Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets

J. Han and D. Sun

Journal of Optimization Theory and Applications, 1997, vol. 94, issue 3, No 6, 659-676

Abstract: Abstract We present a generalized Newton method and a quasi-Newton method forsolving $$H(x): = F(\prod {_c } (x)) + x - \prod {_c } (x) = 0$$ , when C isa polyhedral set. For both the Newton and quasi-Newton methodsconsidered here, the subproblem to be solved is a linear system ofequations per iteration. The other characteristics of the quasi-Newtonmethod include: (i) a Q-superlinear convergencetheorem is established without assuming the existence ofH′ at a solution x * ofH(x)=0; (ii) only oneapproximate matrix is needed; (iii) the linearindependence condition is not assumed; (iv)Q-superlinear convergence is established on the originalvariable x; and (v) from theQR-factorization of the kth iterative matrix, we needat most $$O((1 + 2\left| {J_k } \right| + 2\left| {L_k } \right|)n^2 )$$ arithmetic operations to get the QR-factorization of the(k+1)th iterative matrix.

Keywords: Normal maps; Newton methods; quasi-Newton methods; Q-superlinear convergence (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (3)

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DOI: 10.1023/A:1022653001160

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