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SOR- and Jacobi-type iterative methods for solving 1 2 problems by way of Fenchel duality

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Abstract

We present an SOR-type algorithm and a Jacobi-type algorithm that can effectively be applied to the 1 2 problem by exploiting its special structure. The algorithms are globally convergent and can be implemented in a particularly simple manner. Relations with coordinate minimization methods are discussed.

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References

  1. Auslender A.: Optimisation Méthodes Numériques. Masson, Paris (1976)

    MATH  Google Scholar 

  2. Center Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23, 444–466 (1981)

    Article  MathSciNet  Google Scholar 

  3. Censor Y., Lent A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cottle R.W., Pang J.-S., Stone R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  5. Herman G.T., Lent A.: A family of iterative quadratic optimization algorithms for pairs of inequalities. with application in diagnostic radiology. Math. Progr. Study 9, 15–29 (1979)

    Article  MathSciNet  Google Scholar 

  6. Hildreth C.: A quadratic programming procedure. Naval Res. Logist. Q. 4, 79–85 (1957)

    Article  MathSciNet  Google Scholar 

  7. Iusem A.N.: On the convergence of iterative methods for symmetric linear complementarity problems. Mathem. Progr. 59, 33–48 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lent A., Censor Y.: Extensions of Hildreth’s row action method for quadratic programming. SIAM J. Control Optim. 18, 444–454 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Luo Z.-Q., Tseng P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2, 43–54 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mangasarian O.L.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465–485 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  12. Shimazu Y., Fukushima M., Ibaraki T.: A successive over-relaxation method for quadratic programming problems with interval constraints. J. Oper. Res. Soc. Jpn. 36, 73–89 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Sugimoto T., Fukushima M., Ibaraki T.: A parallel relaxation method for quadratic programming problems with interval constraints. J. Comput. Appl. Math. 60, 219–236 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tropp, J.A., Wright, S.J.: Computational methods for sparse solution of linear inverse problems. In: Proceedings of the IEEE, Special Issue on Applications of Sparse Representation and Compressive Sensing, vol. 98, pp. 948–958 (2010)

  15. Tseng P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109, 475–494 (1991)

    Article  MathSciNet  Google Scholar 

  16. Tseng P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Progr. 125, 263–295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tseng P., Yun S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Progr. 117, 387–423 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yun, S., Toh, K.-C.: A coordinate gradient descent method for 1-regularized convex minimization, Singapore-MIT Alliance, Singapore, (revised January 2009). Comput. Optim. Appl. (to appear)

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Correspondence to Masao Fukushima.

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This work was supported in part by a Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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Fukushima, M. SOR- and Jacobi-type iterative methods for solving 1 2 problems by way of Fenchel duality. Optim Lett 6, 679–686 (2012). https://doi.org/10.1007/s11590-011-0292-4

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  • DOI: https://doi.org/10.1007/s11590-011-0292-4

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