Abstract
We present an approximate bundle method for solving nonsmooth equilibrium problems. An inexact cutting-plane linearization of the objective function is established at each iteration, which is actually an approximation produced by an oracle that gives inaccurate values for the functions and subgradients. The errors in function and subgradient evaluations are bounded and they need not vanish in the limit. A descent criterion adapting the setting of inexact oracles is put forward to measure the current descent behavior. The sequence generated by the algorithm converges to the approximately critical points of the equilibrium problem under proper assumptions. As a special illustration, the proposed algorithm is utilized to solve generalized variational inequality problems. The numerical experiments show that the algorithm is effective in solving nonsmooth equilibrium problems.
Similar content being viewed by others
References
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Bigi, G., Pappalardo, M., Passacantando, M.: Optimization tools for solving equilibrium problems with nonsmooth data. J. Optim. Theory Appl. 170(3), 887–905 (2016)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1), 123–145 (1994)
Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62(1), 261–275 (1993)
Emiel, G., Sagastizábal, C.: Incremental-like bundle methods with applications to energy planning. Comput. Optim. Appl. 46(2), 305–332 (2010)
Han, D., Lo, H.K.: Two new self-adaptive projection methods for variational inequality problems. Comput. Math. Appl. 43(12), 1529–1537 (2002)
Hare, W., Sagastizábal, C., Solodov, M.: A proximal bundle method for nonsmooth nonconvex functions with inexact information. Comput. Optim. Appl. 63(1), 1–28 (2016)
Harker, P.T., Pang, J.S.: A damped-newton method for the linear complementarity problem. Lect. Appl. Math. 26(2), 265–284 (1990)
Hintermüller, M.: A proximal bundle method based on approximate subgradients. Comput. Optim. Appl. 20(3), 245–266 (2001)
Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. J. Optim. 52(3), 301–316 (2003)
Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. Theory Methods Appl. 52(2), 621–635 (2003)
Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. J. Optim. 59(8), 1259–1274 (2010)
Kiwiel, K.C.: An algorithm for nonsmooth convex minimization with errors. Math. Comput. 45(171), 173–180 (1985)
Kiwiel, K.C.: A proximal bundle method with approximate subgradient linearizations. SIAM J. Optim. 16(4), 1007–1023 (2006)
Kiwiel, K.C.: Bundle Methods for Convex Minimization with Partially Inexact Oracles. Technical Report, Systems Research Institute, Polish Academy of Sciences (April 2010)
Konnov, I.V.: The application of a linearization method to solving nonsmooth equilibrium problems. Russ. Math. 40(12), 54–62 (1996)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)
Li, X., Tomasgard, A., Barton, P.: Nonconvex generalized benders decomposition for stochastic separable mixed-integer nonlinear programs. J. Optim. Theory Appl. 151(3), 425–454 (2011)
Li, X.B., Li, S.J., Chen, C.R.: Lipschitz continuity of an approximate solution mapping to equilibrium problems. Taiwan. J. Math. 16(3), 1027–1040 (2012)
Lv, J., Pang, L.P., Wang, J.H.: Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization. Appl. Math. Comput. 265, 635–651 (2015)
Malitsky, Y.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)
Mordukhovich, B.S., Panicucci, B., Pappalardo, M., Passacantando, M.: Hybrid proximal methods for equilibrium problems. Optim. Lett. 6(7), 1535–1550 (2012)
Muu, L.D., Nguyen, V.H., Quy, N.V.: On Nash–Cournot oligopolistic market equilibrium models with concave cost function. J. Glob. Optim. 41(3), 351–364 (2008)
Nguyen, T.T., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving equilibrium problems. Math. Program. Ser. B 116(1), 529–552 (2009)
Oliveira, W., Sagastizabal, C., Scheimberg, S.: Inexact bundle methods for two-stage stochastic programming. SIAM. J. Optim. 21(2), 517–544 (2011)
Pang, L.P., Lv, J., Wang, J.H.: Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems. Comput. Optim. Appl. 64(2), 433–465 (2016)
Salmon, G., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14(3), 869–893 (2004)
Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30(1), 91–107 (2011)
Shen, J., Pang, L.P.: A bundle-type auxiliary problem method for solving generalized variational-like inequalities. Comput. Math. Appl. 55(12), 2993–2998 (2008)
Shen, J., Pang, L.P.: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 1, 1–18 (2012)
Solodov, M.V.: On approximations with finite precision in bundle methods for nonsmooth optimization. J. Optim. Theory Appl. 119(1), 151–165 (2003)
Sun, D.F.: A new step-size skill for solving a class of nonlinear projection equations. J. Comput. Math. 13(4), 357–368 (1995)
Wang, Y.J., Xiu, N.H., Wang, C.Y.: A new version of extragradient method for variational inequality problems. Comput. Math. Appl. 42, 969–979 (2001)
Xiao, B.C., Harker, P.T.: A nonsmooth Newton method for variational inequalities, II: numerical results. Math. Program. 65(1), 195–216 (1994)
Yang, Y., Pang, L.P., Ma, X.F., Shen, J.: Constrained nonsmooth nonsmooth optimization via proximal bundle method. J. Optim. Theory Appl. 163, 900–925 (2014)
Ye, M.L., He, Y.R.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60(1), 141–150 (2015)
Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM. J. Optim. 6(3), 714–726 (1996)
Acknowledgements
The authors thank two anonymous referees for a number of valuable and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research is partially supported by the Natural Science Foundation of China, Grants 11171049, 11301347 and 31271077.
Rights and permissions
About this article
Cite this article
Meng, FY., Pang, LP., Lv, J. et al. An approximate bundle method for solving nonsmooth equilibrium problems. J Glob Optim 68, 537–562 (2017). https://doi.org/10.1007/s10898-016-0490-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-016-0490-9