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Saddle Points Theory of Two Classes of Augmented Lagrangians and Its Applications to Generalized Semi-infinite Programming

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Abstract

In this paper, we develop the sufficient conditions for the existence of local and global saddle points of two classes of augmented Lagrangian functions for nonconvex optimization problem with both equality and inequality constraints, which improve the corresponding results in available papers. The main feature of our sufficient condition for the existence of global saddle points is that we do not need the uniqueness of the optimal solution. Furthermore, we show that the existence of global saddle points is a necessary and sufficient condition for the exact penalty representation in the framework of augmented Lagrangians. Based on these, we convert a class of generalized semi-infinite programming problems into standard semi-infinite programming problems via augmented Lagrangians. Some new first-order optimality conditions are also discussed.

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References

  1. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 2nd edn. Wiley, New York (1993)

    MATH  Google Scholar 

  2. Coope, I.D., Watson, M.A.: A projected Lagrangian algorithm for semi-infinite programming. Math. Program. 32, 337–356 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ehrgott, M., Wiecek, M.M.: Saddle points and Pareto points in multiple objective programming. J. Global Optim. 32, 11–33 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Global Optim. 24, 187–203 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gasimov, R.N., Rubinov, A.M.: On augmented Lagrangians for optimization problems with a single constraint. J. Global Optim. 28, 153–173 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goberna, M.A., López, M.A.: Semi-Infinite Programming Recent Advances. Kluwer Academic, Dordrecht (2001)

    MATH  Google Scholar 

  7. Griva, I., Polyak, R.: Primal-dual nonlinear rescaling method with dynamic scaling parameter update. Math. Program. Ser. A. 106, 237–259 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 524–532 (2003)

    Article  MathSciNet  Google Scholar 

  11. Huang, X.X., Yang, X.Q.: Further study on augmented Lagrangian duality theory. J. Global Optim. 31, 193–210 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, D.: Zero duality gap for a class of nonconvex optimization problems. J. Optim. Theory Appl. 85, 309–323 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Li, D.: Saddle-point generation in nonlinear nonconvex optimization. Nonlinear Anal. 30, 4339–4344 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Li, D., Sun, X.L.: Convexification and existence of saddle point in a pth-power reformulation for nonconvex constrained optimization. Nonlinear Anal. 47, 5611–5622 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, D., Sun, X.L.: Existence of a saddle point in nonconvex constrained optimization. J. Global Optim. 21, 39–50 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, New York (1997)

    MATH  Google Scholar 

  17. Polak, E., Royset, J.O.: Algorithms for finite and semi-infinite min-max problems using adaptive smoothing techniques. J. Optim. Theory Appl. 119, 421–457 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Polak, E., Royset, J.O.: On the use of augmented Lagrangians in the solution of generalized semi-infinite min-max problems. Comput. Optim. Appl. 31, 173–192 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Polak, E., Tits, A.L.: A recursive quadratic programming algorithm for semi-infinite optimization problems. Appl. Math. Optim. 8, 325–349 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. 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 

  22. Qi, L., Wu, S.Y., Zhou, G.: Semismooth Newton methods for solving semi-infinite programming problem. J. Global Optim. 27, 215–232 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Reemtsen, R., Rückamann, J.-J.: Semi-infinite Programming. Kluwer Academic, Boston (1998)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Rockafellar, R.T.: Lagrangian multipliers and optimality. SIAM Rev. 35, 83–238 (1993)

    Article  MathSciNet  Google Scholar 

  26. Rockafellar, R.T., Wets, J.-B.: Variational Analysis. Springer, Berlin (1998)

    MATH  Google Scholar 

  27. Royset, J.O., Polak, E., Kiureghian, A.D.: Adaptive approximations and exact penalization for the solution of generalized semi-infinite min-max problems. SIAM J. Optim. 14, 1–34 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rubinov, A.M., Huang, X.X., Yang, X.Q.: The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res. 27, 775–791 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rubinov, A.M., Yang, X.Q.: Lagrangian Type Functions in Constrained Non-convex Optimization. Kluwer, Boston (2003)

    Google Scholar 

  30. Stein, O.: First order optimality conditions for degenerate index sets in generalized semi-infinite programming. Math. Oper. Res. 26, 565–582 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Still, G.: Solving generalized semi-infinite programs by reduction to simpler problems. Optimization 53, 19–38 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear Lagrange duality theorems and penalty function methods in continuous optimization. J. Global Optim. 27, 473–484 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yang, X.Q., Hung, X.X.: A nonlinear Lagrangian approach to constrained optimization problems. SIAM J. Optim. 11, 1119–1149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Zhou, J.C., Wang, C.Y., Xiu, N.H.: First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Preprint, Department of Mathematics, Beijing Jiaotong University, P.R. China (2007)

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Correspondence to Changyu Wang.

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This research was supported by the National Natural Science Foundation of P.R. China (Grant No. 10571106 and No. 10701047).

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Wang, C., Zhou, J. & Xu, X. Saddle Points Theory of Two Classes of Augmented Lagrangians and Its Applications to Generalized Semi-infinite Programming. Appl Math Optim 59, 413–434 (2009). https://doi.org/10.1007/s00245-008-9060-y

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