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

Graf, Peter A.; Billups, Stephen

doi:10.1007/s10589-017-9922-9

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

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:

Graf, Peter A. ^[1]; Billups, Stephen ^[2]

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

Publication Date:: Mon Jul 24 00:00:00 EDT 2017

Research Org.:: Energy Frontier Research Centers (EFRC) (United States). Center for Inverse Design (CID); National Renewable Energy Laboratory (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:: 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.

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                    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.  https://doi.org/10.1007/s10589-017-9922-9

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                    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.  https://doi.org/10.1007/s10589-017-9922-9.  https://www.osti.gov/servlets/purl/1395085.

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@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}

}

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Works referenced in this record:

A genetic algorithm based inverse band structure method for semiconductor alloys
journal, September 2005

Kim, Kwiseon; Graf, Peter A.; Jones, Wesley B.
Journal of Computational Physics, Vol. 208, Issue 2
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Boyd, Stephen; Vandenberghe, Lieven
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journal, October 1993

Jones, D. R.; Perttunen, C. D.; Stuckman, B. E.
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The inverse band-structure problem of finding an atomic configuration with given electronic properties
journal, November 1999

Franceschetti, Alberto; Zunger, Alex
Nature, Vol. 402, Issue 6757
DOI: 10.1038/46995

Derivative-free optimization: a review of algorithms and comparison of software implementations
journal, July 2012

Rios, Luis Miguel; Sahinidis, Nikolaos V.
Journal of Global Optimization, Vol. 56, Issue 3
DOI: 10.1007/s10898-012-9951-y

Surface passivation optimization using DIRECT
journal, June 2007

Graf, Peter A.; Kim, Kwiseon; Jones, Wesley B.
Journal of Computational Physics, Vol. 224, Issue 2
DOI: 10.1016/j.jcp.2006.10.033

A Sequential Method Seeking the Global Maximum of a Function
journal, September 1972

Shubert, Bruno O.
SIAM Journal on Numerical Analysis, Vol. 9, Issue 3
DOI: 10.1137/0709036

Works referencing / citing this record:

Secant Penalized BFGS: A Noise Robust Quasi-Newton Method Via Penalizing The Secant Condition
text, January 2020

Irwin, Brian; Haber, Eldad
arXiv
DOI: 10.48550/arxiv.2010.01275

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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

Abstract

Citation Formats

A genetic algorithm based inverse band structure method for semiconductor alloys journal, September 2005

Convex Optimization book, January 2004

Lipschitzian optimization without the Lipschitz constant journal, October 1993

The inverse band-structure problem of finding an atomic configuration with given electronic properties journal, November 1999

Derivative-free optimization: a review of algorithms and comparison of software implementations journal, July 2012

Surface passivation optimization using DIRECT journal, June 2007

A Sequential Method Seeking the Global Maximum of a Function journal, September 1972

Secant Penalized BFGS: A Noise Robust Quasi-Newton Method Via Penalizing The Secant Condition text, January 2020

A genetic algorithm based inverse band structure method for semiconductor alloys
journal, September 2005

Convex Optimization
book, January 2004

Lipschitzian optimization without the Lipschitz constant
journal, October 1993

The inverse band-structure problem of finding an atomic configuration with given electronic properties
journal, November 1999

Derivative-free optimization: a review of algorithms and comparison of software implementations
journal, July 2012

Surface passivation optimization using DIRECT
journal, June 2007

A Sequential Method Seeking the Global Maximum of a Function
journal, September 1972

Secant Penalized BFGS: A Noise Robust Quasi-Newton Method Via Penalizing The Secant Condition
text, January 2020