A Modified Fletcher-Reeves-Type Method for Nonsmooth Convex Minimization
AbstractConjugate gradient methods are efficient for smooth optimization problems, while there are rare conjugate gradient based methods for solving a possibly nondifferentiable convex minimization problem. In this paper by making full use of inherent properties of Moreau-Yosida regularization and descent property of modified conjugate gradient method we propose a modified Fletcher-Reeves-type method for nonsmooth convex minimization. It can be applied to solve large-scale nonsmooth convex minimization problem due to lower storage requirement. The algorithm is globally convergent under mild conditions.
A. Auslender, “Numerical methods for nondifferentiable convex optimization,” Mathematical Programming Study, vol. 30, pp. 102-126, 1987.
J. F. Bonnans, J. C. Gilbert, C. Lemarechal and C. Sagastizabal, “A family of variable-metric proximal methods,” Mathematical Programming, vol. 68, pp. 15-47, 1995.
J. V. Burke and M. Qian, “On the superlinear convergence of the variable- metric proximal point-algorithm using Broyden and BFGS matrix secant updating,” Mathematical Programming, vol. 88, pp. 157-181, 2000.
X. Chen, and M. Fukushima, “Proximal quasi-Newton methods for nondifferentiable convex optimization,” Mathematical Programming, vol. 85, pp. 313-334, 1999.
R. Correa and C. Lemarechal, “Convergence of some algorithms for convex minimization,” Mathematical Programming, vol. 62, pp. 261-275, 1993.
M. Fukushima, “A descent algorithm for nonsmooth convex optimization,” Mathematical Programming, vol. 30, pp. 163-175, 1984.
M. Fukushima and L. Qi, “A globally and superlinearly convergent algorithm for nonsmooth convex minimization,” SIAM Journal on Optimization, vol. 6, pp. 1106-1120, 1996.
W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pacific Journal of Optimization, vol. 2, pp. 35-58, 2006.
J.-B. Hiriart-Urruty and C. Lemarechal, “Convex Analysis and Minimization Algorithms,” Springer Verlag, Berlin, Germany, 1993.
C. Lemarechal and C. Sagastizabal, “An approach to variable-metric bundle methods,” Proceedings of the 16th IFIP-TC7 Conference on Systems Modelling and Optimization, edited by J. Henry and J. P. Yuvor, Springer Verlag, New York, NY, pp. 144-162, 1994.
C. Lemarechal and C. Sagastizabal, “Practical aspects of the Moreau-Yosida regularization, I:theoretical preliminaries,” SIAM Journal on Optimization, vol. 7, pp. 367-385, 1997.
S. Lu, Z. Wei and L. Li, “A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization,” Computational Optimization and Applications, vol. 51,no. 2, pp. 551-573, 2012.
R. Mifflin, “A quasi-second-order proximal bundle algorithm,” Mathematical Programming,vol. 73, pp. 51-72, 1996.
R. Mifflin, D. Sun, and L. Qi, “Quasi-Newton bundle-type methods for nondifferentiable convex optimization,” SIAM Journal on Optimization, vol. 8, pp. 583-603, 1998.
L. Qi and X. Chen, “A preconditioned proximal Newton method for nondifferentiable convex optimization,” Mathematical Programming, vol. 76, pp. 411-429, 1997.
A. I. Rauf and M. Fukushima, “Global convergent BFGS method for nonsmooth convex optimization,” Journal of Optimization Theory and Application, vol. 104, no. 3, pp. 539-558,2000.
N. Sagara and M. Fukushima, “A trust region method for nonsmooth convex optimization,” Journal of Industrial and Management Optimization, vol. 1, no. 2, pp. 171-180, 2005.
G. Yuan, Z. Wei, G. Li, “A modified Polak-Ribi` ere-Polyak conjugate gradient algorithm for nonsmooth convex programs,” Journal of Computational and Applied Mathematics, vol. 255,PP. 86-96, 2014.
L. Zhang, “A new trust region algorithm for nonsmooth convex minimization,” Applied Mathematics and Computation, vol. 193, pp. 135-142, 2007.
L. Zhang, W. Zhou and D. Li, “Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search”, Numerische Mathematik, vol. 104, pp. 561-572, 2006.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).