On the Convergence of Some Constrained Minimization Algorithms Based on Recursive Quadratic Programming

This paper considers the convergence of the method of recursiveequality quadratic programming (REQP) for constrained minimization.A theorem of Wolfe (1969) gives conditions on the search directionsand step lengths used by a minimization algorithm which ensurethat it will locate an unconstrained stationary point of a function.It is shown here that, under suitable circumstances, the iterationsof REQP satisfy these conditions both with respect to the conventionalpenalty function P(x, r) and also with respect to the augmentedpenalty function proposed by Fletcher (1969) which has a minimumat the solution to the constrained problem. The behaviour ofREQP in the neighbourhood of the solution is also considered,and it is shown that the algorithm is capable of superlinearconvergence.     

Minimization Algorithms Making Use of Non-quadratic Properties of the Objective Function

This paper proposes modifications to the matrix updating formulaeused in the Fletcher-Powell (1963), Broyden (1970) and Fletcher(1970) methods for function minimization, and discusses someproperties of these modified formulae. A function minimizationalgorithm incorporating the new expressions has been programmedand the results of tests with some well-known functions arereported.     

A Note on Minimization Algorithms which make Use of Non-quardratic Properties of the Objective Function

In an earlier paper (Biggs, 1971) modified forms of some matrixupdating formulae for function minimization were introduced,involving a scalar ^*. This brief note presents a simpler methodof calculating ^*.