Newton’s method and secant method for set-valued mappings. (English) Zbl 1354.65097

Lirkov, Ivan (ed.) et al., Large-scale scientific computing. 8th international conference, LSSC 2011, Sozopol, Bulgaria, June 6–10, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-29842-4/pbk). Lecture Notes in Computer Science 7116, 91-98 (2012).
Summary: For finding zeros or fixed points of set-valued maps, the fact that the space of convex, compact, nonempty sets of \(\mathbb R^{n }\) is not a vector space presents a major disadvantage. Therefore, fixed point iterations or variants of Newton’s method, in which the derivative is applied only to a smooth single-valued part of the set-valued map, are often applied for calculations. We will embed the set-valued map with convex, compact images (i.e. by embedding its images) and shift the problem to the Banach space of directed sets. This Banach space extends the arithmetic operations of convex sets and allows to consider the Fréchet-derivative or divided differences of maps that have embedded convex images. For the transformed problem, Newton’s method and the secant method in Banach spaces are applied via directed sets. The results can be visualized as usual nonconvex sets in \(\mathbb R^{n}\).
For the entire collection see [Zbl 1241.65001].


65H10 Numerical computation of solutions to systems of equations
47J20 Variational and other types of inequalities involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
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