Skip to main content
SpringerLink
Log in

Approximate Optimality Conditions for Composite Convex Optimization Problems

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces, where all functions involved are not necessarily lower semi-continuous. By using the properties of the epigraph of conjugate functions, we introduce a new regularity condition and give its equivalent characterizations. Under this new regularity condition, we derive necessary and sufficient optimality conditions of \(\varepsilon \)-optimal solutions for the composite convex optimization problem. As applications of our results, we derive approximate optimality conditions to cone-convex optimization problems. Our results extend or cover many known results in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gao, Y., Hou, S.H., Yang, X.M.: Existence and optimality conditions for approximate solutions to vector optimization problems. J. Optim. Theory Appl. 152, 97–120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Sov. Math. Dokl. 20, 391–393 (1979)

    MATH  Google Scholar 

  3. Long, X.J., Li, X.B., Zeng, J.: Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems. Optim. Lett. 7, 1847–1856 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bot, R.I., Hodrea, I.B., Wanka, G.: \(\varepsilon \)-Optimality conditions for composed convex optimization problems. J. Approx. Theory 153, 108–121 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bot, R.I., Grad, S.M., Wanka, G.: Generalized Moreau-Rockafellar results for composed convex functions. Optimization 58, 917–933 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Combari, C., Laghdir, M., Thibault, L.: A note subdifferentials of convex composite functionals. Arch. Math. 67, 239–252 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang, D.H., Li, C., Yang, X.Q.: Stable and total Fenchel duality for DC optimization problems in locally convex spaces. SIAM J. Optim. 21, 730–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang, D.H., Li, C., Yang, X.Q.: Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces. Nonlinear Anal. 75, 3672–3681 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C., Fang, D.H., Lopez, G., Lopez, M.A.: Stable and total Fenchel duality for convex optimization problems in locally convex spaces. SIAM J. Optim. 20, 1032–1051 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Long, X.J., Huang, N.J., O’Regan, D.: Farkas-type results for general composed convex optimization problems with inequality constraints. Math. Inequal. Appl. 13, 135–143 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Sun, X.K., Guo, X.L., Zhang, Y.: Fenchel-Lagrange duality for DC programs with composite functions. J. Nonlinear Convex Anal. 16, 1607–1618 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Sun, X.K., Long, X.J., Zeng, J.: Constraint qualifications characterizing Fenchel duality in composed convex optimization. J. Nonlinear Convex Anal. 17, 325–347 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Bot, R.I., Grad, S.M., Wanka, G.: New constraint qualification and conjugate duality for composed convex optimization problems. J. Optim. Theory Appl. 135, 241–255 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dinh, N., Nghia, T.T.A., Vallet, G.: A closedness condition and its applications to DC programs with convex constraints. Optimization 59, 541–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  16. Jeyakumar, V.: Asymptotic dual conditions characterizing optimality for infinite convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas’s lemmas and Lagranging dulaities in locally convex spaces. SIAM J. Optim. 20, 1311–1332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Flores-Bazan, F., Masteoeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeyakumar, V., Kum, S., Lee, G.M.: Necessary and sufficient conditions for Farkas’ lemma for cone systems and second-order cone programming duality. J. Convex Anal. 15, 63–71 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Jeyakumar, V., Lee, G.M.: Complete characterizations of stable Farkas’ lemma and cone-convex programming duality. Math. Program. 114, 335–347 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and suggestions, which have improved the presentation of the paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Xian-Jun Long & Xiang-Kai Sun

  2. College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing, 400074, China

    Zai-Yun Peng

Authors
  1. Xian-Jun Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang-Kai Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zai-Yun Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian-Jun Long.

Additional information

This research was supported by the National Natural Science Foundation of China (Nos. 11471059, 11301571, and 11301570), the Chongqing Research Program of Basic Research and Frontier Technology (Nos. cstc2014jcyjA00037, cstc2015jcyjB00001, cstc2015jcyjA00025, and cstc2015jcyjA00002), the Education Committee Project Research Foundation of Chongqing (Nos. KJ1400618 and KJ1500626), the Postdoctoral Science Foundation of China (Nos. 2015M580774 and 2016T90837) and the Program for University Innovation Team of Chongqing (CXTDX201601026 and CXTDX201601022).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Long, XJ., Sun, XK. & Peng, ZY. Approximate Optimality Conditions for Composite Convex Optimization Problems. J. Oper. Res. Soc. China 5, 469–485 (2017). https://doi.org/10.1007/s40305-016-0140-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-016-0140-4

Keywords

Mathematics Subject Classification

Search

Navigation