Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Calcolo
Published in: Calcolo 1/2019

01-03-2019

A new class of conjugate gradient methods for unconstrained smooth optimization and absolute value equations

Authors: Farzad Rahpeymaii, Keyvan Amini, Tofigh Allahviranloo, Mohsen Rostamy Malkhalifeh

Published in: Calcolo | Issue 1/2019

Login to get access

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we introduce a new three-term conjugate gradient (NTTCG) method to solve unconstrained smooth optimization problems. NTTCG is based on conjugate gradient methods proposed by Dai and Yuan (SIAM J Optim 10:177–182, 1999) and Polak and Ribière (Rev Francaise Inform Rech Oper 3(16):35–43, 1969). The descent property of the direction generated by NTTCG in each iteration is established. Under some standard assumptions, the global convergence results of the new methods are investigated. The extension of this algorithm is proposed to solve absolute value equations (AVE), called three-term conjugate subgradient (NTTCS) method. Numerical experiments are reported for unconstrained CUTEst problems and AVE.
previous article A new generalized shrinkage conjugate gradient method for sparse recovery
next article A new splitting method for monotone inclusions of three operators
Literature
1.
Al-Bayati, A.Y., Sharif, W.H.: A new three-term conjugate gradient method for unconstrained optimization. Can. J. Sci. Eng. Math. 1(5), 108–124 (2010)
2.
Andrei, N.: A modified Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization. Optimization 60, 1457–1471 (2011)MathSciNetCrossRef
3.
Andrei, N.: A simple three-term conjugate gradient algorithm for unconstrained optimization. J. Comput. Appl. Math. 241, 19–29 (2013)MathSciNetCrossRef
4.
Andrei, N.: On three-term conjugate gradient algorithms for unconstrained optimization. Appl. Math. Comput. 219, 6316–6327 (2013)MathSciNetMATH
5.
Aslam Noor, M., Iqbal, J., Inayat Noor, Kh, Al-Said, E.: On an iterative method for solving absolute value equations. Optim. Lett. 6, 1027–1033 (2012)MathSciNetCrossRef
6.
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetCrossRef
7.
Bioucas-Dias, J., Figueiredo, M.: A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE. Trans. Image Process. 16, 2992–3004 (2007)MathSciNetCrossRef
8.
Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48(1), 45–58 (2011)MathSciNetCrossRef
9.
Conn, A.R., Gould, N.I.M., Toint, P.L.: Convergence of quasi-Newton matrices generated by the symmetric rank one update. Math. Program. 50, 177–195 (1991)MathSciNetCrossRef
10.
Dai, Y.H., Yuan, Y.: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10, 177–182 (1999)MathSciNetCrossRef
11.
Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F., Yin, H.X., Yuan, Y.: Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10, 345–358 (1999)MathSciNetCrossRef
12.
Deng, S., Wan, Z.: A three-term conjugate gradient algorithm for large-scale unconstrained optimization problems. Appl. Numer. Math. 92, 70–81 (2015)MathSciNetCrossRef
13.
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRef
14.
Dong, X.L., Li, H.W., He, Y.B.: New version of the three-term conjugate gradient method based on spectral scaling conjugacy condition that generates descent search direction. Appl. Math. Comput. 269, 606–617 (2015)MathSciNet
15.
Esmaeili, H., Kimiaei, M.: An improved adaptive trust-region method for unconstrained optimization. Math. Model. Anal. 19(4), 469–490 (2014)MathSciNetCrossRef
16.
17.
Griewank, A.: The global convergence of Broyden-like methods with a suitable line search. J. Austr. Methods Soc. Ser. B 28, 75–92 (1986)MathSciNetCrossRef
18.
Gould, N.I.M., Orban, D., Toint, PhL: CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Optim. Appl. 60(3), 545–557 (2015)MathSciNetCrossRef
19.
20.
Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)MathSciNetCrossRef
21.
Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)MathSciNetMATH
22.
Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation applied to compressed sensing: implementation and numerical experiment. J. Comput. Math. 28(2), 170–194 (2010)MathSciNetCrossRef
23.
Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952)MathSciNetCrossRef
24.
Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)MathSciNetCrossRef
25.
Kimiaei, M., Ghaderi, S.: A new restarting adaptive trust-region method for unconstrained optimization. J. Oper. Res. Soc. China 5(4), 487–507 (2017)MathSciNetCrossRef
26.
Lu, Z., Chen, X.: Generalized conjugate gradient methods for \(\ell _1\) regularized convex quadratic programming with finite convergence. Math. Oper. Res. 43(1), 275–303 (2017)CrossRef
27.
28.
29.
Moosaei, H., Ketabchi, S., Jafari, H.: Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afr. Mat. 26(7–8), 1221–1228 (2015)MathSciNetCrossRef
30.
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, NewYork (2006)MATH
31.
Moosaei, H., Ketabchi, S., Noor, A., Iqbal, J., Hooshyarbakhsh, V.: Some techniques for solving absolute value equations. Appl. Math. Comput. 268, 696–705 (2015)MathSciNet
32.
More, J.J., Thuente, D.J.: Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20(3), 286–307 (1994)MathSciNetCrossRef
33.
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATH
34.
Polak, E., Ribière, G.: Note sur la convergence de directions conjugées. Rev. Francaise Inform. Rech. Oper. 3(16), 35–43 (1969)MATH
35.
36.
Rockafellar, R.T.: New applications of duality in convex programming. In: Proceedings Fourth Conference on Probability, Brasov, Romania (1971)
37.
Rohn, J.: A theorem of the alternatives for the equation \(Ax+B|x|=b\). Linear Multilinear Algebra 52, 421–426 (2004)MathSciNetCrossRef
38.
Sorber, L., Barel, M.V., Lathauwer, L.D.: Unconstrained optimization of real functions in complex variables. SIAM J. Optim. 22(3), 879–898 (2012)MathSciNetCrossRef
39.
Wright, S.J., Nowak, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE. Trans. Signal Process. 57(7), 2479–2493 (2009)MathSciNetCrossRef
40.
Yuan, G., Wei, Z., Li, G.: A modified Polak–Ribière–Polyak conjugate gradient algorithm for nonsmooth convex programs. J. Comput. Appl. Math. 255, 86–96 (2014)MathSciNetCrossRef
41.
42.
Zhang, L., Zhou, W., Li, D.H.: A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)MathSciNetCrossRef
43.
Zhang, L., Zhou, W., Li, D.H.: Some descent three-term conjugate gradient methods and their global convergence. Optim. Methods Softw. 22, 697–711 (2007)MathSciNetCrossRef
44.
45.
Zhang, C., Wei, Q.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143(2), 391–403 (2009)MathSciNetCrossRef
46.
Zhou, Q., Zhou, F., Cao, F.: A nonmonotone trust region method based on simple conic models for unconstrained optimization. Appl. Math. Comput. 225, 295–305 (2013)MathSciNetMATH
47.
Ujevic, N.: A new iterative method for solving linear systems. Appl. Math. Comput. 179, 725–730 (2006)MathSciNetMATH
Metadata
Title
A new class of conjugate gradient methods for unconstrained smooth optimization and absolute value equations
Authors
Farzad Rahpeymaii
Keyvan Amini
Tofigh Allahviranloo
Mohsen Rostamy Malkhalifeh
Publication date
01-03-2019
Publisher
Springer International Publishing
Published in
Calcolo / Issue 1/2019
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-018-0298-8

Other articles of this Issue 1/2019

Calcolo 1/2019 Go to the issue

OriginalPaper

A new splitting method for monotone inclusions of three operators

OriginalPaper

Analysis of collocation methods for nonlinear Volterra integral equations of the third kind

OriginalPaper

Iterative method with inertial for variational inequalities in Hilbert spaces

OriginalPaper

Structure-preserving stochastic conformal exponential integrator for linearly damped stochastic differential equations

OriginalPaper

A new generalized shrinkage conjugate gradient method for sparse recovery

OriginalPaper

Finite element discretization of local minimization schemes for rate-independent evolutions

Premium Partner

    Image Credits