Behavior of Newton-Type Methods Near Critical Solutions of Nonlinear Equations with Semismooth Derivatives

Andreas Fischer (), Alexey F. Izmailov () and Mario Jelitte ()
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Andreas Fischer: Technische Universität Dresden
Alexey F. Izmailov: Lomonosov Moscow State University
Mario Jelitte: Technische Universität Dresden

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 3, No 4, 2179-2205

Abstract: Abstract Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the analysis and results on local linear convergence and asymptotic acceptance of the full step, the issues addressed in this work. Moreover, we consider not only the basic Newton method, but also some stabilized versions of it intended for tackling singular (including nonisolated) solutions. Applications to nonlinear complementarity problems are also dealt with.

Keywords: Nonlinear equation; Constrained equation; Strongly semismooth derivative; Singular solution; Critical solution; 2-Regularity; Perturbed Newton method; Acceptance of the full step; Extrapolation; Nonlinear complementarity problem; 49J52; 65J15; 65K15; 90C33 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-023-02350-w

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