Variable sample size method for equality constrained optimization problems

Nataša Krejić; Nataša Krklec Jerinkić; Andrea Rožnjik

doi:10.1007/s11590-017-1143-8

Optimization Letters

May 2018, Volume 12, Issue 3, pp 485–497 | Cite as

Variable sample size method for equality constrained optimization problems

Authors
Authors and affiliations

Nataša Krejić
Nataša Krklec Jerinkić
Andrea RožnjikEmail author

Original Paper

First Online: 08 April 2017

Received: 12 September 2016

Accepted: 03 April 2017

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Abstract

An equality constrained optimization problem with a deterministic objective function and constraints in the form of mathematical expectation is considered. The constraints are transformed into the Sample Average Approximation form resulting in deterministic problem. A method which combines a variable sample size procedure with line search is applied to a penalty reformulation. The method generates a sequence that converges towards first-order critical points. The final stage of the optimization procedure employs the full sample and the SAA problem is eventually solved with significantly smaller cost. Preliminary numerical results show that the proposed method can produce significant savings compared to SAA method and some heuristic sample update counterparts while generating a solution of the same quality.

Keywords

Stochastic optimization Equality constraints Variable sample size Penalty method Line search

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Notes

Acknowledgements

We are grateful to the Associate Editor and reviewers whose comments helped us to improve the paper. N. Krejić and N. Krklec Jerinkić are supported by Serbian Ministry of Education, Science and Technological Development, Grant No. 174030.

Appendix

Algorithm 2

Step 0

Input parameters: $dm_{k}$, $\epsilon _{\delta }^{N_{k}}(x_{k})$, $x_k$, $N_k$, $N_{k}^{min}$, $\nu _{1}\in (0,1)$.

Step 1

Determine $N_{k+1}$

1)

If $dm_{k}=\epsilon _{\delta }^{N_{k}}(x_{k})$ set $N_{k+1}=N_{k}$.

2)

If $dm_{k}>\epsilon _{\delta }^{N_{k}}(x_{k})$

Starting with $N=N_{k}$, while $dm_{k}>\frac{N_k}{N}\;\epsilon _{\delta }^{N}(x_{k})$ and $N>N_{k}^{min}$, decrease N by 1 and calculate $\varepsilon _{\delta }^{N}(x_{k}). $ Set $N_{k+1}=N$.

3)

$dm_{k}<\epsilon _{\delta }^{N_{k}}(x_{k})$

i): If $dm_{k}\ge \nu _{1} \epsilon _{\delta }^{N_{k}}(x_{k})$ Starting with $N=N_{k}$, while $dm_{k}<\frac{N_k}{N}\;\epsilon _{\delta }^{N}(x_{k})$ and $N<N_{max}$, increase N by 1 and calculate $\epsilon _{\delta }^{N}(x_{k})$. Set $N_{k+1}=N$.
ii): If $dm_{k}< \nu _{1}\epsilon _{\delta }^{N_{k}}(x_{k})$ set $N_{k+1}=N_{max}$.

Algorithm 3

We say that we have not made big enough decrease of the function $\hat{\theta }_{N_{k+1}}$ if the following inequality is true

$$\begin{aligned} \frac{\hat{\theta }_{N_{k+1}}(x_{l(k)})-\hat{\theta }_{N_{k+1}}(x_{k+1})}{k+1-l(k)}< \frac{N_{k+1}}{N_{max}}\,\epsilon _{\delta }^{N_{k+1}}(x_{k+1}), \end{aligned}$$

where l(k) is the iteration at which we started to use the sample size $N_{k+1}$ for the last time.

Step 0

Input parameters: $N_{k}$, $N_{k+1}$, $N_{k}^{min}$.

Step 1

Determine $N_{k+1}^{min}$:

1)

If $N_{k+1}\le N_{k}$ then $N_{k+1}^{min}=N_{k}^{min}$.

2)

If $N_{k+1}> N_{k}$ and

i): if $N_{k+1}$ is a sample size which has not been used so far then $N_{k+1}^{min}=N_{k}^{min}$.
ii): if $N_{k+1}$ is a sample size which had been used and if we have made big enough decrease of the function $\hat{\theta }_{N_{k+1}}$ since the last time we used it, then $N_{k+1}^{min}=N_{k}^{min}$.
iii): if $N_{k+1}$ is a sample size which had been used and if we have not made big enough decrease of the function $\hat{\theta }_{N_{k+1}}$ since the last time we used it, then $N_{k+1}^{min}=N_{k+1}$.

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Nataša Krejić
- 1
Nataša Krklec Jerinkić
- 1
Andrea Rožnjik
- 2
Email authorView author's OrcID profile

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

Variable sample size method for equality constrained optimization problems

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Acknowledgements

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