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Technical Note—A General Inner Approximation Algorithm for Nonconvex Mathematical Programs

Author

Listed:
  • Barry R. Marks

    (Saint Mary's University, Halifax, Nova Scotia)

  • Gordon P. Wright

    (Purdue University, West Lafayette, Indiana)

Abstract

Inner approximation algorithms have had two major roles in the mathematical programming literature. Their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle. However, recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs. Avriel and Williams' (Avriel, M., A. C. Williams. 1970. Complementary geometric programming. SIAM J. Appl. Math. 19 125–141.) complementary geometric programming algorithm, Duffin and Peterson's (Duffin, R. J., E. L. Peterson. 1972. Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J. 22 531–550.) reversed geometric programming algorithms, Reklaitis and Wilde’s (Reklaitis, G. V., D. J. Wilde. 1974. Geometric programming via a primal auxiliary problem. AIIE Trans. 6 308–317.) primal reversed geometric programming algorithm, and Bitran and Novaes' (Bitran, G. R., A. G. Novaes. 1973. Linear programming with a fractional objective function. Opns. Res. 21 22–29.) linear fractional programming algorithm are all examples of this class of inner approximation algorithms. A sequence of approximating convex programs are solved in each of these algorithms. Rosen's (Rosen, J. B. 1966. Iterative solution of nonlinear optimal control problems. SIAM J. Control 4 223–244.) inner approximation algorithm is a special case of the general inner approximation algorithm presented in this note.

Suggested Citation

  • Barry R. Marks & Gordon P. Wright, 1978. "Technical Note—A General Inner Approximation Algorithm for Nonconvex Mathematical Programs," Operations Research, INFORMS, vol. 26(4), pages 681-683, August.
    • Handle: RePEc:inm:oropre:v:26:y:1978:i:4:p:681-683
    DOI: 10.1287/opre.26.4.681
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    Cited by:

    1. Shiyong Li & Wei Sun & Huan Liu, 2022. "Optimal resource allocation for multiclass services in peer-to-peer networks via successive approximation," Operational Research, Springer, vol. 22(3), pages 2605-2630, July.
    2. Ziping Zhao & Rui Zhou & Zhongju Wang & Daniel P. Palomar, 2018. "Optimal Portfolio Design for Statistical Arbitrage in Finance," Papers 1803.02974, arXiv.org.
    3. Xu, Gongxian, 2014. "Global optimization of signomial geometric programming problems," European Journal of Operational Research, Elsevier, vol. 233(3), pages 500-510.
    4. Huang, Jinbo & Li, Yong & Yao, Haixiang, 2022. "Partial moments and indexation investment strategies," Journal of Empirical Finance, Elsevier, vol. 67(C), pages 39-59.
    5. Ata Allah Taleizadeh & Leila Aliabadi & Park Thaichon, 2022. "A sustainable inventory system with price-sensitive demand and carbon emissions under partial trade credit and partial backordering," Operational Research, Springer, vol. 22(4), pages 4471-4516, September.
    6. Xu, Gongxian, 2013. "Steady-state optimization of biochemical systems through geometric programming," European Journal of Operational Research, Elsevier, vol. 225(1), pages 12-20.
    7. Y. Shi & H. D. Tuan & H. Tuy & S. Su, 2017. "Global optimization for optimal power flow over transmission networks," Journal of Global Optimization, Springer, vol. 69(3), pages 745-760, November.
    8. Kha-Hung Nguyen & Hieu V. Nguyen & Mai T. P. Le & Tuan X. Cao & Oh-Soon Shin, 2020. "Rate Fairness and Power Consumption Optimization for NOMA-Assisted Downlink Networks," Energies, MDPI, vol. 14(1), pages 1-18, December.

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