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Approximate norm descent methods for constrained nonlinear systems. (English) Zbl 1383.65051
Summary: We address the solution of convex-constrained nonlinear systems of equations where the Jacobian matrix is unavailable or its computation/storage is burdensome. In order to efficiently solve such problems, we propose a new class of algorithms which are “derivative-free” both in the computation of the search direction and in the selection of the steplength. Search directions comprise the residuals and quasi-Newton directions while the steplength is determined by using a new linesearch strategy based on a nonmonotone approximate norm descent property of the merit function. We provide a theoretical analysis of the proposed algorithm and we discuss several conditions ensuring convergence to a solution of the constrained nonlinear system. Finally, we illustrate its numerical behaviour also in comparison with existing approaches.

MSC:
 65H10 Numerical computation of solutions to systems of equations 65K05 Numerical mathematical programming methods 90C25 Convex programming 90C53 Methods of quasi-Newton type 90C56 Derivative-free methods and methods using generalized derivatives
STRSCNE; levmar
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