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A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

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Abstract

In the second part of our study, we introduce the concept of global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in an extension of the original constrained optimization problem by adding some extra variables, and then construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended merit functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of a merit function based on sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.

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References

  1. Dolgopolik, M.V.: A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness. arXiv: 1709.07073 (2017)

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Di Pillo, G., Grippo, L., Lampariello, F.: A method for solving equality constrained optimization problems by unconstrained minimization. In: Iracki, K., Malanowski, K., Walukiewicz, S. (eds.) Optimization Techniques: Proceedings of the 9th IFIP Conference on Optimization Techniques, pp. 96–105. Springer, Berlin (1980)

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Di Pillo, G., Lucidi, S., Palagi, L.: An exact penalty-Lagrangian approach for a class of constrained optimization problems with bounded variables. Optimization 28, 129–148 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Di Pillo, G., Lucidi, S.: On exact augmented Lagrangian functions in nonlinear programming. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 85–100. Plenum Press, New York (1996)

    Chapter  Google Scholar 

  8. 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 

  9. Di Pillo, G., Liuzzi, G., Lucidi, S., Palagi, L.: Fruitful uses of smooth exact merit functions in constrained optimization. In: Di Pillo, G., Murli, A. (eds.) High Performance Algorithms and Software for Nonlinear Optimization, pp. 201–225. Kluwer Academic Publishers, Dordrecht (2003)

    Chapter  Google Scholar 

  10. Di Pillo, G., Liuzzi, G., Lucidi, S., Palagi, L.: An exact augmented Lagrangian function for nonlinear programming with two-sided constraints. Comput. Optim. Appl. 25, 57–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, X., Zhang, L., Gao, Y.: A class of augmented Lagrangians for equality constraints in nonlinear programming problems. Appl. Math. Comput. 172, 644–663 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Du, X., Liang, Y., Zhang, L.: Further study on a class of augmented Lagrangians of Di Pillo and Grippo in nonlinear programming. J. Shanghai Univ. (Engl. Ed.) 10, 293–298 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Di Pillo, G., Luizzi, G., Lucidi, S.: An exact penalty-Lagrangian approach for large-scale nonlinear programming. Optimization 60, 223–252 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fukuda, E.H., Lourenco, B.F.: Exact augmented Lagrangian functions for nonlinear semidefinite programming. arXiv: 1705.06551 (2017)

  16. Dolgopolik, M.V.: Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property. J. Glob. Optim. (2018). https://doi.org/10.1007/s10898-017-0603-0

    Google Scholar 

  17. Huyer, W., Neumaier, A.: A new exact penalty function. SIAM J. Optim. 13, 1141–1158 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bingzhuang, L., Wenling, Z.: A modified exact smooth penalty function for nonlinear constrained optimization. J. Inequal. Appl. 1, 173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, C., Ma, C., Zhou, J.: A new class of exact penalty functions and penalty algorithms. J. Glob. Optim. 58, 51–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, B., Yu, C.J., Teo, K.L., Duan, G.R.: An exact penalty function method for continuous inequality constrained optimal control problem. J. Optim. Theory Appl. 151, 260–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, C., Li, X., Cedric Yiu, K.-F., Zhang, L.-S.: New exact penalty function for solving constrained finite min-max problems. Appl. Math. Mech. Engl. Ed. 33, 253–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H., Yu, C.: A new exact penalty method for semi-infinite programming problems. J. Comput. Appl. Math. 261, 271–286 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiang, C., Lin, Q., Yu, C., Teo, K.L., Duan, G.-R.: An exact penalty method for free terminal time optimal control problem with continuous inequality constraints. J. Optim. Theory Appl. 154, 30–53 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H.: Optimal feedback control for dynamic systems with state constraints: an exact penalty approach. Optim. Lett. 8, 1535–1551 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ma, C., Zhang, L.: On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems. Appl. Math. Comput. 271, 642–656 (2015)

    MathSciNet  Google Scholar 

  26. Zheng, F., Zhang, L.: Constrained global optimization using a new exact penalty function. In: Gao, D., Ruan, N., Xing, W. (eds.) Advances in Global Optimization, pp. 69–76. Springer, Cham (2015)

    Google Scholar 

  27. Dolgopolik, M.V.: Smooth exact penalty functions: a general approach. Optim. Lett. 10, 635–648 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dolgopolik, M.V.: Smooth exact penalty function II: a reduction to standard exact penalty functions. Optim. Lett. 10, 1541–1560 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions II: parametric penalty functions. Optimization 66, 1577–1622 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65, 1167–1202 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dolgopolik, M.V.: A unified approach to the global exactness of penalty and augmented Lagrangian functions II: extended exactness. arXiv: 1710.01961 (2017)

  32. Giannessi, F.: Constrained Optimization and Image Space Analysis. Volume 1: Separation of Sets and Optimality Conditions. Springer, New York (2005)

    MATH  Google Scholar 

  33. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  34. Shapiro, A., Sun, J.: Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29, 479–491 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun, J., Zhang, L.W., Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 129, 437–456 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, D., Sun, J., Zhang, L.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349–391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhao, X.Y., Sun, D., Toh, K.-C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luo, H.Z., Wu, H.X., Chen, G.T.: On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. J. Glob. Optim. 54, 599–618 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wu, H., Luo, H., Ding, X., Chen, G.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefintie programming. Comput. Optim. Appl. 56, 531–558 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, H.X., Luo, H.Z., Yang, J.F.: Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming. J. Glob. Optim. 59, 695–727 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg State University, Saint Petersburg, Russia

    M. V. Dolgopolik

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Dolgopolik, M.V. A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness. J Optim Theory Appl 176, 745–762 (2018). https://doi.org/10.1007/s10957-018-1239-z

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