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Penalty variable sample size method for solving optimization problems with equality constraints in a form of mathematical expectation

  • Nataša Krklec Jerinkić
  • Andrea RožnjikEmail author
Original Paper
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Abstract

Equality-constrained optimization problems with deterministic objective function and constraints in the form of mathematical expectation are considered. The constraints are approximated by employing the sample average where the sample size varies throughout the iterations in an adaptive manner. The proposed method incorporates variable sample size scheme with cumulative and unbounded sample into the well- known quadratic penalty iterative procedure. Line search is used for globalization and the sample size is updated in a such way to preserve the balance between two types of errors—errors coming from the sample average approximation and the approximation of the optimal point. Moreover, the penalty parameter is also updated in an adaptive way. We prove that the proposed algorithm pushes the sample size and the penalty parameter to infinity which further allows us to prove the almost sure convergence towards a Karush-Kuhn-Tucker optimal point of the original problem under the rather standard assumptions. Numerical comparison on a set of relevant problems shows the advantage of the proposed adaptive scheme over the heuristic (predetermined) sample scheduling in terms of number of function evaluations as a measure of the optimization cost.

Keywords

Stochastic optimization Equality constraints Sample average approximation Variable sample size Quadratic penalty method Line search 

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Notes

Acknowledgements

We are grateful to the anonymous referee whose comments and suggestions helped us to improve the quality of this paper.

Funding

Nataša Krklec Jerinkić is supported by the Serbian Ministry of Education, Science and Technological Development, grant no. 174030.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia
  2. 2.Faculty of Civil EngineeringUniversity of Novi SadSuboticaSerbia

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