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## A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints

## Abstract

We approach the problem of parallelizing state-of-the-art nonlinear programming optimization algorithms.Specifically, we focus on parallelizing quasi-Newton interior-point methods that use limited-memory secant Hessian approximations. Such interior-point methods are known to have better convergence properties and to be more effective on large-scale problems than gradient-based and derivative-free optimization algorithms. We target nonlinear and potentially nonconvex optimization problems with an arbitrary number of bound constraints and a small number of general equality and inequality constraints on the optimization variables. These problems occur for example in the form of optimal control, optimal design, and inverse problems governed by ordinary or partial differential equations, whenever they are expressed in a “reduced-space” optimization approach. We introduce and analyze the time and space complexity of a decomposition method for solving the quasi-Newton linear systems that leverages the fact that the quasi-Newton Hessian matrix has a small number of dense blocks that border a low-rank update of a diagonal matrix. This enables an efficient parallelization on memory-distributed computers of the iterations of the optimization algorithm, a state-of-the-art filter line-search interior-point algorithm by Wächter et. al. We illustrate the efficiency of the proposed method by solving structural topology optimization problems on up to 4608 cores on a parallelmore »

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1562383

- Report Number(s):
- LLNL-JRNL-739001

Journal ID: ISSN 0743-7315; 892535

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Parallel and Distributed Computing

- Additional Journal Information:
- Journal Volume: 133; Journal Issue: C; Journal ID: ISSN 0743-7315

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Parallel optimization; Parallel interior-point; Quasi-Newton

### Citation Formats

```
Petra, Cosmin G. A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints. United States: N. p., 2019.
Web. doi:10.1016/j.jpdc.2018.10.009.
```

```
Petra, Cosmin G. A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints. United States. doi:10.1016/j.jpdc.2018.10.009.
```

```
Petra, Cosmin G. Thu .
"A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints". United States. doi:10.1016/j.jpdc.2018.10.009.
```

```
@article{osti_1562383,
```

title = {A memory-distributed quasi-Newton solver for nonlinear programming problems with a small number of general constraints},

author = {Petra, Cosmin G.},

abstractNote = {We approach the problem of parallelizing state-of-the-art nonlinear programming optimization algorithms.Specifically, we focus on parallelizing quasi-Newton interior-point methods that use limited-memory secant Hessian approximations. Such interior-point methods are known to have better convergence properties and to be more effective on large-scale problems than gradient-based and derivative-free optimization algorithms. We target nonlinear and potentially nonconvex optimization problems with an arbitrary number of bound constraints and a small number of general equality and inequality constraints on the optimization variables. These problems occur for example in the form of optimal control, optimal design, and inverse problems governed by ordinary or partial differential equations, whenever they are expressed in a “reduced-space” optimization approach. We introduce and analyze the time and space complexity of a decomposition method for solving the quasi-Newton linear systems that leverages the fact that the quasi-Newton Hessian matrix has a small number of dense blocks that border a low-rank update of a diagonal matrix. This enables an efficient parallelization on memory-distributed computers of the iterations of the optimization algorithm, a state-of-the-art filter line-search interior-point algorithm by Wächter et. al. We illustrate the efficiency of the proposed method by solving structural topology optimization problems on up to 4608 cores on a parallel machine.},

doi = {10.1016/j.jpdc.2018.10.009},

journal = {Journal of Parallel and Distributed Computing},

number = C,

volume = 133,

place = {United States},

year = {2019},

month = {11}

}