# Nonlinearly-constrained optimization using asynchronous parallel generating set search.

## Abstract

Many optimization problems in computational science and engineering (CS&E) are characterized by expensive objective and/or constraint function evaluations paired with a lack of derivative information. Direct search methods such as generating set search (GSS) are well understood and efficient for derivative-free optimization of unconstrained and linearly-constrained problems. This paper addresses the more difficult problem of general nonlinear programming where derivatives for objective or constraint functions are unavailable, which is the case for many CS&E applications. We focus on penalty methods that use GSS to solve the linearly-constrained problems, comparing different penalty functions. A classical choice for penalizing constraint violations is {ell}{sub 2}{sup 2}, the squared {ell}{sub 2} norm, which has advantages for derivative-based optimization methods. In our numerical tests, however, we show that exact penalty functions based on the {ell}{sub 1}, {ell}{sub 2}, and {ell}{sub {infinity}} norms converge to good approximate solutions more quickly and thus are attractive alternatives. Unfortunately, exact penalty functions are discontinuous and consequently introduce theoretical problems that degrade the final solution accuracy, so we also consider smoothed variants. Smoothed-exact penalty functions are theoretically attractive because they retain the differentiability of the original problem. Numerically, they are a compromise between exact and {ell}{sub 2}{sup 2}, i.e., theymore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Laboratories

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 909393

- Report Number(s):
- SAND2007-3257

TRN: US200722%%1088

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; NONLINEAR PROGRAMMING; OPTIMIZATION; CALCULATION METHODS; NONLINEAR PROBLEMS; CONVERGENCE; Nonlinear programming.; Optimization methods; Mathematical optimization.; Optimization.

### Citation Formats

```
Griffin, Joshua D., and Kolda, Tamara Gibson.
```*Nonlinearly-constrained optimization using asynchronous parallel generating set search.*. United States: N. p., 2007.
Web. doi:10.2172/909393.

```
Griffin, Joshua D., & Kolda, Tamara Gibson.
```*Nonlinearly-constrained optimization using asynchronous parallel generating set search.*. United States. doi:10.2172/909393.

```
Griffin, Joshua D., and Kolda, Tamara Gibson. Tue .
"Nonlinearly-constrained optimization using asynchronous parallel generating set search.". United States.
doi:10.2172/909393. https://www.osti.gov/servlets/purl/909393.
```

```
@article{osti_909393,
```

title = {Nonlinearly-constrained optimization using asynchronous parallel generating set search.},

author = {Griffin, Joshua D. and Kolda, Tamara Gibson},

abstractNote = {Many optimization problems in computational science and engineering (CS&E) are characterized by expensive objective and/or constraint function evaluations paired with a lack of derivative information. Direct search methods such as generating set search (GSS) are well understood and efficient for derivative-free optimization of unconstrained and linearly-constrained problems. This paper addresses the more difficult problem of general nonlinear programming where derivatives for objective or constraint functions are unavailable, which is the case for many CS&E applications. We focus on penalty methods that use GSS to solve the linearly-constrained problems, comparing different penalty functions. A classical choice for penalizing constraint violations is {ell}{sub 2}{sup 2}, the squared {ell}{sub 2} norm, which has advantages for derivative-based optimization methods. In our numerical tests, however, we show that exact penalty functions based on the {ell}{sub 1}, {ell}{sub 2}, and {ell}{sub {infinity}} norms converge to good approximate solutions more quickly and thus are attractive alternatives. Unfortunately, exact penalty functions are discontinuous and consequently introduce theoretical problems that degrade the final solution accuracy, so we also consider smoothed variants. Smoothed-exact penalty functions are theoretically attractive because they retain the differentiability of the original problem. Numerically, they are a compromise between exact and {ell}{sub 2}{sup 2}, i.e., they converge to a good solution somewhat quickly without sacrificing much solution accuracy. Moreover, the smoothing is parameterized and can potentially be adjusted to balance the two considerations. Since many CS&E optimization problems are characterized by expensive function evaluations, reducing the number of function evaluations is paramount, and the results of this paper show that exact and smoothed-exact penalty functions are well-suited to this task.},

doi = {10.2172/909393},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Tue May 01 00:00:00 EDT 2007},

month = {Tue May 01 00:00:00 EDT 2007}

}