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Decomposition Descent Method for Limit Optimization Problems

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10556)


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.


  • Optimization problems
  • Limit problems
  • Non-smooth functions
  • Mixed variational inequality
  • Decomposition descent method
  • Coercivity conditions

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  • DOI: 10.1007/978-3-319-69404-7_12
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  1. Burges, C.J.C.: A tutorial on support vector machines for pattern recognition. Data Mining Know. Disc. 2, 121–167 (1998)

    CrossRef  Google Scholar 

  2. Cevher, V., Becker, S., Schmidt, M.: Convex optimization for big data. Signal Process. Magaz. 31, 32–43 (2014)

    CrossRef  Google Scholar 

  3. Facchinei, F., Scutari, G., Sagratella, S.: Parallel selective algorithms for nonconvex big data optimization. IEEE Trans. Sig. Process. 63, 1874–1889 (2015)

    CrossRef  MathSciNet  Google Scholar 

  4. Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Progr. 117, 387–423 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. Math. Program. 156, 433–484 (2016)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Konnov, I.V.: Sequential threshold control in descent splitting methods for decomposable optimization problems. Optim. Meth. Softw. 30, 1238–1254 (2015)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Alart, P., Lemaire, B.: Penalization in non-classical convex programming via variational convergence. Math. Program. 51, 307–331 (1991)

    CrossRef  MATH  Google Scholar 

  8. Cominetti, R.: Coupling the proximal point algorithm with approximation methods. J. Optim. Theor. Appl. 95, 581–600 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. 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)

    CrossRef  MATH  Google Scholar 

  10. Konnov, I.V.: An inexact penalty method for non stationary generalized variational inequalities. Set-Valued Variat. Anal. 23, 239–248 (2015)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  12. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Czarnecki, M.-O., Rifford, L.: Approximation and regularization of lipschitz functions: convergence of the gradients. Trans. Amer. Math. Soc. 358, 4467–4520 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. 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)

    Google Scholar 

  15. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. The Math. Stud. 63, 127–149 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)

    CrossRef  MATH  Google Scholar 

  17. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. Ser. B. 58, 267–288 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain non-convex minimization problems. Int. J. Syst. Sci. 12, 989–1000 (1981)

    CrossRef  MATH  Google Scholar 

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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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Correspondence to Igor Konnov .

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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.

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