IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v538y2020ics0378437119315171.html
   My bibliography  Save this article

Indicator of power convex and exponential transformations for solving nonlinear problems containing posynomial terms

Author

Listed:
  • Lu, Hao-Chun

Abstract

Posynomial terms frequently appear in many nonlinear problems and are the core components of geometric and generalized geometric programming problems. The most popular method to treat nonconvex posynomial terms for obtaining global optimization is to convert nonconvex posynomial terms as convex underestimators using transformation techniques. Among the transformation techniques, exponential transformation (ET) and power convex transformation (PCT) can yield the tightest underestimators of posynomial terms. However, the current literature has rarely discussed which to select between ET and PCT. This study employs the definite integral with piecewise linear technique to calculate the error between the original posynomial and the corresponding ET/PCT underestimators. Lastly, this study aims to identify an indicator that can choose the appropriate transformation between ET and PCT and analyze the correctness of the proposed indicator for posynomial terms in nonlinear problems. The proposed indicator can efficiently solve nonlinear problems containing posynomial terms. Numerical examples are used to demonstrate the efficacy of the proposed indicator.

Suggested Citation

  • Lu, Hao-Chun, 2020. "Indicator of power convex and exponential transformations for solving nonlinear problems containing posynomial terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    • Handle: RePEc:eee:phsmap:v:538:y:2020:i:c:s0378437119315171
    DOI: 10.1016/j.physa.2019.122658
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437119315171
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2019.122658?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Han-Lin Li & Hao-Chun Lu, 2009. "Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables," Operations Research, INFORMS, vol. 57(3), pages 701-713, June.
    2. Jung-Fa Tsai & Ming-Hua Lin, 2011. "An Efficient Global Approach for Posynomial Geometric Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 483-492, August.
    3. Stephen P. Boyd & Seung-Jean Kim & Dinesh D. Patil & Mark A. Horowitz, 2005. "Digital Circuit Optimization via Geometric Programming," Operations Research, INFORMS, vol. 53(6), pages 899-932, December.
    4. Hao-Chun Lu & Han-Lin Li & Chrysanthos Gounaris & Christodoulos Floudas, 2010. "Convex relaxation for solving posynomial programs," Journal of Global Optimization, Springer, vol. 46(1), pages 147-154, January.
    5. Hao Cheng & Shu-Cherng Fang & John Lavery, 2005. "A Geometric Programming Framework for Univariate Cubic L 1 Smoothing Splines," Annals of Operations Research, Springer, vol. 133(1), pages 229-248, January.
    6. Hao-Chun Lu & Liming Yao, 2019. "Efficient Convexification Strategy for Generalized Geometric Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 226-234, April.
    7. Lin, Ming-Hua & Tsai, Jung-Fa, 2012. "Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 216(1), pages 17-25.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hao-Chun Lu & Liming Yao, 2019. "Efficient Convexification Strategy for Generalized Geometric Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 226-234, April.
    2. Hao-Chun Lu, 2017. "Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms," Journal of Global Optimization, Springer, vol. 68(1), pages 95-123, May.
    3. Xu, Gongxian, 2014. "Global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 233(3), pages 500-510.
    4. Tseng, Chung-Li & Zhan, Yiduo & Zheng, Qipeng P. & Kumar, Manish, 2015. "A MILP formulation for generalized geometric programming using piecewise-linear approximations," European Journal of Operational Research, Elsevier, vol. 245(2), pages 360-370.
    5. Warren P. Adams & Stephen M. Henry, 2012. "Base-2 Expansions for Linearizing Products of Functions of Discrete Variables," Operations Research, INFORMS, vol. 60(6), pages 1477-1490, December.
    6. Lin, Ming-Hua & Tsai, Jung-Fa, 2012. "Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 216(1), pages 17-25.
    7. Rashed Khanjani-Shiraz & Salman Khodayifar & Panos M. Pardalos, 2021. "Copula theory approach to stochastic geometric programming," Journal of Global Optimization, Springer, vol. 81(2), pages 435-468, October.
    8. Rashed Khanjani Shiraz & Hirofumi Fukuyama, 2018. "Integrating geometric programming with rough set theory," Operational Research, Springer, vol. 18(1), pages 1-32, April.
    9. Andreas Lundell & Anders Skjäl & Tapio Westerlund, 2013. "A reformulation framework for global optimization," Journal of Global Optimization, Springer, vol. 57(1), pages 115-141, September.
    10. Han-Lin Li & Hao-Chun Lu, 2009. "Global Optimization for Generalized Geometric Programs with Mixed Free-Sign Variables," Operations Research, INFORMS, vol. 57(3), pages 701-713, June.
    11. Yiduo Zhan & Qipeng P. Zheng & Chung-Li Tseng & Eduardo L. Pasiliao, 2018. "An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming," Journal of Global Optimization, Springer, vol. 70(3), pages 579-599, March.
    12. Jiao, Hongwei & Liu, Sanyang & Lu, Nan, 2015. "A parametric linear relaxation algorithm for globally solving nonconvex quadratic programming," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 973-985.
    13. Rashed Khanjani Shiraz & Madjid Tavana & Debora Di Caprio & Hirofumi Fukuyama, 2016. "Solving Geometric Programming Problems with Normal, Linear and Zigzag Uncertainty Distributions," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1075-1078, September.
    14. O’Brien, Thomas & Reeb, Tyler, 2016. "Freight Efficiency Strategies: Operational Modernization at Distribution Nodes," Institute of Transportation Studies, Working Paper Series qt4fg9f0gv, Institute of Transportation Studies, UC Davis.
    15. Qingwei Jin & Lu Yu & John Lavery & Shu-Cherng Fang, 2012. "Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties," Computational Optimization and Applications, Springer, vol. 51(2), pages 575-600, March.
    16. Chiu, Nan-Chieh & Fang, Shu-Cherng & Lavery, John E. & Lin, Jen-Yen & Wang, Yong, 2008. "Approximating term structure of interest rates using cubic L1 splines," European Journal of Operational Research, Elsevier, vol. 184(3), pages 990-1004, February.
    17. Han-Lin Li & Yao-Huei Huang & Shu-Cherng Fang, 2017. "Linear Reformulation of Polynomial Discrete Programming for Fast Computation," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 108-122, February.
    18. Armin Jabbarzadeh & Leyla Aliabadi & Reza Yazdanparast, 2021. "Optimal payment time and replenishment decisions for retailer’s inventory system under trade credit and carbon emission constraints," Operational Research, Springer, vol. 21(1), pages 589-620, March.
    19. Consiglio, Andrea & Carollo, Angelo & Zenios, Stavros A., 2014. "Generating Multi-factor Arbitrage-Free Scenario Trees with Global Optimization," Working Papers 13-35, University of Pennsylvania, Wharton School, Weiss Center.
    20. Xinlei Wang & Johan Lim & Seung-Jean Kim & Kyu Hahn, 2015. "Estimating cell probabilities in contingency tables with constraints on marginals/conditionals by geometric programming with applications," Computational Statistics, Springer, vol. 30(1), pages 107-129, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:538:y:2020:i:c:s0378437119315171. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.