Newton-type methods near critical solutions of piecewise smooth nonlinear equations

• Published in: Computational optimization and applications Vol. 80; no. 2; pp. 587 - 615
• Main Authors: Fischer, A, Izmailov, A. F, Jelitte, M
• Format: Journal Article
• Language: English
• Published: New York, NY Springer US 01-11-2021; Springer Nature B.V
• Subjects: 2-regularity; Complementarity problem; Constrained equation; Convergence; Convex and Discrete Geometry; Critical solution; Globalization; Management Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Operations Research; Operations Research/Decision Theory; Optimization; Piecewise smooth equation; Singular solution; Statistics; Constrained equation; 65K15; 49J53; 49J52; 90C33; Singular solution; Piecewise smooth equation; Complementarity problem; Critical solution; 2-regularity
• ISSN: 1573-2894; 0926-6003; 1573-2894
• DOI: 10.1007/s10589-021-00306-2

It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems.