# Stochastic derivative-free optimization using a trust region framework

## Abstract

This study presents a trust region algorithm to minimize a function f when one has access only to noise-corrupted function values f¯. The model-based algorithm dynamically adjusts its step length, taking larger steps when the model and function agree and smaller steps when the model is less accurate. The method does not require the user to specify a fixed pattern of points used to build local models and does not repeatedly sample points. If f is sufficiently smooth and the noise is independent and identically distributed with mean zero and finite variance, we prove that our algorithm produces iterates such that the corresponding function gradients converge in probability to zero. As a result, we present a prototype of our algorithm that, while simplistic in its management of previously evaluated points, solves benchmark problems in fewer function evaluations than do existing stochastic approximation methods.

- Authors:

- Argonne National Lab. (ANL), Lemont, IL (United States)
- Univ. of Colorado, Denver, CO (United States)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1366470

- Grant/Contract Number:
- AC02-06CH11357

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computational Optimization and applications

- Additional Journal Information:
- Journal Volume: 64; Journal Issue: 3; Journal ID: ISSN 0926-6003

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; derivative-free optimization; model-based trust region methods; stochastic optimization

### Citation Formats

```
Larson, Jeffrey, and Billups, Stephen C. Stochastic derivative-free optimization using a trust region framework. United States: N. p., 2016.
Web. doi:10.1007/s10589-016-9827-z.
```

```
Larson, Jeffrey, & Billups, Stephen C. Stochastic derivative-free optimization using a trust region framework. United States. doi:10.1007/s10589-016-9827-z.
```

```
Larson, Jeffrey, and Billups, Stephen C. Wed .
"Stochastic derivative-free optimization using a trust region framework". United States. doi:10.1007/s10589-016-9827-z. https://www.osti.gov/servlets/purl/1366470.
```

```
@article{osti_1366470,
```

title = {Stochastic derivative-free optimization using a trust region framework},

author = {Larson, Jeffrey and Billups, Stephen C.},

abstractNote = {This study presents a trust region algorithm to minimize a function f when one has access only to noise-corrupted function values f¯. The model-based algorithm dynamically adjusts its step length, taking larger steps when the model and function agree and smaller steps when the model is less accurate. The method does not require the user to specify a fixed pattern of points used to build local models and does not repeatedly sample points. If f is sufficiently smooth and the noise is independent and identically distributed with mean zero and finite variance, we prove that our algorithm produces iterates such that the corresponding function gradients converge in probability to zero. As a result, we present a prototype of our algorithm that, while simplistic in its management of previously evaluated points, solves benchmark problems in fewer function evaluations than do existing stochastic approximation methods.},

doi = {10.1007/s10589-016-9827-z},

journal = {Computational Optimization and applications},

number = 3,

volume = 64,

place = {United States},

year = {2016},

month = {2}

}

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