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Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions

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Abstract

In this paper, we will study optimality conditions of semi-infinite programs and generalized semi-infinite programs by employing lower order exact penalty functions and the condition that the generalized second-order directional derivative of the constraint function at the candidate point along any feasible direction for the linearized constraint set is non-positive. We consider three types of penalty functions for semi-infinite program and investigate the relationship among the exactness of these penalty functions. We employ lower order integral exact penalty functions and the second-order generalized derivative of the constraint function to establish optimality conditions for semi-infinite programs. We adopt the exact penalty function technique in terms of a classical augmented Lagrangian function for the lower-level problems of generalized semi-infinite programs to transform them into standard semi-infinite programs and then apply our results for semi-infinite programs to derive the optimality condition for generalized semi-infinite programs. We will give various examples to illustrate our results and assumptions.

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References

  1. Krabs, W.: Optimization and Approximation. Translated from the German by Philip M. Anselone and Ronald B. Guenther. A Wiley-Interscience Publication, Chichester (1979)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Polak, E.: Optimization: algorithms and consistent approximations. Appl. Math. Sci. 124. Springer-Verlag, New York (1997)

  4. Reemtsen, R., Rückmann, J.-J. (eds.): Semi-Infinite Programming. Nonconvex Optim. Appl. 25. Kluwer Academic Publishers, Boston, MA (1998)

  5. Goberna, M.Á., López, M.A. (eds.): Semi-Infinite Programming: Recent Advances. Nonconvex Optim. Appl. 57. Kluwer Academic Publishers, Dordrecht (2001)

  6. Jongen, H.T., Rückmann, J.-J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Stein, O.: Bi-level Strategies for Semi-Infinite Programming. Kluwer Academic Publishers, Boston (2003)

    Book  MATH  Google Scholar 

  8. Still, G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, 301–313 (1999)

    Article  MATH  Google Scholar 

  9. John, F.: Extremum problems with inequalities as subsidiary conditions. In: Friederichs, K.O., Neugebauer, O.E., Stocker, J.J. (eds.) Studies and Essays, Courant Anniversary Volume, pp. 187–204. Wiley, New York (1948)

    Google Scholar 

  10. Pschenichnyi, B.N.: Necessary Conditions for an Extremum. Marcel Dekker, New York (1971)

    Google Scholar 

  11. Hettich, R., Jongen, H.T.: Semi-infinite programming: conditions of optimality and applications. In: Optimization techniques (Proceedings of the 8th IFIP Conference, Würzburg, 1977), Part 2, Lecture Notes in Control Information. Sci. 7, pp. 1–11. Springer, Berlin (1978)

  12. Borwein, J.M.: Direct theorems in semi-infinite convex programming. Math. Program. 21, 301–318 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, W., Nahak, C., Singer, I.: Constraint qualifications for semi-infinite systems of convex inequalities. SIAM J. Optim. 11, 31–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. López, M.A., Vercher, E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27, 307–319 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stein, O.: On constraint qualifications in nonsmooth optimization. J. Optim. Theory Appl. 121, 647–671 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zheng, X.Y., Yang, X.Q.: Lagrange multipliers in nonsmooth semi-infinite optimization problems. Math. Oper. Res. 32, 168–181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Weber, G.-W.: Generalized semi-infinite optimization: on some foundations. Vychisl. Tekhnol. 4, 41–61 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Rückmann, J.J., Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101, 677–691 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Ye, J.J., Wu, S.Y.: First order optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 137, 419–434 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)

    Google Scholar 

  22. Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, X.Q., Meng, Z.Q.: Lagrange multipliers and calmness conditions of order \(p\). Math. Oper. Res. 32, 95–101 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3208–3231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rubinov, A., Yang, X.Q.: Lagrange-type Functions in Constrained Nonconvex Optimization. Kluwer Academic Publishers, Dordrecht (2003)

    Book  MATH  Google Scholar 

  26. Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  29. Pietrzykowski, T.: A generalization of the potential method for conditional maxima on the Banach, reflexive spaces. Numer. Math. 18, 367–372 (1971/72)

  30. Conn, A.R., Gould, N.I.M.: An exact penalty function for semi-infinite programming. Math. Program. 37, 19–40 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)

    MATH  Google Scholar 

  32. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  33. Levitin, E.S.: Reduction of generalized semi-infinite programing problems to semi-infinite or piece-wise smooth programming problems. Preprint, 8-2001, University of Trier (2001)

  34. 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  MathSciNet  MATH  Google Scholar 

  35. Royset, J.O., Polak, E., Der Kiureghian, A.: Adaptive approximations and exact penalization for the solution of generalized semi-infinite min-max problems. SIAM J. Optim. 14, 1–33 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lang, S.: Real and Functional Analysis Volume 142 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (1993)

    Google Scholar 

  37. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)

    Book  MATH  Google Scholar 

  38. Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28, 789–809 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hiriart-Urruty, J.-B., Strodiot, J.-J., Nguyen, V.H.: Generalized hessian matrix and second-order optimality conditions for problems with \({C}^{1,1}\) data. Appl. Math. Optim. 11, 43–56 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, X.Q.: Generalized second-order derivatives and optimality conditions. Nonlinear Anal. 23, 767–784 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Burachik, R.S., Rubinov, A.: Abstract convexity and augmented Lagrangians. SIAM J. Optim. 18, 413–436 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nedic, A., Ozdaglar, A.: Separation of nonconvex sets with general augmenting functions. Math. Oper. Res. 33, 587–605 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are grateful to the referees for providing very detailed, constructive and helpful comments and suggestions, from which the presentation of the paper has been significantly improved. The research of the first author was supported by the Research Grants Council of Hong Kong with grant numbers: PolyU 5334/08E and PolyU 5292/13E, and National Science Foundation of China (11431004). The research of the third author was partly supported by National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2012AM016).

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

    Xiaoqi Yang

  2. College of Mathematics and Statistics, South-West Jiao Tong University, Chengdu, China

    Zhangyou Chen

  3. Department of Mathematics, School of Science, Shandong University of Technology, Zibo, China

    Jinchuan Zhou

Authors
  1. Xiaoqi Yang
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  2. Zhangyou Chen
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  3. Jinchuan Zhou
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Corresponding author

Correspondence to Xiaoqi Yang.

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This paper is dedicated to Professor Elijah (Lucien) Polak for his 85th birthday.

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Yang, X., Chen, Z. & Zhou, J. Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions. J Optim Theory Appl 169, 984–1012 (2016). https://doi.org/10.1007/s10957-016-0914-1

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  • DOI: https://doi.org/10.1007/s10957-016-0914-1

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