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An Augmented Lagrangian method for quasi-equilibrium problems

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Abstract

In this paper, we propose an Augmented Lagrangian algorithm for solving a general class of possible non-convex problems called quasi-equilibrium problems (QEPs). We define an Augmented Lagrangian bifunction associated with QEPs, introduce a secondary QEP as a measure of infeasibility and we discuss several special classes of QEPs within our theoretical framework. For obtaining global convergence under a new weak constraint qualification, we extend the notion of an Approximate Karush–Kuhn–Tucker (AKKT) point for QEPs (AKKT-QEP), showing that in general it is not necessarily satisfied at a solution, differently from its counterpart in optimization. We study some particular cases where AKKT-QEP does hold at a solution, while discussing the solvability of the subproblems of the algorithm. We also present illustrative numerical experiments.

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References

  1. Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60(5), 627–641 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135, 255–273 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22, 1109–1135 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125(2), 473–483 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: Strict constraint qualifications and sequential optimality conditions for constrained optimization. Math. Oper. Res. 43(3), 693–717 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26(1), 96–110 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Anh, P.K., Hai, T.N.: Novel self-adaptive algorithm for non-Lipschitz equilibrium problems with applications. J. Glob. Optim. (2018). https://doi.org/10.1007/s10898-018-0722-2

    Article  Google Scholar 

  8. Aussel, D., Cotrina, J., Iusem, A.: An existence result for quasi-equilibrium problems. J. Convex Anal. 24, 55–66 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  10. Bigi, G., Passacantando, M.: Gap functions for quasi-equilibria. J. Glob. Optim. 66(4), 791–810 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Bigi, G., Passacantando, M.: Twelve monotonicity conditions arising from algorithms for equilibrium problems. Optim. Methods Soft. 30(2), 323–337 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM Publications, Philadelphia (2014)

    MATH  Google Scholar 

  13. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1993)

    MathSciNet  Google Scholar 

  14. Bueno, L.F., Haeser, G., Rojas, F.N.: Optimality conditions and constraint qualifications for generalized Nash equilibrium problems and their practical implications. SIAM J. Optim. 29(1), 31–54 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Castellani, M., Giuli, M.: An existence result for quasiequilibrium problems in separable Banach spaces. J. Math. Anal. Appl. 425, 85–95 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    MathSciNet  MATH  Google Scholar 

  17. Chen, H., Wang, Y., Xu, Y.: An alternative extragradient projection method for quasi-equilibrium problems. J. Inequal. Appl. 1, 26 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM Publications, Philadelphia (1990)

    MATH  Google Scholar 

  19. Cojocaru, M., Migot, T.: A decomposition method for the convex generalized Nash equilibrium problem, submitted, (2019)

  20. Cotrina, J., Garcia, Y.: Equilibrium problems: existence results and applications. Set-Valued Var. Anal. 26, 159–177 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Cubiotti, P.: Existence of solutions for lower semicontinuous quasi-equilibrium problems. Comput. Math. Appl. 30, 11–22 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Facchinei, F., Kanzow, C.: Generalized nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems (with complete test problems). SIAM J. Optim. 20(5), 2228–2253 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Facchinei, F., Kanzow, C., Sagratella, S.: Solving quasi-variational inequalities via their KKT conditions. Math. Program. 50(1), 369–412 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Fischer, A., Herrich, M., Schönefeld, K.: Generalized nash equilibrium problems—recent advances and challenges. Pesq. Oper. 34, 521–558 (2014)

    Google Scholar 

  26. Goeleven, D.: Complementarity and Variational Inequalities in Electronics. Academic Press, London (2017)

    MATH  Google Scholar 

  27. Gowda, M.: Pseudomonotone and copositive star matrices. Linear Algebra Appl. 113, 107–118 (1989)

    MathSciNet  MATH  Google Scholar 

  28. Hadjisavvas, N., Komlosi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Boston (2005)

    MATH  Google Scholar 

  29. Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)

    MATH  Google Scholar 

  30. Iusem, A., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116, 259–273 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Iusem, A., Lara, F.: Optimality conditions for vector equilibrium problems with applications. J. Optim. Theory Appl. 180, 187–206 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Iusem, A., Lara, F.: Existence results for noncoercive mixed variational inequalities in finite dimensional spaces. J. Optim. Theory Appl. 183, 122–138 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Iusem, A., Nasri, M.: Augmented Lagrangian methods for variational inequality problems. RAIRO Oper. Res. 44, 5–24 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Jiang, H.Y., Qi, L.: Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities. J. Math. Anal. Appl. 196, 314–331 (1995)

    MathSciNet  MATH  Google Scholar 

  35. Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)

    MathSciNet  MATH  Google Scholar 

  36. Kanzow, C.: On the multiplier-penalty-approach for quasi-variational inequalities. Math. Program. 160(1), 353–377 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Kanzow, C., Steck, D.: Augmented lagrangian methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 26(4), 2034–2058 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Kanzow, C., Steck, D.: Augmented Lagrangian and exact penalty methods for quasi-variational inequalities. Comput. Optim. Appl. 69(3), 801–824 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Kravchuk, A.S., Neittaanmäki, P.J.: Variational and quasi-variational inequalities in mechanics. In: Solid Mechanics and Its Applications, vol. 147. Springer, Dordrecht, (2007)

  40. Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl. 118(1), 117–133 (2003)

    MathSciNet  MATH  Google Scholar 

  41. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005)

    MathSciNet  MATH  Google Scholar 

  42. Qi, L., Wei, Z.: On the constant positive linear dependence conditions and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    MathSciNet  MATH  Google Scholar 

  43. Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)

    MathSciNet  MATH  Google Scholar 

  44. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56(2), 373–397 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Van, N.T.T., Strodiot, J.J., Nguyen, V.H., Vuong, P.T.: An extragradient-type method for solving nonmonotone quasi-equilibrium problems. Optimization 67(5), 651–664 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to express gratitude to the anonymous referees whose valuable suggestions improved this paper. This work was supported by FAPESP (2013/05475-7, 2018/24293-0 and 2017/18308-2), CAPES and CNPq. For F. Lara, this research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320.

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Authors and Affiliations

  1. Institute of Science and Technology, Federal University of São Paulo, São José dos Campos, SP, Brazil

    L. F. Bueno

  2. Department of Applied Mathematics, University of São Paulo, São Paulo, SP, Brazil

    G. Haeser

  3. Departamento de Matemática, Universidad de Tarapacá, Arica, Chile

    F. Lara

  4. Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil

    F. N. Rojas

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  1. L. F. Bueno
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  2. G. Haeser
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  3. F. Lara
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  4. F. N. Rojas
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Correspondence to F. Lara.

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Bueno, L.F., Haeser, G., Lara, F. et al. An Augmented Lagrangian method for quasi-equilibrium problems. Comput Optim Appl 76, 737–766 (2020). https://doi.org/10.1007/s10589-020-00180-4

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