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

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  • J. Han
  • D. Sun

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.

Suggested Citation

  • J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    • Handle: RePEc:spr:joptap:v:94:y:1997:i:3:d:10.1023_a:1022653001160
    DOI: 10.1023/A:1022653001160
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    References listed on IDEAS

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    1. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    2. Stephen M. Robinson, 1992. "Normal Maps Induced by Linear Transformations," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 691-714, August.
    3. Jong-Shi Pang & Daniel Ralph, 1996. "Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 401-426, May.
    4. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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    Cited by:

    1. Shenglong Hu & Guoyin Li, 2021. "$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex," Computational Optimization and Applications, Springer, vol. 80(3), pages 915-941, December.
    2. Youyicun Lin & Shenglong Hu, 2022. "$${\text {B}}$$ B -Subdifferential of the Projection onto the Generalized Spectraplex," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 702-724, February.

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