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Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions

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Abstract

The paper is devoted to the study of a new class of conic constrained optimization problems with objectives given as differences of a composite function and a convex function. We first introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we provide necessary and sufficient conditions for several versions of Farkas lemmas to hold. Similarly, we provide characterizations for conic constrained optimization problems to have the strong or stable strong dualities such as Lagrange, Fenchel–Lagrange or Toland–Fenchel–Lagrange duality.

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References

  1. Ban, L., Song, W.: Duality gap of the conic convex constrained optimization problems in normed spaces. Math. Program. Ser. A 119, 195–214 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boţ, R.I., Grad, S.M., Wanka, G.: New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces. Nonlinear Anal. 69, 323–336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boţ, R.I., Grad, S.M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boţ, 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 

  5. Jeyakumar, V.: Farkas’ Lemma and Extensions, Encyclopaedia of Optimization. Kluwer, Boston (2001)

    Google Scholar 

  6. Jeyakumar, V., Lee, G.M.: Complete characterization of stable Farkas’ Lemma and cone-convex programming duality. Math. Program. Ser. A 114, 335–347 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zǎlinescu, C.: On zero duality gap and the Farkas Lemma for conic programming. Math. Oper. Res. 34, 991–1001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dinh, N., Goberna, M.A., Lopez, M.A., Mo, T.H.: From the Farkas lemma to the Hahn–Banach theorem. SIAM J. Optim. 24, 678–701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dinh, N., Goberna, M.A., Lopez, M.A., Mo, T.H.: Farkas-type results for vector-valued functions with applications. J. Optim. Theory Appl. 173, 357–390 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dinh, N., Mordukhovich, B., Nghia, T.T.A.: Qualification and optimality conditions for convex and DC programs with infinite constraints. Acta Math. Vietnam. 34, 125–155 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Dinh, N., Mordukhovich, B., Nghia, T.T.A.: Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs. Math. Program. 123, 101–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming. SIAM J. Optim. 20, 1311–1332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming. Nonlinear Anal. 73, 1143–1159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19, 163–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Boţ, R.I., Grad, S.M., Wanka, G.: A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces. Math. Nachr. 281, 1088–1107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Boţ, 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 

  19. 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 

  20. Dinh, N., Vallet, G., Nghia, T.T.A.: Farkas-type results and duality for DC programs with convex constraints. J. Convex Anal. 2, 235–262 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Zhou, Y.Y., Li, G.: The Toland–Fenchel–Lagrange duality of DC programs for composite convex functions. Numer. Algebra Control Optim. 4, 9–23 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Fang, D.H., Gong, X.: Extended Farkas lemma and strong duality for composite optimization problems with DC functions. Optimization 66, 179–196 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fang, D.H., Lee, G.M., Li, C., Yao, J.C.: Extended Farkas’s lemmas and strong Lagrange dualities for DC infinite programming. J. Nonlinear Convex Anal. 14, 747–767 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Fang, D.H., Wang, M.D., Zhao, X.P.: The strong duality for DC optimization problems with composited convex functions. J. Nonlinear Convex Anal. 16, 1337–1352 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Sun, X.K., Li, S.J., Zhao, D.: Duality and Farkas-type results for DC infinite programming with inequality constraints. Taiwan. J. Math. 4, 1227–1244 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dinh, N., Vallet, G., Volle, M.: Functional inequalities and theorems of the alternative involving composite functions. J. Glob. Optim. 59, 837–863 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Horst, R., Thoat, N.V.: DC programming: overview. J. Optim. Theory Appl. 103, 1–43 (1999)

    Article  MathSciNet  Google Scholar 

  28. Flores-Bazán, F.: On minima of the difference of functions. J. Optim. Theory Appl. 93, 525–531 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zǎlinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)

    Book  MATH  Google Scholar 

  30. Long, X.J., Sun, X.K., Peng, Z.Y.: Approximate optimality conditions for composite convex optimization problems. J. Oper. Res. Soc. China 5, 469–485 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the editor and the referees for their valuable comments and constructive suggestions which improved the presentation of this manuscript. Research work of the first author is supported in part by the National Natural Science Foundation of China (Grant 11461027), Hunan Provincial Natural Science Foundation of China (Grant 2016JJ2099) and the Scientific Research Fund of Hunan Provincial Education Department (Grant 17A172). The second authors is supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant 15C1156).

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  1. College of Mathematics and Statistics, Jishou University, Jishou, 416000, People’s Republic of China

    D. H. Fang & Y. Zhang

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  1. D. H. Fang
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  2. Y. Zhang
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Correspondence to D. H. Fang.

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Fang, D.H., Zhang, Y. Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions. J Optim Theory Appl 176, 351–376 (2018). https://doi.org/10.1007/s10957-018-1219-3

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  • DOI: https://doi.org/10.1007/s10957-018-1219-3

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