the approximations become unsuitable, which stimulated the development of trust region techniques that make partial corrections to the constraints. That work is also introduced, noting that quadratic models of the objective or Lagrange function do not have to be convex. We consider the sequence of models that is given by the symmetric Broyden updating formula in unconstrained optimization, including the case when first derivatives are not available. The emphasis of the paper is on algorithms that can be applied without good initial estimates of the variables.


Report DAMTP 2008/NA03, University of Cambridge, UK. January/2008 (to be published in The Birth of Numerical Analysis, editors A. Bultheel and R. Cools, World Scientific Publishing Company).



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