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Convexification for a Class of Global Optimization Problems with C\(^{1,1}\) Functions

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Abstract

A general convexification method via domain transformation scheme is presented for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems with C\(^{1,1}\) functions can be converted into equivalent convex optimization problems by using the proposed convexification method. Finally, an example is shown to illustrate how a monotone non-convex optimization problem can be transformed into an equivalent convex minimization problem.

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References

  1. Tzafestas, S.G.: Optimization of system reliability: a survey of problems and techniques. Int. J. Syst. Sci. 11, 455–486 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ibaraki, T., Katoh, N.: Resource Allocation Problems: Algorithmic Approaches. MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  3. Li, D., Sun, X.L., Biswal, M.P., Gao, F.: Convexification, concavification, and monotonization in global optimization. Ann. Oper. Res. 105, 213–226 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Sun, X.L., Mckinnon, K.I.M., Li, D.: A convexification method for a class of global optimization problems with applications to reliability optimization. J. Glob. Optim. 21, 185–199 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Xia, Y.: A survey of hidden convex optimization. J. Oper. Res. Soc. China 8, 1–28 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benson, H.P.: Deterministic algorithms for constrained concave minimization: a unified critical survey. Naval Res. Log. 43, 765–795 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoffman, K.L.: A method for globally minimizing concave functions over convex sets. Math. Program. 20, 22–32 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Horst, R., Pardalos, P.M., Van Thoai, N.: Introduction to Global Optimization. Springer (2000)

  9. Tuy, H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tuy, H.: Convex Analysis and Global Optimization. Spring, Berlin (2016)

    Book  MATH  Google Scholar 

  11. Horst, R.: On the convexification of nonlinear programming problems: an applications-oriented survey. Eur. J. Oper. Res. 15, 382–392 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xu, G.X.: Global optimization of signomial geometric programming problems. Eur. J. Oper. Res. 233, 500–510 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu, H.C., Yao, L.M.: Efficient convexification strategy for generalized geometric programming problems. INFORMS J. Comput. 31, 226–234 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wu, Z.Y., Zhang, L.S., Bai, F.S., Yang, X.M.: Convexification and convexification methods for some global optimization problems. J. Syst. Sci. Complex 17, 421–436 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Wu, Z.Y., Bai, F.S., Zhang, L.S.: Convexification and concavification for a general class of global optimization problems. J. Glob. Optim. 31, 45–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, Z.Y., Lee, H.W.J., Yang, X.M.: A class of convexification and concavification methods for non-monotone optimization problems. Optimization 54, 605–625 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, D., Wu, Z.Y., Lee, H.W.J., Yang, X.M., Zhang, L.S.: Hidden convex minimization. J. Glob. Optim. 31, 211–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sun, X.L., Luo, H.Z., Li, D.: Convexification of nonsmooth monotone functions. J. Optim. Theory Appl. 132, 339–351 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, D.: Convexification of a noninferior frontier. J. Optim. Theory Appl. 88, 177–196 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, L.S., Wu, D.H.: Convexification, concavification and monotonization in nonlinear programming. Chin. Ann. Math. Ser. A 23, 537–544 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Yan, Q., Yang, X.M., Wu, Z.Y.: On convexification for a class of global optimization problems. J. Oper. Res. Soc. China. https://doi.org/10.1007/s40305-021-00379-0

  22. Rockafellar, R.T.: Solving a nonlinear programming problem by way of a dual problem. Symp. Math. 19, 135–160 (1976)

    MathSciNet  Google Scholar 

  23. Beck, A.: First-Order Methods in Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2017)

    Book  MATH  Google Scholar 

  24. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  25. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chang, T.J., Meade, N., Beasley, J.E., Sharaiha, Y.M.: Heuristics for cardinality constrained portfolio optimisation. Comput. Oper. Res. 27, 1271–1302 (2000)

    Article  MATH  Google Scholar 

  27. La Torre, D., Vercellis, C.: C\(^{1,1}\) approximations of generalized support vector machines. J. Concr. Appl. Math. 3, 41–54 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer (2009)

  29. Ginchev, I., La Torre, D., Rocca, M.: C\(^{k,1}\) functions, characterization, Taylor’s formula and optimization: a survey. Real Anal. Exch. 35, 311–342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, X.Q.: Generalized second-order directional derivatives and optimization conditions. Ph.D. Disseration, University of New South Wales, Australia (1994)

  31. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  32. Yang, X.Q., Jeyakumar, V.: Generalized second-order directional derivatives and optimization with C\(^{1,1}\) functions. Optimization 26, 165–185 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Sciences, Chongqing University of Technology, Chongqing, 400054, China

    Qian Yan

  2. College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China

    Qian Yan

  3. National Center for Applied Mathematics of Chongqing, Chongqing, 401331, China

    Xin-Min Yang

  4. College of Mathematical Science, Chongqing Normal University, Chongqing, 401331, China

    Xin-Min Yang & Zhi-You Wu

Authors
  1. Qian Yan
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  2. Xin-Min Yang
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  3. Zhi-You Wu
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Contributions

Q. Yan has contributed to analysis, writing and revision. X.-M. Yang has contributed to proposing the idea and method and revision of the manuscript. Z.-Y. Wu has contributed to analysis and revision.

Corresponding author

Correspondence to Qian Yan.

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The authors declare no conflict of interest.

Additional information

This work was supported by the Major Program of the National Natural Science Foundation of China (Nos. 11991020, 11991024), the National Natural Science Foundation of China (No. 11871128), Scientific Research Foundation of Chongqing University of Technology (No. 2023ZDZ021), Youth project of science and technology research program of Chongqing Education Commission of China (No. KJQN202301160).

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Yan, Q., Yang, XM. & Wu, ZY. Convexification for a Class of Global Optimization Problems with C\(^{1,1}\) Functions. J. Oper. Res. Soc. China (2023). https://doi.org/10.1007/s40305-023-00521-0

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  • DOI: https://doi.org/10.1007/s40305-023-00521-0

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