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Weak via strong Stackelberg problem: New results

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Abstract

We are concerned with weak Stackelberg problems such as those considered in [19], [23] and [25]. Based on a method due to Molodtsov, we present new results to approximate such problems by sequences of strong Stackelberg problems. Results related to convergence of marginal functions and approximate solutions are given. The case of data perturbations is also considered.

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References

  1. Addoune, S., Loridan, P. and Morgan, J. (1992), Sous équilibres de Stackelberg. Communication présentée aux Journées d'Optimisation, Perpignan.

  2. Aiyoshi, E. and Shimizu, K.A. (1982), Solution method for the static constrained Stackelberg problem via penalty method, IEEE Trans. Auto Control., AC-29: 1111–1114.

    Google Scholar 

  3. Attouch, H. (1984) Variational convergences for functions and operators. Pitman, Boston.

    Google Scholar 

  4. Bard, J. (1983), An efficient point algorithm for a linear two-stage optimization problem, Operations Research: 670–684.

  5. Basar, T. and Olsder, G.J. (1982), Dynamic noncooperative game theory, Academic Press, New York.

    Google Scholar 

  6. Bard, J. and Falk, J. (1982), An explicit solution to the multi-level programming, Compt. and Operation Research 9: 77–100.

    Google Scholar 

  7. Buttazzo, G. and dal Maso, G. (1982), Г-convergence and optimal control problems, Journal of Optimization Theory and Applications 38, 3: 385–407.

    Google Scholar 

  8. Chen, Y. and Florian, M. (1991), The nonlinear bilevel programming problem: formulations, regularity and optimality conditions. CRT-794, Centre de Recherche sur les Transports, Université de Montreal, Canada.

    Google Scholar 

  9. de Giorgi, E. and Franzoni, T. (1975), Su un tipo di convergenza variazionale, Atti Accademia Nazionale dei Lincei, Classe di Scienze Fisiche Matematiche e Naturali 58: 842–850.

    Google Scholar 

  10. Dempe, S. (1987), A simple algorithm for the linear bilevel programming problem, Optimization 18: 373–385.

    Google Scholar 

  11. Dempe, S. (1992), A necessary and sufficient optimality condition for Bilevel Programming Problems, Optimization 25: 341–354.

    Google Scholar 

  12. Dempe, S. (1993), On an algorithm solving two-level programming problems with nonunique lower level solutions. Preprint.

  13. Falk, J. and Liu, J. (1993), On bilevel programming. Part I: General non linear cases. Preprint.

  14. Fedorov, V. (1978), A method solving linear hierarchical games, USSR Comput. Maths. Math. Phys. 17: 97–103.

    Google Scholar 

  15. Fedorov, V. and Molodtsov, D.A. (1973), Approximation of two-person games with information exchange, USSR Comput. Maths. Math. Phys. 13: 123–142.

    Google Scholar 

  16. Fiacco, A.V. and Hutzler, W.P. (1982), Basic results in the development of sensitivity analysis in nonlinear programming, Computers and Operations Research 9: 9–28.

    Google Scholar 

  17. Ishikuza, Y. and Aiyoshi, E. (1992), Double penalty method for bilevel optimization problem, Annals of Operations Research 34: 73–88.

    Google Scholar 

  18. Laurent, J.L. (1972), Approximation et optimisation. Hermann, Paris.

    Google Scholar 

  19. Leitmann, G. (1978), On generalized Stackelberg strategies, Journal of Optimization Theory and Applications 59: 637–643.

    Google Scholar 

  20. Lignola, M.B. and Morgan, J. (1992), Semicontinuities of marginal functions in a sequential setting, Optimization 24: 241–252.

    Google Scholar 

  21. Lignola, M.B. and Morgan, J. (1992), Convergence of marginal functions with dependent constraints, Optimization 23: 189–213.

    Google Scholar 

  22. Lignola, M.B. and Morgan, J. (1992), Existence and approximation for Min Sup problems. Proceedings of the International Conference on Operations Research 90 in Vienna, Edited by W. Buhler, G. Feichtinger, F. Hartl, F.J. Radermacher and P. Stahly. Springer Verlag, Berlin, Germany: 157–164.

    Google Scholar 

  23. Lignola, M.B. and Morgan, J. (1995), Topological existence and stability for Stackelberg problems, Journal of Optimization Theory and Applications 84 n.1: 145–169.

    Google Scholar 

  24. Lignola, M.B. and Morgan, J. (1993), Regularized bilevel programming problem. Preprint n.22/1993, Dip. di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”.

  25. Loridan, P. and Morgan, J. (1988), Approximate solutions for two-level optimization problems, in K. Hoffmann, J. Hiriart-Urruty, C. Lemarechal, J. Zowe (eds.): Trends in Mathematical Optimization, International Series of Numerical Mathematics, 84, Birkhauser Verlag: 181–196.

  26. Loridan, P. and Morgan, J. (1989), New results on approximate solutions in two-level optimization, Optimization 20: 819–836.

    Google Scholar 

  27. Loridan, P. and Morgan, J. (1990), Quasi convex lower level problems and applications in two-level optimization. Lecture Notes in Economic and Mathematical Systems, Springer Verlag, n. 345: 325–341.

  28. Loridan, P. and Morgan, J. (1992), On strict ε-solutions for a two-level optimization problem. Proceedings of the International Conference on Operations Research 90 in Vienna, Edited by W. Buhler, G.F. Feichtinger, F. Hartl, F.J. Radermacher and P. Stahly, Springer Verlag, Berlin, Germany: 165–172.

    Google Scholar 

  29. Loridan, P. and Morgan, J. (1992), Weak two-level optimization problems and Molodtsov regularization. Working paper.

  30. Lucchetti, R., Mignanego, F., Pieri (1987), Existence theorems of equilibrium points in Stackelberg games with constraints, Optimization 18: 857–866.

    Google Scholar 

  31. Mallozzi, L. and Morgan, J. (1993), ε-mixed strategies for static continuous Stackelberg problem, Journal of Optimization Theory and Applications 78, n.2.

    Google Scholar 

  32. Mangasarian, O.L. (1969), Non linear programming. McGraw-Hill, New York.

    Google Scholar 

  33. Molodtsov, D.A. (1976), The solution of a class of non antagonistic games; USSR Comput. Maths. Math. Phys. 16: 1451–1456.

    Google Scholar 

  34. Morgan, J. (1989), Constrained well-posed two-level optimization problems, in Non-smooth Optimization and Related Topics, Edited by F.H. Clarke, V.F. Dem'yanov and F. Giannessi, Plenum Press, New York.

    Google Scholar 

  35. Outrata, J.V. (1990), On the numerical solution of a class of Stackelberg problems, Zeit. Oper. Res. 34: 255–277.

    Google Scholar 

  36. Outrata, J.V. (1993), Necessary optimality conditions for Stackelberg problems, Journal of Optimization Theory and Appl. 76: 305–320.

    Google Scholar 

  37. Szymanowski, J. and Ruszczynski, A. (1979), Convergence analysis for two-level algorithms of mathematical programming, Mathematical Programming Study 10: 158–171.

    Google Scholar 

  38. Tuy, H., Migdalas, A. and Varbrand, P. (1993), A global optimization approach for the linear two-level program, Journal of Global Optimization 3: 1–23.

    Google Scholar 

  39. Ye, J.J. (1993), Necessary conditions for bilevel dynamic optimization problems. Preprint.

  40. Ye, J.J. and Zhu, D.L. (1992), Optimality conditions for bilevel programming problems. Preprint.

  41. Zangwill, W.I. (1967), Non linar programming via penalty functions, Management Science 13: 344–358.

    Google Scholar 

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Authors and Affiliations

  1. CERMSEM Université de Paris-Panthéon Sorbonne, 90 Rue de Tolbiac, 75634, Paris Cedex 13

    Pierre Loridan

  2. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Compl. univ. Monte S. Angelo, Via Cintia, 80126, Napoli

    Jacqueline Morgan

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  1. Pierre Loridan
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  2. Jacqueline Morgan
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Loridan, P., Morgan, J. Weak via strong Stackelberg problem: New results. J Glob Optim 8, 263–287 (1996). https://doi.org/10.1007/BF00121269

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