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Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

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Abstract

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.

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References

  1. Benson, H.P.: Deterministic algorithm for constrained concave minimization: a unified critical survey. Nav. Res. Logist. 43, 765–795 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Floudas, C.A.: Deterministic Global Optimization: Theory, Methods and Application. Kluwer Academic, Dordrecht (1999)

    Google Scholar 

  3. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic, Dordrecht (2000)

    Book  MATH  Google Scholar 

  4. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  5. Pinter, J.D.: Global Optimization in Action. Kluwer Academic, Dordrecht (1996)

    Book  MATH  Google Scholar 

  6. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed- Integer Nonlinear Programming. Kluwer Academic, Dordrecht (2002)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Neumaier, A., Shcherbina, O., Huyer, W., Vinkó, T.: A comparison of complete global optimization solvers. Math. Program. 103, 335–356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Birgin, E.G., Floudas, C.A., Martınez, J.M.: Global minimization using an augmented Lagrangian method with variable lower-level constraints. Math. Program. 125, 139–162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Luo, H.Z., Sun, X.L., Li, D.: On the convergence of augmented Lagrangian methods for constrained global optimization. SIAM J. Optim. 18, 1209–1230 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Luo, H.Z., Sun, X.L., Wu, H.X.: Convergence properties of augmented Lagrangian methods for constrained global optimization. Optim. Methods Softw. 23, 763–778 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, C.Y., Li, D.: Unified theory of augmented Lagrangian methods for constrained global optimization. J. Glob. Optim. 44, 433–458 (2009)

    Article  MATH  Google Scholar 

  14. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  15. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)

    Google Scholar 

  16. Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bertsekas, D.P.: Constrained Optimization and Lagrangian Multiplier Methods. Academic Press, New York (1982)

    Google Scholar 

  18. Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, X.L., Li, D., McKinnon, K.I.M.: On saddle points of augmented Lagrangians for constrained nonconvex optimization. SIAM J. Optim. 15, 1128–1146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wu, H.X., Luo, H.Z.: A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization. Optimization 61, 1331–1345 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, H.X., Luo, H.Z.: Saddle points of general augmented Lagrangians for constrained nonconvex optimization. J. Glob. Optim. 53, 683–697 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5–32 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Conn, A.R., Gould, N.I.M., Sartenaer, A., Toint, P.L.: Convergence properties of an augmented Lagrangian algorithm for optimization with a combination of general equality and linear constraints. SIAM J. Optim. 6, 674–703 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545–572 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Polyak, R.: Modified barrier functions: theory and methods. Math. Program. 54, 177–222 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Di Pillo, G., Grippo, L.: A new class of augmented Lagrangian function in nonlinear programming. SIAM J. Control Optim. 17, 618–628 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Di Pillo, G., Grippo, L.: A new augmented Lagrangian function for inequality constraints in nonlinear programming problems. J. Optim. Theory Appl. 36, 495–519 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lucidi, S.: New results on a class of exact augmented Lagrangians. J. Optim. Theory Appl. 58, 259–282 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Di Pillo, G., Lucidi, S.: An augmented Lagrangian function with improved exactness properties. SIAM J. Optim. 12, 376–406 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Di Pillo, G., Lucidi, S., Palagi, L.: Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming. Math. Oper. Res. 30, 897–915 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  34. Giannessi, F., Mastroeni, G., Yao, J.C.: On maximum and variational principles via image space analysis. Positivity 16, 405–427 (2012)

    Article  MathSciNet  Google Scholar 

  35. Mastroeni, G.: Nonlinear separation in the image space with applications to penalty methods. Appl. Anal. 91, 1901–1914 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31, 143–165 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ben-Tal, A., Zowe, J.: Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47, 483–490 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the two anonymous referees for the constructive comments and suggestions, which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China under grant 11071219, the Zhejiang Provincial Natural Science Foundation of China under grants LY13A010012 and LY13A010017, and the Postdoctoral Key Research Foundation of China under grant 201003242.

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, Zhejiang, 310032, P.R. China

    Hezhi Luo

  2. Department of Mathematics, College of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, P.R. China

    Huixian Wu & Jianzhen Liu

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  1. Hezhi Luo
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  3. Jianzhen Liu
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Correspondence to Hezhi Luo.

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Luo, H., Wu, H. & Liu, J. Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis. J Optim Theory Appl 159, 360–385 (2013). https://doi.org/10.1007/s10957-013-0358-9

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