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A derivative-free line search and DFP method for symmetric equations with global and superlinear convergence. (English) Zbl 0927.65067

Authors’ summary: This paper is concerned with the open problem as to whether the Dennis-Fletcher-Powell (DFP) method with inexact line search converges globally to the minimum of a uniformly convex function. We study this problem by way of a Gauss-Newton approach rather than an ordinary Newton approach. We also propose a derivative-free line search that can be implemented conveniently by a backtracking process and has such an attractive property that any iterative method with this line search generates a sequence of iterates that is approximately norm descent. Moreover, if the Jacobian matrices are uniformly nonsingular, then the generated sequence converges. Under appropriate conditions, we establish global and superlinear convergence of the proposed Gauss-Newton based DFP method, which supports the open problem positively.

MSC:

65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
90C25 Convex programming

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References:

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