# A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION

## Abstract

We present a structured quasi-Newton algorithm for unconstrained optimization problems that have unavailable second-order derivatives or Hessian terms. We provide a formal derivation of the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) secant update formula that approximates only the missing Hessian terms, and we propose a linesearch quasi-Newton algorithm based on a modification of Wolfe conditions that converges to first-order optimality conditions. We also analyze the local convergence properties of the structured BFGS algorithm and show that it achieves superlinear convergence under the standard assumptions used by quasi-Newton methods using secant updates. We provide a thorough study of the practical performance of the algorithm on the CUTEr suite of test problems and show that our structured BFGS-based quasi-Newton algorithm outperforms the unstructured counterpart(s).

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)

- OSTI Identifier:
- 1573037

- DOE Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article

- Journal Name:
- SIAM Journal on Optimization

- Additional Journal Information:
- Journal Volume: 29; Journal Issue: 2

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Petra, Cosmin G., Chiang, Nai-Yuan, and Anitescu, Mihai.
```*A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION*. United States: N. p., 2019.
Web. doi:10.1137/18M1167942.

```
Petra, Cosmin G., Chiang, Nai-Yuan, & Anitescu, Mihai.
```*A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION*. United States. doi:10.1137/18M1167942.

```
Petra, Cosmin G., Chiang, Nai-Yuan, and Anitescu, Mihai. Tue .
"A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION". United States. doi:10.1137/18M1167942.
```

```
@article{osti_1573037,
```

title = {A STRUCTURED QUASI-NEWTON ALGORITHM FOR OPTIMIZING WITH INCOMPLETE HESSIAN INFORMATION},

author = {Petra, Cosmin G. and Chiang, Nai-Yuan and Anitescu, Mihai},

abstractNote = {We present a structured quasi-Newton algorithm for unconstrained optimization problems that have unavailable second-order derivatives or Hessian terms. We provide a formal derivation of the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) secant update formula that approximates only the missing Hessian terms, and we propose a linesearch quasi-Newton algorithm based on a modification of Wolfe conditions that converges to first-order optimality conditions. We also analyze the local convergence properties of the structured BFGS algorithm and show that it achieves superlinear convergence under the standard assumptions used by quasi-Newton methods using secant updates. We provide a thorough study of the practical performance of the algorithm on the CUTEr suite of test problems and show that our structured BFGS-based quasi-Newton algorithm outperforms the unstructured counterpart(s).},

doi = {10.1137/18M1167942},

journal = {SIAM Journal on Optimization},

number = 2,

volume = 29,

place = {United States},

year = {2019},

month = {1}

}