Abstract
We consider a general limit optimization problem whose goal function need not be smooth in general and only approximation sequences are known instead of exact values of this function. We suggest to apply a two-level approach where approximate solutions of a sequence of mixed variational inequality problems are inserted in the iterative scheme of a selective decomposition descent method. Its convergence is attained under coercivity type conditions.
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References
Burges, C.J.C.: A tutorial on support vector machines for pattern recognition. Data Mining Know. Disc. 2, 121–167 (1998)
Cevher, V., Becker, S., Schmidt, M.: Convex optimization for big data. Signal Process. Magaz. 31, 32–43 (2014)
Facchinei, F., Scutari, G., Sagratella, S.: Parallel selective algorithms for nonconvex big data optimization. IEEE Trans. Sig. Process. 63, 1874–1889 (2015)
Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Progr. 117, 387–423 (2010)
Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. Math. Program. 156, 433–484 (2016)
Konnov, I.V.: Sequential threshold control in descent splitting methods for decomposable optimization problems. Optim. Meth. Softw. 30, 1238–1254 (2015)
Alart, P., Lemaire, B.: Penalization in non-classical convex programming via variational convergence. Math. Program. 51, 307–331 (1991)
Cominetti, R.: Coupling the proximal point algorithm with approximation methods. J. Optim. Theor. Appl. 95, 581–600 (1997)
Salmon, G., Nguyen, V.H., Strodiot, J.J.: Coupling the auxiliary problem principle and epiconvergence theory for solving general variational inequalities. J. Optim. Theor. Appl. 104, 629–657 (2000)
Konnov, I.V.: An inexact penalty method for non stationary generalized variational inequalities. Set-Valued Variat. Anal. 23, 239–248 (2015)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Ermoliev, Y.M., Norkin, V.I., Wets, R.J.B.: The minimization of semicontinuous functions: mollifier subgradient. SIAM J. Contr. Optim. 33, 149–167 (1995)
Czarnecki, M.-O., Rifford, L.: Approximation and regularization of lipschitz functions: convergence of the gradients. Trans. Amer. Math. Soc. 358, 4467–4520 (2006)
Gwinner, J.: On the penalty method for constrained variational inequalities. In: Hiriart-Urruty, J.-B., Oettli, W., Stoer, J. (eds.) Optimization: Theory and Algorithms, pp. 197–211. Marcel Dekker, New York (1981)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. The Math. Stud. 63, 127–149 (1994)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. Ser. B. 58, 267–288 (1996)
Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain non-convex minimization problems. Int. J. Syst. Sci. 12, 989–1000 (1981)
Acknowledgement
The results of this work were obtained within the state assignment of the Ministry of Science and Education of Russia, project No. 1.460.2016/1.4. In this work, the author was also supported by Russian Foundation for Basic Research, project No. 16-01-00109 and by grant No. 297689 from Academy of Finland.
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Konnov, I. (2017). Decomposition Descent Method for Limit Optimization Problems. In: Battiti, R., Kvasov, D., Sergeyev, Y. (eds) Learning and Intelligent Optimization. LION 2017. Lecture Notes in Computer Science(), vol 10556. Springer, Cham. https://doi.org/10.1007/978-3-319-69404-7_12
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DOI: https://doi.org/10.1007/978-3-319-69404-7_12
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