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The Fundamental Relations between Geometric Programming Duality, Parametric Programming Duality, and Ordinary Lagrangian Duality

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  • Elmor Peterson

Abstract

Every formulation of mathematical programming duality (known to the author) for continuous finite-dimensional optimization can easily be viewed as a special case of at least one of the following three formulations: the geometric programming formulation (of the generalized geometric programming type), the parametric programming formulation (of the generalized Rockafellar-perturbation type), and the ordinary Lagrangian formulation (of the generalized Falk type). The relative strengths and weaknesses of these three duality formulations are described herein, as are the fundamental relations between them. As a theoretical application, the basic duality between Fenchel's hypothesis and the existence of recession directions in convex programming is established and then expressed within each of these three duality formulations Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • Elmor Peterson, 2001. "The Fundamental Relations between Geometric Programming Duality, Parametric Programming Duality, and Ordinary Lagrangian Duality," Annals of Operations Research, Springer, vol. 105(1), pages 109-153, July.
    • Handle: RePEc:spr:annopr:v:105:y:2001:i:1:p:109-153:10.1023/a:1013353515966
    DOI: 10.1023/A:1013353515966
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    Citations

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    Cited by:

    1. Liu, Shiang-Tai, 2006. "Posynomial geometric programming with parametric uncertainty," European Journal of Operational Research, Elsevier, vol. 168(2), pages 345-353, January.
    2. G. S. Mahapatra & T. K. Mandal, 2012. "Posynomial Parametric Geometric Programming with Interval Valued Coefficient," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 120-132, July.
    3. Frenk, J.B.G. & Kassay, G., 2004. "Introduction to Convex and Quasiconvex Analysis," ERIM Report Series Research in Management ERS-2004-075-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    4. Wasim Akram Mandal, 2021. "Weighted Tchebycheff Optimization Technique Under Uncertainty," Annals of Data Science, Springer, vol. 8(4), pages 709-731, December.
    5. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    6. Rashed Khanjani Shiraz & Hirofumi Fukuyama, 2018. "Integrating geometric programming with rough set theory," Operational Research, Springer, vol. 18(1), pages 1-32, April.
    7. 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(1), pages 243-265, July.
    8. S. K. Mishra & Vinay Singh & Vivek Laha, 2016. "On duality for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 243(1), pages 249-272, August.
    9. R. R. Ota & J. C. Pati & A. K. Ojha, 2019. "Geometric programming technique to optimize power distribution system," OPSEARCH, Springer;Operational Research Society of India, vol. 56(1), pages 282-299, March.
    10. Liu, Shiang-Tai, 2008. "Posynomial geometric programming with interval exponents and coefficients," European Journal of Operational Research, Elsevier, vol. 186(1), pages 17-27, April.

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