Abstract

A novel approach to exploiting the log-convex structure present in many design problems is developed by modifying the classical Sequential Quadratic Programming (SQP) algorithm. The modified algorithm, Logspace Sequential Quadratic Programming (LSQP), inherits some of the computational efficiency exhibited by log-convex methods such as Geometric Programing and Signomial Programing, but retains the natural integration of black-box analysis methods from SQP. As a result, significant computational savings is achieved without the need to invasively modify existing black-box analysis methods prevalent in practical design problems. In the cases considered here, the LSQP algorithm shows a 40–70% reduction in number of iterations compared to SQP.

References

[1] Boyd S., Kim S.-J., Vandenberghe L. and Hassibi A., “A Tutorial on Geometric Programming,” Optimization and Engineering, Vol. 8, No. 1, 2007, pp. 67–127. https://doi.org/10.1007/s11081-007-9001-7 Crossref Google Scholar
[2] Clasen R. J., “The Solution of the Chemical Equilibrium Programming Problem with Generalized Benders Decomposition,” Operations Research, Vol. 32, No. 1, 1984, pp. 70–79. https://doi.org/10.1287/opre.32.1.70 Google Scholar
[3] Greenberg H. J., “Mathematical Programming Models for Environmental Quality Control,” Operations Research, Vol. 43, No. 4, 1995, pp. 578–622. https://doi.org/10.1287/opre.43.4.578 Google Scholar
[4] Boyd S. P., Kim S.-J., Patil D. D. and Horowitz M. A., “Digital Circuit Optimization via Geometric Programming,” Operations Research, Vol. 53, No. 6, 2005, pp. 899–932. https://doi.org/10.1287/opre.1050.0254 Crossref Google Scholar
[5] del Mar Hershenson M., Boyd S. P. and Lee T. H., “Optimal Design of a CMOS op-amp via Geometric Programming,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 20, No. 1, 2001, pp. 1–21. https://doi.org/10.1109/43.905671 Google Scholar
[6] Li X., Gopalakrishnan P., Xu Y. and Pileggi T., “Robust Analog/RF Circuit Design with Projection-Based Posynomial Modeling,” IEEE/ACM International Conference on Computer Aided Design, 2004. ICCAD-2004., Inst. of Electrical and Electronics Engineers, New York, 2004, pp. 855–862. https://doi.org/10.1109/ICCAD.2004.1382694 Google Scholar
[7] Xu Y., Pileggi L. T. and Boyd S. P., “ORACLE: Optimization with Recourse of Analog Circuits Including Layout Extraction,” Proceedings of the 41st Annual Design Automation Conference, June 2004, pp. 151–154. https://doi.org/10.1145/996566.996611 Google Scholar
[8] Jabr R. A., “Application of Geometric Programming to Transformer Design,” IEEE Transactions on Magnetics, Vol. 41, No. 11, 2005, pp. 4261–4269. https://doi.org/10.1109/TMAG.2005.856921 Google Scholar
[9] Chiang M., Geometric Programming for Communication Systems, Now Publishers Inc, Boston, MA, 2005, pp. 115–131. Google Scholar
[10] Chiang M., Tan C. W., Palomar D. P., O’Neill D. and Julian D., “Power Control by Geometric Programming,” IEEE Transactions on Wireless Communications, Vol. 6, No. 7, 2007, pp. 2640–2651. https://doi.org/10.1109/TWC.2007.05960 Google Scholar
[11] Kandukuri S. and Boyd S., “Optimal Power Control in Interference-Limited Fading Wireless Channels with Outage-Probability Specifications,” IEEE Transactions on Wireless Communications, Vol. 1, No. 1, 2002, pp. 46–55. https://doi.org/10.1109/7693.975444 Google Scholar
[12] Marin-Sanguino A., Voit E. O., Gonzalez-Alcon C. and Torres N. V., “Optimization of Biotechnological Systems Through Geometric Programming,” Theoretical Biology and Medical Modelling, Vol. 4, No. 1, 2007, p. 38. https://doi.org/10.1186/1742-4682-4-38 Google Scholar
[13] Vera J., González-Alcón C., Marn-Sanguino A. and Torres N., “Optimization of Biochemical Systems Through Mathematical Programming: Methods and Applications,” Computers & Operations Research, Vol. 37, No. 8, 2010, pp. 1427–1438. https://doi.org/10.1016/j.cor.2009.02.021 Google Scholar
[14] Preciado V. M., Zargham M., Enyioha C., Jadbabaie A. and Pappas G., “Optimal Resource Allocation for Network Protection: A Geometric Programming Approach,” IEEE Transactions on Control of Network Systems, Vol. 1, No. 1, 2014, pp. 99–108. https://doi.org/10.1109/TCNS.2014.2310911 Crossref Google Scholar
[15] Misra S., Fisher M. W., Backhaus S., Bent R., Chertkov M. and Pan F., “Optimal Compression in Natural Gas Networks: A Geometric Programming Approach,” IEEE Transactions on Control of Network Systems, Vol. 2, No. 1, 2014, pp. 47–56. https://doi.org/10.1109/TCNS.2014.2367360 Google Scholar
[16] Sela Perelman L. and Amin S., “Control of Tree Water Networks: A Geometric Programming Approach,” Water Resources Research, Vol. 51, No. 10, 2015, pp. 8409–8430. 10.1002/2014 WR016756 Google Scholar
[17] Agrawal A., Diamond S. and Boyd S., “Disciplined Geometric Programming,” Optimization Letters, Vol. 13, No. 5, 2019, pp. 961–976. https://doi.org/10.1007/s11590-019-01422-z Crossref Google Scholar
[18] Hoburg W. and Abbeel P., “Geometric Programming for Aircraft Design Optimization,” AIAA Journal, Vol. 52, No. 11, 2014, pp. 2414–2426. https://doi.org/10.2514/1.J052732 Link Google Scholar
[19] Torenbeek E., Advanced Aircraft Design: Conceptual Design, Analysis and Optimization of Subsonic Civil Airplanes, 2nd ed., Wiley, Chichester, U. K., 2013. Crossref Google Scholar
[20] Hoburg W. and Abbeel P., “Fast Wind Turbine Design via Geometric Programming,” 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2013-1532, 2013. https://doi.org/10.2514/6.2013-1532 Google Scholar
[21] Kirschen P. G., Burnell E. E. and Hoburg W. W., “Signomial Programming Models for Aircraft Design,” 54th AIAA Aerospace Sciences Meeting, AIAA Paper 2016-2003, 2016. https://doi.org/10.2514/6.2016-2003 Google Scholar
[22] Brown A. and Harris W., “A Vehicle Design and Optimization Model for On-Demand Aviation,” 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2018-0105, 2018. https://doi.org/10.2514/6.2018-0105 Google Scholar
[23] York M. A., Öztürk B., Burnell E. and Hoburg W. W., “Efficient Aircraft Multidisciplinary Design Optimization and Sensitivity Analysis via Signomial Programming,” AIAA Journal, Vol. 56, No. 11, 2018, pp. 4546–4561. https://doi.org/10.2514/1.J057020 Link Google Scholar
[24] Burton M. and Hoburg W., “Solar and Gas Powered Long-Endurance Unmanned Aircraft Sizing via Geometric Programming,” Journal of Aircraft, Vol. 55, No. 1, 2018, pp. 212–225. https://doi.org/10.2514/1.C034405 Link Google Scholar
[25] Lin B., Carpenter M. and de Weck O., “Simultaneous Vehicle and Trajectory Design Using Convex Optimization,” AIAA Scitech 2020 Forum, AIAA Paper 2020-0160, 2020. https://doi.org/10.2514/6.2020-0160 Google Scholar
[26] Kirschen P. G., York M. A., Ozturk B. and Hoburg W. W., “Application of Signomial Programming to Aircraft Design,” Journal of Aircraft, Vol. 55, No. 3, 2018, pp. 965–987. https://doi.org/10.2514/1.C034378 Link Google Scholar
[27] York M. A., Hoburg W. W. and Drela M., “Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming,” Journal of Aircraft, Vol. 55, No. 3, 2018, pp. 988–1003. https://doi.org/10.2514/1.C034463 Link Google Scholar
[28] Saab A., Burnell E. and Hoburg W. W., “Robust Designs via Geometric Programming,” arXiv, Aug. 2018, https://arxiv.org/abs/1808.07192. Google Scholar
[29] Hall D. K., Dowdle A., Gonzalez J., Trollinger L. and Thalheimer W., “Assessment of a Boundary Layer Ingesting Turboelectric Aircraft Configuration Using Signomial Programming,” 2018 Aviation Technology, Integration, and Operations Conference, AIAA Paper 2018-3973, 2018. https://doi.org/10.2514/6.2018-3973 Google Scholar
[30] Martins J. R. and Lambe A. B., “Multidisciplinary Design Optimization: A Survey of Architectures,” AIAA Journal, Vol. 51, No. 9, 2013, pp. 2049–2075. https://doi.org/10.2514/1.J051895 Link Google Scholar
[31] Wakayama S., “Multidisciplinary Design Optimization of the Blended-Wing-Body,” AIAA Paper 1998-4938, 1998. https://doi.org/10.2514/6.1998-4938 Link Google Scholar
[32] Kroo I. and Takai M., “A Quasi-Procedural, Knowledge-Based System for Aircraft Design,” AIAA Paper 1988-6502, 1988. https://doi.org/10.2514/6.1988-4428 Google Scholar
[33] Boggs P. T. and Tolle J. W., “Sequential Quadratic Programming,” Acta Numerica, Vol. 4, Jan. 1995, pp. 1–51. https://doi.org/10.1017/S0962492900002518 Crossref Google Scholar
[34] Nocedal J. and Wright S., Numerical Optimization, Springer Science & Business Media, New York, 2006, pp. 1–664. Google Scholar
[35] Kraft D., “A Software Package for Sequential Quadratic Programming,” Institut für Dynamik der Flugsysteme, Rept. DFVLR-FB 88-28, Oberpfaffenhofen, Germany, 1988. Google Scholar
[36] Diamond S. and Boyd S., “CVXPY: A Python-Embedded Modeling Language for Convex Optimization,” Journal of Machine Learning Research, Vol. 17, No. 1, 2016, pp. 2909–2913. Google Scholar
[37] Kirschen P. G. and Hoburg W. W., “The Power of Log Transformation: A Comparison of Geometric and Signomial Programming with General Nonlinear Programming Techniques for Aircraft Design Optimization,” 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2018-0655, 2018. https://doi.org/10.2514/6.2018-0655 Link Google Scholar
[38] Hoburg W., Kirschen P. and Abbeel P., “Data Fitting with Geometric-Programming-Compatible Softmax Functions,” Optimization and Engineering, Vol. 17, No. 4, 2016, pp. 1–22. https://doi.org/10.1007/s11081-016-9332-3 Google Scholar
[39] Floudas C. A., Pardalos P. M., Adjiman C., Esposito W. R., Gümüs Z. H., Harding S. T., Klepeis J. L., Meyer C. A. and Schweiger C. A., Handbook of Test Problems in Local and Global Optimization, Vol. 33, Springer Science & Business Media, New York, 2013, pp. 90–92. https://doi.org/10.1007/978-1-4757-3040-1 Google Scholar
[40] Reuther A., Kepner J., Byun C., Samsi S., Arcand W., Bestor D., Bergeron B., Gadepally V., Houle M., Hubbell M., Jones M., Klein A., Milechin L., Mullen J., Prout A., Rosa A., Yee C. and Michaleas P., “Interactive Supercomputing on 40,000 Cores for Machine Learning and Data Analysis,” 2018 IEEE High Performance extreme Computing Conference (HPEC), Inst. of Electrical and Electronics Engineers, New York, 2018, pp. 1–6. https://doi.org/10.1109/HPEC.2018.8547629 Google Scholar

Volume 60, Number 3March 2022

Crossmark

Information

Copyright © 2021 by Cody J. Karcher. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp.

Topics

Keywords

Acknowledgments

This material is based on research sponsored by the U.S. Air Force under agreement number FA8650-20-2-2002. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Air Force or the U.S. Government. The author would like to thank Bob Haimes and Mark Drela for their mentorship and support and for their notes on this paper, the EnCAPS Technical Monitor Ryan Durscher, Philippe Kirschen who laid some of the foundation for this work, Berk Ozturk who originally authored the fourth test problem, Devon Jedamski for his review and comments, and the two anonymous reviewers who provided comments and suggestions. The author also acknowledges the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing high performance computing, database, and consultation resources that have contributed to the research results reported within this paper.

PDF

Received30 May 2021
Accepted13 December 2021
Published online9 February 2022

Logspace Sequential Quadratic Programming for Design Optimization

Abstract

References

What's Popular

Information