Abstract

Parametric optimization is the process of solving an optimization problem as a function of currently unknown or changing variables known as parameters. Parametric optimization methods have been shown to benefit engineering design and optimal morphing applications through its specialized problem formulation and specialized algorithms. Despite its benefits to engineering design, existing parametric optimization algorithms (similar to many optimization algorithms) can be inefficient when design analyses are expensive. Since many engineering design problems involve some level of expensive analysis, a more efficient parametric optimization algorithm is needed. Therefore, the multi-objective efficient parametric optimization (MO-EPO) algorithm is developed to allow for the efficient optimization of problems with multiple parameters and objectives. This technique relies on the parametric hypervolume indicator (pHVI) which measures the space dominated by a given set of data involving both objectives and parameters. The novel MO-EPO algorithm is demonstrated on a number of analytical benchmarking problems with various numbers of objectives and parameters. Additionally, a morphing airfoil case study is examined. In each case, MO-EPO is shown to find solutions that are as good as or better than those found from the existing P3GA (i.e., equal or greater pHVI value) when the number of design evaluations is limited.

References

1.
Pistikopoulos
,
E. N.
,
for Process Systems Engineering C
, eds.,
2007
, Multi-parametric Programming: Theory, Algorithms, and Applications, No. Centre for Process Systems Engineering, E. N. Pistikopoulos, M. C. Georgiadis, and V. Dua, eds. (Process systems engineering, Vol. 1), Wiley-VCH, Weinheim. OCLC: 180943251.
2.
Johansen
,
T.
,
2002
, “
On Multi-Parametric Nonlinear Programming and Explicit Nonlinear Model Predictive Control
,”
Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 3
,
Las Vegas, NV
,
Dec. 10–13
,
IEEE
, pp.
2768
2773
.
3.
Fiacco
,
A. V.
,
1976
, “
Sensitivity Analysis for Nonlinear Programming Using Penalty Methods
,”
Math. Program.
,
10
(
1
), pp.
287
311
.
4.
Gal
,
T.
,
1984
, “Linear Parametric Programming—A Brief Survey,”
Sensitivity, Stability and Parametric Analysis
,
Fiacco
,
A. V.
, ed.,
Springer
,
Berlin/Heidelberg
, pp.
43
68
.
5.
Kojima
,
M.
, and
Hirabayashi
,
R.
,
1984
, “Continuous Deformation of Nonlinear Programs,”
Sensitivity, Stability and Parametric Analysis
,
R. W.
,
Cottle
,
L. C. W.
,
Dixon
,
B.
,
Korte
,
M. J.
,
Todd
,
E. L.
,
Allgower
,
W. H.
,
Cunningham
,
J. E.
,
Dennis
, et al., eds.,
Vol. 21
,
Springer, Berlin/Heidelberg
, pp.
150
198
.
6.
Weaver-Rosen
,
J. M.
,
Leal
,
P. B. C.
,
Hartl
,
D. J.
, and
Malak
,
R. J.
,
2020
, “
Parametric Optimization for Morphing Structures Design: Application to Morphing Wings Adapting to Changing Flight Conditions
,”
Struct. Multidiscipl. Optim.
,
62
(
6
), pp.
2995
3007
.
7.
Summers
,
C.
,
Weaver-Rosen
,
J. M.
,
Karakalas
,
A. A.
,
Malak
,
R. J.
, and
Lagoudas
,
D. C.
,
2021
, “
Parametric Optimization of SMA Torsional Actuators for Aircraft Morphing Applications
,” Volume 3: Advanced Materials: Design, Processing, Characterization, and Applications,
American Society of Mechanical Engineers
, Paper No. V003T03A056.
8.
Hartl
,
D. J.
,
Galvan
,
E.
,
Malak
,
R. J.
, and
Baur
,
J. W.
,
2016
, “
Parameterized Design Optimization of a Magnetohydrodynamic Liquid Metal Active Cooling Concept
,”
ASME J. Mech. Des.
,
138
(
3
), p.
031402
.
9.
Tsai
,
Y.-K.
, and
Malak
,
R. J.
,
2021
, “
A Methodology for Designing a Nonlinear Feedback Controller Via Parametric Optimization: State-Parameterized Nonlinear Programming Control
,”
ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Virtual Online
,
Aug. 17–19
, p. V03AT03A011.
10.
Pistikopoulos
,
E. N.
,
Dua
,
V.
,
Bozinis
,
N. A.
,
Bemporad
,
A.
, and
Morari
,
M.
,
2004
, “
On-Line Optimization Via Off-Line Parametric Optimization Tools
,”
Comput. Chem. Eng.
,
24
(
2–7
), pp.
183
188
.
11.
Wang
,
Y.
,
Seki
,
H.
,
Ohyama
,
S.
,
Akamatsu
,
K.
,
Ogawa
,
M.
, and
Ohshima
,
M.
,
2000
, “
Optimal Grade Transition Control for Polymerization Reactors
,”
Comput. Chem. Eng.
,
24
(
2–7
), pp.
1555
1561
.
12.
Spiegel
,
M.
, and
Subrahmanyam
,
A.
,
1992
, “
Informed Speculation and Hedging in a Noncompetitive Securities Market
,”
Rev. Financ. Stud.
,
5
(
2
), pp.
307
329
.
13.
Malak
,
R. J.
, and
Paredis
,
C. J. J.
,
2010
, “
Using Parameterized Pareto Sets to Model Design Concepts
,”
ASME J. Mech. Des.
,
132
(
4
), p.
041007
.
14.
Weaver-Rosen
,
J.
,
Tsai
,
Y.-K.
,
Schoppe
,
J.
,
Terada
,
Y.
,
Malak
,
R.
,
Cizmas
,
P. G.
, and
Lazzara
,
D. S.
,
2022
, “
Surrogate Modeling and Parametric Optimization Strategy for Minimizing Sonic Boom in a Morphing Aircraft
,”
AIAA SCITECH 2022 Forum
,
San Diego, CA
,
Jan. 3–7
,
American Institute of Aeronautics and Astronautics
, pp.
1
15
.
15.
Gamboa
,
P.
,
Vale
,
J.
, and
Suleman
,
A.
,
2009
, “
Optimization of a Morphing Wing Based on Coupled Aerodynamic and Structural Constraints
,”
AIAA J.
,
47
(
9
), pp.
2087
2104
.
16.
Beyer
,
H.-G.
, and
Sendhoff
,
B.
,
2007
, “
Robust Optimization – A Comprehensive Survey
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
33–34
), pp.
3190
3218
.
17.
Weaver-Rosen
,
J. M.
, and
Malak
,
R. J.
,
2021
, “
Efficient Parametric Optimization for Expensive Single Objective Problems
,”
ASME J. Mech. Des.
,
143
(
3
), p.
031711
.
18.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J.
,
1998
, “
Efficient Global Optimization of Expensive Black-Box Functions
,”
J. Global Optim.
,
13
(
4
), pp.
455
492
.
19.
Emmerich
,
M.
,
Giannakoglou
,
K.
, and
Naujoks
,
B.
,
2006
, “
Single- and Multiobjective Evolutionary Optimization Assisted by Gaussian Random Field Metamodels
,”
IEEE Trans. Evol. Comput.
,
10
(
4
), pp.
421
439
.
20.
Gutmann
,
H. M.
,
2001
, “
A Radial Basis Function Method for Global Optimization
,”
J. Global Optim.
,
19
(
3
), pp.
201
227
.
21.
Wang
,
D.
,
Wu
,
Z.
,
Fei
,
Y.
, and
Zhang
,
W.
,
2014
, “
Structural Design Employing a Sequential Approximation Optimization Approach
,”
Comput. Struct.
,
134
(
Apr.
), pp.
75
87
.
22.
Knowles
,
J.
,
2006
, “
ParEGO: A Hybrid Algorithm With On-line Landscape Approximation for Expensive Multiobjective Optimization Problems
,”
IEEE Trans. Evol. Comput.
,
10
(
1
), pp.
50
66
.
23.
Keane
,
A. J.
,
2006
, “
Statistical Improvement Criteria for Use in Multiobjective Design Optimization
,”
AIAA J.
,
44
(
4
), pp.
879
891
.
24.
Hernández-Lobato
,
D.
,
Hernández-Lobato
,
J.
,
Shah
,
A.
, and
Adams
,
R.
,
2016
, “
Predictive Entropy Search for Multi-Objective Bayesian Optimization
,” International Conference on Machine Learning, PMLR, pp.
1492
1501
.
25.
Sun
,
G.
,
Tian
,
Y.
,
Wang
,
R.
,
Fang
,
J.
, and
Li
,
Q.
,
2020
, “
Parallelized Multiobjective Efficient Global Optimization Algorithm and Its Applications
,”
Struct. Multidiscipl. Optim.
,
61
(
2
), pp.
763
786
.
26.
Weaver-Rosen
,
J. M.
,
2021
, “
Multi-objective Efficient Parametric Optimization
,” Ph.D. thesis,
Texas A&M University
,
College Station, TX
.
27.
Galvan
,
E.
, and
Malak
,
R. J.
,
2015
, “
P3GA: An Algorithm for Technology Characterization
,”
ASME J. Mech. Des.
,
137
(
1
), p.
011401
.
28.
Filippi
,
C.
,
2004
, “
An Algorithm for Approximate Multiparametric Linear Programming
,”
J. Optim. Theory Appl.
,
120
(
1
), pp.
73
95
.
29.
Spjøtvold
,
J.
,
Tøndel
,
P.
, and
Johansen
,
T. A.
,
2007
, “
Continuous Selection and Unique Polyhedral Representation of Solutions to Convex Parametric Quadratic Programs
,”
J. Optim. Theory Appl.
,
134
(
2
), pp.
177
189
.
30.
Bemporad
,
A.
, and
Filippi
,
C.
,
2003
, “
Suboptimal Explicit Receding Horizon Control Via Approximate Multiparametric Quadratic Programming
,”
J. Optim. Theory Appl.
,
117
(
1
), pp.
9
38
.
31.
Acevedo
,
J.
, and
Salgueiro
,
M.
,
2003
, “
An Efficient Algorithm for Convex Multiparametric Nonlinear Programming Problems
,”
Ind. Eng. Chem. Res.
,
42
(
23
), pp.
5883
5890
.
32.
Dua
,
V.
, and
Pistikopoulos
,
E. N.
,
1999
, “
Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems
,”
Ind. Eng. Chem. Res.
,
38
(
10
), pp.
3976
3987
.
33.
Weaver-Rosen
,
J. M.
, and
Malak
,
R. J.
,
2021
, “
A Novel Method for Calculating the Parametric Hypervolume Indicator
,” Volume 3B: 47th Design Automation Conference (DAC),
American Society of Mechanical Engineers
, Paper No. V03BT03A027.
34.
Zitzler
,
E.
,
Brockhoff
,
D.
, and
Thiele
,
L.
,
2007
, “The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration,”
Evolutionary Multi-criterion Optimization
,
Vol. 4403
,
Obayashi
,
S.
,
Deb
,
K.
,
Poloni
,
C.
,
Hiroyasu
,
T.
, and
Murata
,
T.
, eds.,
Springer
,
Berlin/Heidelberg
, pp.
862
876
.
35.
Emmerich
,
M. T. M.
,
Deutz
,
A. H.
, and
Klinkenberg
,
J. W.
,
2011
, “
Hypervolume-Based Expected Improvement: Monotonicity Properties and Exact Computation
,”
2011 IEEE Congress of Evolutionary Computation (CEC)
,
New Orleans, LA
,
June 5–8
,
IEEE
, pp.
2147
2154
.
36.
Golub
,
G. H.
, and
Welsch
,
J. H.
,
1969
, “
Calculation of Gauss Quadrature Rules
,”
Math. Comput.
,
23
(
106
), pp.
221
230
.
37.
Ralston
,
A.
, and
Rabinowitz
,
P.
,
2001
,
A First Course in Numerical Analysis
, 2nd ed.,
Dover Publications
,
Mineola, NY
.
38.
Tax
,
D. M.
, and
Duin
,
R. P.
,
1999
, “
Support Vector Domain Description
,”
Pattern Recognit. Lett.
,
20
(
11–13
), pp.
1191
1199
.
39.
Galvan
,
E.
,
Malak
,
R. J.
,
Hartl
,
D. J.
, and
Baur
,
J. W.
,
2018
, “
Performance Assessment of a Multi-objective Parametric Optimization Algorithm With Application to a Multi-physical Engineering System
,”
Struct. Multidiscipl. Optim.
,
58
(
2
), pp.
489
509
.
40.
Lophaven
,
S. N.
,
Nielsen
,
H. B.
, and
Sondergaard
,
J.
,
2002
, “Dace – A Matlab Kriging Toolox,” Version 2.0.
41.
Von Winckel
,
G.
,
2004
, “
Legendre–Gauss Quadrature Weights and Nodes
,”
Matlab Function lgwt
. http://www.mathworks.com/matlabcentral/fileexchange/4540 (
Wessel
,
P.
, and
Smith
,
W. H. F.
, 1998, New, Improved Version of the Generic Mapping Tools Released, EOS Trans. AGU, 79, p.
579
).
42.
Deb
,
K.
,
Thiele
,
L.
,
Laumanns
,
M.
, and
Zitzler
,
E.
,
2005
, “Scalable Test Problems for Evolutionary Multiobjective Optimization,”
Evolutionary Multiobjective Optimization
,
A.
,
Abraham
,
L.
,
Jain
, and
R.
,
Goldberg
, eds.,
Springer-Verlag
,
London
, pp.
105
145
.
43.
Audet
,
C.
,
Bigeon
,
J.
,
Cartier
,
D.
,
Le Digabel
,
S.
, and
Salomon
,
L.
,
2021
, “
Performance Indicators in Multiobjective Optimization
,”
Eur. J. Oper. Res.
,
292
(
2
), pp.
397
422
.
44.
Zitzler
,
E.
,
Thiele
,
L.
,
Laumanns
,
M.
,
Fonseca
,
C.
, and
da Fonseca
,
V.
,
2003
, “
Performance Assessment of Multiobjective Optimizers: An Analysis and Review
,”
IEEE Trans. Evol. Comput.
,
7
(
2
), pp.
117
132
.
45.
Lizarraga-Lizarraga
,
G.
,
Hernandez-Aguirre
,
A.
, and
Botello-Rionda
,
S.
,
2008
, “
G-Metric: An M-ary Quality Indicator for the Evaluation of Non-dominated Sets
,”
Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation – GECCO ’08
,
Atlanta, GA
,
July 12–16
,
ACM Press
, p.
665
.
46.
Li
,
M.
,
Yang
,
S.
, and
Liu
,
X.
,
2015
, “
A Performance Comparison Indicator for Pareto Front Approximations in Many-Objective Optimization
,”
Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation
,
Madrid, Spain
,
July 11–15
,
ACM
, pp.
703
710
.
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