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A projected Newton method for minimization problems with nonlinear inequality constraints

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Summary

Recently developed projected Newton methods for minimization problems in polyhedrons and Cartesian products of Euclidean balls are extended here to general convex feasible sets defined by finitely many smooth nonlinear inequalities. Iterate sequences generated by this scheme are shown to be locally superlinearly convergent to nonsingular extremals\(\bar u\), and more specifically, to local minimizers\(\bar u\) satisfying the standard second order Kuhn-Tucker sufficient conditions; moreover, all such convergent iterate sequences eventually enter and remain within the smooth manifold defined by the active constraints at\(\bar u\). Implementation issues are considered for large scale specially structured nonlinear programs, and in particular, for multistage discrete-time optimal control problems; in the latter case, overall per iteration computational costs will typically increase only linearly with the number of stages. Sample calculations are presented for nonlinear programs in a right circular cylinder in ℝ3.

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Authors and Affiliations

  1. Mathematics Department, North Carolina State University, Box 8205, 27695-8205, Raleigh, NC, USA

    J. C. Dunn

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  1. J. C. Dunn
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Investigation supported by NSF Research Grant #DMS-85-03746

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Dunn, J.C. A projected Newton method for minimization problems with nonlinear inequality constraints. Numer. Math. 53, 377–409 (1988). https://doi.org/10.1007/BF01396325

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  • DOI: https://doi.org/10.1007/BF01396325

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