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Superlinearly Convergent Norm-Relaxed SQP Method Based on Active Set Identification and New Line Search for Constrained Minimax Problems

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Abstract

In this paper, the minimax problems with inequality constraints are discussed, and an alternative fast convergent method for the discussed problems is proposed. Compared with the previous work, the proposed method has the following main characteristics. First, the active set identification which can reduce the scale and the computational cost is adopted to construct the direction finding subproblems. Second, the master direction and high-order correction direction are computed by solving a new type of norm-relaxed quadratic programming subproblem and a system of linear equations, respectively. Third, the step size is yielded by a new line search which combines the method of strongly sub-feasible direction with the penalty method. Fourth, under mild assumptions without any strict complementarity, both the global convergence and rate of superlinear convergence can be obtained. Finally, some numerical results are reported.

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References

  1. Zhou, J.L., Tits, A.L.: Nonmonotone line search for minimax problems. J. Optim. Theory Appl. 76, 455–476 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Rustem, B., Nguyen, Q.: An algorithm for the inequality-constrained discrete minimax problem. SIAM J. Optim. 8, 265–283 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yu, Y.H., Gao, L.: Nonmonotone line search algorithm for constrained minimax problems. J. Optim. Theory Appl. 115, 419–446 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jian, J.B., Quan, R., Zhang, X.L.: Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints. J. Comput. Appl. Math. 205, 406–429 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Reemtsen, R.: A cutting plane method for solving minimax problems in the complex plane. Numer. Algorithms 2, 409–436 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20, 267–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Polak, E., Womersley, R.S., Yin, H.X.: An algorithm based on active sets and smoothing for discretized semi-infinite minimax problems. J. Math. Anal. Appl. 138, 311–328 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Zhu, Z.B., Cai, X., Jian, J.B.: An improved SQP algorithm for solving minimax problems. Appl. Math. Lett. 22, 464–469 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jian, J.B., Zhang, X.L., Quan, R., Ma, Q.: Generalized monotone line search SQP algorithm for constrained minimax problems. Optimization 58, 101–131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, D.L., Jian, J.B., Li, J.: On the accurate identification of active set for constrained minimax problems. Nonlinear Anal., Real World Appl. 74, 3022–3032 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Royset, J.O., Pee, E.Y.: Rate of convergence analysis of discretization and smoothing algorithms for semi-infinite minimax problems. J. Optim. Theory Appl. 155, 855–882 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bagirov, A.M., Al Nuaimat, A., Sultanova, N.: Hyperbolic smoothing function method for minimax problems. Optimization 62, 759–782 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hare, W., Macklem, M.: Derivative-free optimization methods for finite minimax problems. Optim. Methods Softw. 28, 300–312 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hare, W., Nutini, J.: A derivative-free approximate gradient sampling algorithm for finite minimax problems. Comput. Optim. Appl. 56, 1–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, F.S.: A hybrid algorithm for linearly constrained minimax problems. Ann. Oper. Res. 206, 501–525 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jian, J.B., Mo, X.D., Qiu, L.J., Yang, S.M., Wang, F.S.: Simple sequential quadratically constrained quadratic programming feasible algorithm with active identification sets for constrained minimax problems. J. Optim. Theory Appl. (2013, in press). doi:10.1007/s10957-013-0339-z

  17. Zoutendijk, G.: Methods of Feasible Directions. Elsevier, Amsterdam (1960)

    MATH  Google Scholar 

  18. Topkis, D.M., Veinott, A.F.: On the convergence of some feasible direction algorithms for nonlinear programming. SIAM J. Control 5, 268–279 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pironneau, O., Polak, E.: Rate of convergence of a class of methods of feasible directions. SIAM J. Numer. Anal. 10, 161–173 (1973)

    Article  MathSciNet  Google Scholar 

  20. Cawood, M.E., Kostreva, M.M.: Norm-relaxed method of feasible direction for solving the nonlinear programming problems. J. Optim. Theory Appl. 83, 311–320 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, X., Kostreva, M.M.: A generalization of the norm-relaxed method of feasible directions. Appl. Math. Comput. 102, 257–273 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kostreva, M.M., Chen, X.: A superlinearly convergent method of feasible directions. Appl. Math. Comput. 116, 231–244 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lawrence, C.T., Tits, A.L.: A computationally efficient feasible sequential quadratic programming algorithm. SIAM J. Optim. 11, 1092–1118 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, X., Kostreva, M.M.: Global convergence analysis of algorithm for finding feasible points in norm-relaxed method of feasible directions. J. Optim. Theory Appl. 100, 287–309 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jian, J.B.: Strong combined Phase I–Phase II methods of sub-feasible directions. Math. Econ. 12, 64–70 (1995) (in Chinese)

    Google Scholar 

  26. Jian, J.B., Zheng, H.Y., Hu, Q.J., Tang, C.M.: A new norm-relaxed method of strongly sub-feasible direction for inequality constrained optimization. Appl. Math. Comput. 168, 1–28 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jian, J.B., Zheng, H.Y., Hu, Q.J., Tang, C.M.: A new superlinearly convergent norm-relaxed method of strongly sub-feasible direction for inequality constrained optimization. Appl. Math. Comput. 182, 955–976 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jian, J.B.: Fast Algorithms for Smooth Constrained Optimization-Theoretical Analysis and Numerical Experiments. Science Press, Beijing (2010)

    Google Scholar 

  29. Solodov, M.V.: Global convergence of an SQP method without boundedness assumptions on any of iterative sequences. Math. Program. 118, 1–12 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zheng, H.Y., Jian, J.B., Tang, C.M., Quan, R.: A new norm-relaxed SQP algorithm with global convergence. Appl. Math. Lett. 23, 670–675 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Burke, J.V., More, J.J.: On the identification of active constraints. SIAM J. Numer. Anal. 25, 1197–1211 (1998)

    Article  MathSciNet  Google Scholar 

  32. Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hare, W.L., Lewis, A.S.: Identifying active constraints via partial smoothness and prox-regularity. J. Convex Anal. 11, 251–266 (2004)

    MathSciNet  MATH  Google Scholar 

  34. Jian, J.B., Liu, Y.: A superlinearly convergent method of quasi-strongly sub-feasible direction with active set identifying for constrained optimization. Nonlinear Anal., Real World Appl. 12, 2717–2729 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chao, M.T., Wang, Z.X., Liang, Y.M., Hu, Q.J.: Quadratically constraint quadratical algorithm model for nonlinear minimax problems. Appl. Math. Comput. 205, 247–262 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Karmitsa, N.: Test problems for large-scale nonsmooth minimization. Reports of the Department of Mathematical Information Technology, Series B. Scientific Computing, No. B.4 (2007)

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Acknowledgements

Project supported by NSFC (No. 11271086 and 11171250), the Natural Science Foundation of Guangxi Province (No. 2011GXNSFD018022), and Innovation Group of Talents Highland of Guangxi Higher School.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Yulin Normal University, Yulin, 537000, China

    Jin-bao Jian

  2. Luoshan Secondary Vocational School, Luoshan, 464200, China

    Qing-juan Hu

  3. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Chun-ming Tang

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  1. Jin-bao Jian
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  2. Qing-juan Hu
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  3. Chun-ming Tang
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Correspondence to Jin-bao Jian.

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Jian, Jb., Hu, Qj. & Tang, Cm. Superlinearly Convergent Norm-Relaxed SQP Method Based on Active Set Identification and New Line Search for Constrained Minimax Problems. J Optim Theory Appl 163, 859–883 (2014). https://doi.org/10.1007/s10957-013-0503-5

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