Some leading experts in optimization take the view that convergence theory is necessary for the development of a successful new algorithm. The speaker has found, however, that this view has two severe disadvantages. Firstly, several algorithms have become highly useful without suitable convergence theory. Secondly, available techniques for proving convergence may impose restrictions on new algorithms that exclude huge improvements in practice over existing software. Examples of these situations will be presented and discussed, most of them being from unconstrained optimization.
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