Abstract
Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples.
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Acknowledgments
The authors gratefully acknowledge support by the Office of Naval Research under grant N00014-11-1-0068, and by the Defense Advanced Research Project Agency under contract HR0011-12-C-0011. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the US Government. Approved for public release, distribution unlimited.
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November 23, 2013, Revised March 25, 2014.
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Hager, W.W., Mico-Umutesi, D. Error estimation in nonlinear optimization. J Glob Optim 59, 327–341 (2014). https://doi.org/10.1007/s10898-014-0186-y
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DOI: https://doi.org/10.1007/s10898-014-0186-y