Abstract
We present an SOR-type algorithm and a Jacobi-type algorithm that can effectively be applied to the ℓ 1 − ℓ 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.
Similar content being viewed by others
References
Auslender A.: Optimisation Méthodes Numériques. Masson, Paris (1976)
Center Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev. 23, 444–466 (1981)
Censor Y., Lent A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–353 (1981)
Cottle R.W., Pang J.-S., Stone R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)
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)
Hildreth C.: A quadratic programming procedure. Naval Res. Logist. Q. 4, 79–85 (1957)
Iusem A.N.: On the convergence of iterative methods for symmetric linear complementarity problems. Mathem. Progr. 59, 33–48 (1993)
Lent A., Censor Y.: Extensions of Hildreth’s row action method for quadratic programming. SIAM J. Control Optim. 18, 444–454 (1980)
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)
Mangasarian O.L.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465–485 (1977)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
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)
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)
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)
Tseng P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109, 475–494 (1991)
Tseng P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Progr. 125, 263–295 (2010)
Tseng P., Yun S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Progr. 117, 387–423 (2009)
Yun, S., Toh, K.-C.: A coordinate gradient descent method for ℓ 1-regularized convex minimization, Singapore-MIT Alliance, Singapore, (revised January 2009). Comput. Optim. Appl. (to appear)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by a Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.
Rights and permissions
About this article
Cite this article
Fukushima, M. SOR- and Jacobi-type iterative methods for solving ℓ 1 − ℓ 2 problems by way of Fenchel duality. Optim Lett 6, 679–686 (2012). https://doi.org/10.1007/s11590-011-0292-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0292-4