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Global and local convergence of a nonmonotone SQP method for constrained nonlinear optimization

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Abstract

In this paper, we propose a robust sequential quadratic programming (SQP) method for nonlinear programming without using any explicit penalty function and filter. The method embeds the modified QP subproblem proposed by Burke and Han (Math Program 43:277–303, 1989) for the search direction, which overcomes the common difficulty in the traditional SQP methods, namely the inconsistency of the quadratic programming subproblems. A non-monotonic technique is employed further in a framework in which the trial point is accepted whenever there is a sufficient relaxed reduction of the objective function or the constraint violation function. A forcing sequence possibly tending to zero is introduced to control the constraint violation dynamically, which is able to prevent the constraint violation from over-relaxing and plays a crucial role in global convergence and the local fast convergence as well. We prove that the method converges globally without the Mangasarian–Fromovitz constraint qualification (MFCQ). In particular, we show that any feasible limit point that satisfies the relaxed constant positive linear dependence constraint qualification is also a Karush–Kuhn–Tucker point. Under the strict MFCQ and the second order sufficient condition, furthermore, we establish the superlinear convergence. Preliminary numerical results show the efficiency of our method.

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References

  1. Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 10(4), 1116–1135 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andreani, R., Haeser, G., Schuverdt, M., Silva, P.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. (2011). doi:10.1007/s10107-011-0456-0

  3. Andreani, R., Martinez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bielschowsky, R.H., Gomes, F.A.M.: Dynamical control of infeasibility in nonlinearly constrained optimization. SIAM J. Optim. 19(3), 1299–1325 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bongartz, I., Conn, A.R., Gould, N.I.M., Toint, PhL: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21, 123–160 (1995)

    Article  MATH  Google Scholar 

  6. Bonnans, J.F., Gilbert, JCh., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  7. Boggs, P.T., Tolle, J.W., Wang, P.: On the local convergence of quasi-newton methods for constrained optimization. SIAM J. Control Optim. 20, 161–171 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonnans, J.F., Ioffe, A.: Second-order sufficiency and quadratic growth for nonisolated minima. Math. Oper. Res. 20(4), 801–817 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Burke, J.V., Han, S.P.: A robust sequential quadratic programming method. Math. Program. 43, 277–303 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Byrd, R., Lopez-Calva, G., Nocedal, J.: A line search exact penalty method using steering rules. Math. Program. 133(1–2), 39–73 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chamberlain, R.M., Powell, M.J.D., Lemarechal, C., Pedersen, H.C.: The watchdog technique for forcing convergence in algorithms for constrained optimization. Math. Program. Study 16, 1–17 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dolan, E.D., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fernández, D., Izmailov, A.F., Solodov, M.V.: Sharp primal superlinear convergence results for some Newtonian methods for constrained optimization. SIAM J. Optim. 20(6), 3312–3334 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91, 239–269 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fletcher, R., Leyffer, S., Toint, PhL: On the global convergence of a filter-SQP algorithm. SIAM J. Optim. 13(1), 44–59 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fletcher, R.: A sequential linear constraints programming algorithm for NLP. SIAM J. Optim. 22(3), 772–794 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4), 979–1006 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gould, N., Toint, PhL: Nonlinear programming without a penalty function or a filter. Math. Program. 122(1), 155–196 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gould, N., Robinson, D.P., Toint, Ph.L.: Corrigendum: Nonlinear Programming Without a Penalty Function or a Filter. NAXYS Technical Report naxys-07-2011, Namur Center for Complex Systems (NAXYS) (2011)

  20. Gould, N., Robinson, D.P.: A second derivative SQP method: global convergence. SIAM J. Optim. 20(4), 2023–2048 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gould, N., Robinson, D.P.: A second derivative SQP method: local convergence. SIAM J. Optim. 20(4), 2049–2079 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hager, W.W.: Stabilized sequential quadratic programming. Comput. Optim. Appl. 12, 253–273 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Han, S.P.: Superlinearly convergent variable metric algorithms for general nonlinear programming problems. Math. Program. 11, 263–282 (1976)

    Article  Google Scholar 

  25. Han, S.P.: A globally convergent method for nonlinear programming. J. Optim. Theory Appl. 22, 297–309 (1977)

    Article  MATH  Google Scholar 

  26. Hestenes, M.R.: Optimization Theory: The Finite-Dimensional Case. Wiley, New York, NY (1975)

    MATH  Google Scholar 

  27. Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  28. Kyparisis, J.: On uniqueness of Kuhn–Tucker multipliers in nonlinear programming. Math. Program. 32(2), 242–246 (1984)

    Article  MathSciNet  Google Scholar 

  29. Liu, X.W., Yuan, Y.X.: A sequential quadratic programming method without a penalty function or a filter for nonlinear equality constrained optimization. SIAM J. Optim. 21(2), 545–571 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Morales, J.L., Nocedal, J., Wu, Y.: A sequential quadratic programming algorithm with an additional equality constrained phase. IMA J. Numer. Anal. (2011). doi:10.1093/imanum/drq037

  31. Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  32. Powell, M.J.D.: A fast algorithm for nonlinearly constrained optimization calculations. In: Waston, G.A. (ed.) Numerical Analysis, Proceedings, Biennial Conference, Dundee 1977. Lecture Notes in Mathematics 630, pp. 144–157. Springer, Berlin (1978)

  33. Powell, M.J.D.: Variable metric methods for constrained optimization. In: Bachem, A., Grotschel, M., Korte, B. (eds.) Mathematical Programming: The State of Art, Bonn, 1982. Springer, Berlin (1983)

    Google Scholar 

  34. Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shen, C.G., Leyffer, S., Fletcher, R.: A nonmonotone filter method for nonlinear optimization. Comput. Optim. Appl. 52, 583–607 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ulbrich, M., Ulbrich, S.: Nonmonotone trust-region methods for nonlinear equality constrained optimization without a penalty function. Math. Program. 95, 103–135 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wright, S.J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11(3), 253–275 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wright, S.J.: Modifying SQP for degenerate problems. SIAM J. Optim. 13(2), 470–497 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wright, S.J.: Constraint identification and algorithm stabilization for degenerate nonlinear programs. Math. Program. 95(1), 137–160 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wright, S.J.: An algorithm for degenerate nonlinear programming with rapid local convergence. SIAM J. Optim. 15(3), 673–696 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16, 1–31 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: local convergence. SIAM J. Optim. 16, 32–48 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  44. Xue, W.J., Shen, C.G., Pu, D.G.: A penalty-function-free line search SQP method for nonlinear programming. J. Comput. Appl. Math. 228(1), 313–325 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  45. Zhang, J.L., Zhang, X.S.: A modified SQP method with nonmonotone linsearch technique. J. Glob. Optim. 21, 201–218 (2001)

    Article  MATH  Google Scholar 

  46. Zhou, G.L.: A modified SQP method and its global convergence. J. Glob. Optim. 11, 193–205 (1997)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

We are indebted to the editor and two anonymous referees for their many valuable comments and suggestions that have improved the quality of this paper significantly. This research is supported by National Natural Science Foundation of China (Nos. 11101281, 11101257 and 11271259) and Innovation Program of Shanghai Municipal Education Commission (No. 12YZ172).

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai Finance University, Shanghai , 201209, China

    Chungen Shen

  2. Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai , 200433, China

    Lei-Hong Zhang

  3. School of Applied Mathematics, Nanjing University of Finance and Economics, Shanghai , 200092, China

    Bo Wang

  4. Industrial and Systems Engineering Department, Wayne State University, Detroit, MI , 48202, USA

    Wenqiong Shao

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  1. Chungen Shen
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Correspondence to Chungen Shen.

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Shen, C., Zhang, LH., Wang, B. et al. Global and local convergence of a nonmonotone SQP method for constrained nonlinear optimization. Comput Optim Appl 59, 435–473 (2014). https://doi.org/10.1007/s10589-014-9675-7

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