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Duality Results for Conic Convex Programming

Author

Listed:
  • Luo, Z-Q.
  • Sturm, J.F.
  • Zhang, S.

Abstract

This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone in finite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include Gordon-Stiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.

Suggested Citation

  • Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1997. "Duality Results for Conic Convex Programming," Econometric Institute Research Papers EI 9719/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:1412
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    References listed on IDEAS

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    1. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1996. "Duality and Self-Duality for Conic Convex Programming," Econometric Institute Research Papers EI 9620-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. NESTEROV, Yurii & TODD, Michael & YE, Yinyu, 1996. "Primal-Dual Methods and Infeasibility Detectors for Nonlinear Programming Problems," LIDAM Discussion Papers CORE 1996037, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV , Yurii & TODD , Michael, 1995. "Primal-Dual Interior-Point Methods for Self-Scaled Cones," LIDAM Discussion Papers CORE 1995044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Epelman, Marina A., 1973-. & Freund, Robert Michael, 1997. "Condition number complexity of an elementary algorithm for resolving a conic linear system," Working papers WP 3942-97., Massachusetts Institute of Technology (MIT), Sloan School of Management.
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    Citations

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    Cited by:

    1. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    2. Brinkhuis, J. & Zhang, S., 2003. "A D-induced duality and its applications," Econometric Institute Research Papers EI 2003-42, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. O. I. Kostyukova & T. V. Tchemisova, 2022. "On strong duality in linear copositive programming," Journal of Global Optimization, Springer, vol. 83(3), pages 457-480, July.
    4. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1998. "Conic convex programming and self-dual embedding," Econometric Institute Research Papers EI 9815, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Kenneth O. Kortanek & Guolin Yu & Qinghong Zhang, 2021. "Strong duality for standard convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 413-436, December.
    6. Zhang, S., 1998. "Global error bounds for convex conic problems," Econometric Institute Research Papers EI 9830, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Zhi-Quan Luo & Shuzhong Zhang, 1997. "On the extensions of Frank-Wolfe theorem," Tinbergen Institute Discussion Papers 97-122/4, Tinbergen Institute.
    8. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    9. Yao, D.D. & Zhang, S. & Zhou, X.Y., 1999. "LQ control without Ricatti equations: deterministic systems," Econometric Institute Research Papers EI 9913-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. Hayato Waki & Masakazu Muramatsu, 2013. "Facial Reduction Algorithms for Conic Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 188-215, July.

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