# MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys: A DIRECT-inspired optimization algorithm for experimentally accessible computational material design

## Abstract

Computational materials design has suffered from a lack of algorithms formulated in terms of experimentally accessible variables. Here we formulate the problem of (ternary) alloy optimization at the level of choice of atoms and their composition that is normal for synthesists. Mathematically, this is a mixed integer problem where a candidate solution consists of a choice of three elements, and how much of each of them to use. This space has the natural structure of a set of equilateral triangles. We solve this problem by introducing a novel version of the DIRECT algorithm that (1) operates on equilateral triangles instead of rectangles and (2) works across multiple triangles. We demonstrate on a test case that the algorithm is both robust and efficient. Lastly, we offer an explanation of the efficacy of DIRECT -- specifically, its balance of global and local search -- by showing that 'potentially optimal rectangles' of the original algorithm are akin to the Pareto front of the 'multi-component optimization' of global and local search.

- Authors:

- National Renewable Energy Lab. (NREL), Golden, CO (United States)
- Univ. of Colorado Denver, Denver, CO (United States)

- Publication Date:

- Research Org.:
- National Renewable Energy Lab. (NREL), Golden, CO (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1395085

- Report Number(s):
- NREL/JA-2C00-60589

Journal ID: ISSN 0926-6003

- Grant/Contract Number:
- AC36-08GO28308

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computational Optimization and applications

- Additional Journal Information:
- Journal Volume: 68; Journal Issue: 3; Journal ID: ISSN 0926-6003

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; mixed integer optimization; DIRECT optimization; Pareto front; Sierpinski triangle; computational material design

### Citation Formats

```
Graf, Peter A., and Billups, Stephen.
```*MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys: A DIRECT-inspired optimization algorithm for experimentally accessible computational material design*. United States: N. p., 2017.
Web. doi:10.1007/s10589-017-9922-9.

```
Graf, Peter A., & Billups, Stephen.
```*MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys: A DIRECT-inspired optimization algorithm for experimentally accessible computational material design*. United States. doi:10.1007/s10589-017-9922-9.

```
Graf, Peter A., and Billups, Stephen. Mon .
"MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys: A DIRECT-inspired optimization algorithm for experimentally accessible computational material design". United States.
doi:10.1007/s10589-017-9922-9. https://www.osti.gov/servlets/purl/1395085.
```

```
@article{osti_1395085,
```

title = {MDTri: robust and efficient global mixed integer search of spaces of multiple ternary alloys: A DIRECT-inspired optimization algorithm for experimentally accessible computational material design},

author = {Graf, Peter A. and Billups, Stephen},

abstractNote = {Computational materials design has suffered from a lack of algorithms formulated in terms of experimentally accessible variables. Here we formulate the problem of (ternary) alloy optimization at the level of choice of atoms and their composition that is normal for synthesists. Mathematically, this is a mixed integer problem where a candidate solution consists of a choice of three elements, and how much of each of them to use. This space has the natural structure of a set of equilateral triangles. We solve this problem by introducing a novel version of the DIRECT algorithm that (1) operates on equilateral triangles instead of rectangles and (2) works across multiple triangles. We demonstrate on a test case that the algorithm is both robust and efficient. Lastly, we offer an explanation of the efficacy of DIRECT -- specifically, its balance of global and local search -- by showing that 'potentially optimal rectangles' of the original algorithm are akin to the Pareto front of the 'multi-component optimization' of global and local search.},

doi = {10.1007/s10589-017-9922-9},

journal = {Computational Optimization and applications},

number = 3,

volume = 68,

place = {United States},

year = {Mon Jul 24 00:00:00 EDT 2017},

month = {Mon Jul 24 00:00:00 EDT 2017}

}