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An accelerated augmented Lagrangian method for multi-criteria optimization problem
School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China |
By virtue of the Nesterov's acceleration technique, we establish an accelerated augmented Lagrangian method for solving linearly constrained multi-criteria optimization problem. For this method, we establish its global convergence under suitable condition. Further, we show that its iteration-complexity is $O(1/k^2)$ which improves the original ALM whose iteration-complexity is $O(1/k)$.
References:
[1] |
C. J. Y. Bello, P. L. R. Lucambio and J. G. Melo,
Convergence of the projected gradient method for quasiconvex multiobjective optimization, Nonlinear Anal., 74 (2011), 5268-5273.
doi: 10.1016/j.na.2011.04.067. |
[2] |
A. Chinchuluun and P. M. Pardalos,
A survey of recent developments in multiobjective optimization, Ann.Oper.Res., 154 (2007), 29-50.
doi: 10.1007/s10479-007-0186-0. |
[3] |
C. R. Chen, S. J. Li and X. Q. Yang,
Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[4] |
G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Analysis, Berlin: Spring, 2005. |
[5] |
H. Chen and Y. Wang,
A Family of higher-order convergent iterative methods for computing the Moore-Penrose inverse, Appl. Math. Comput., 218 (2011), 4012-4016.
doi: 10.1016/j.amc.2011.05.066. |
[6] |
H. Che, Y. Wang and M. Li,
A smoothing inexact Newton method for P-0 nonlinear complementarity problem, Front. Math. China, 7 (2012), 1043-1058.
doi: 10.1007/s11464-012-0245-y. |
[7] |
H. Chen, Y. Wang and G. Wang, Strong convergence of extragradient method for generalized variational inequalities in Hilbert space, J. Inequal. Appl., 2014 (2014), 11pp.
doi: 10.1186/1029-242X-2014-223. |
[8] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[9] |
H. Chen, Y. Wang and Y. Xu, An alternative extragradient projection method for quasi-equilibrium problems, J. Inequal. Appl., 26 (2018), 15pp.
doi: 10.1186/s13660-018-1619-9. |
[10] |
D. Feng, M. Sun and X. Wang, A family of conjugate gradient methods for large-scale nonlinear equation, J. Inequal. Appl., 236 (2017), 8pp.
doi: 10.1186/s13660-017-1510-0. |
[11] |
J. Fliege and B. F. Svaiter,
Steepest descent methods for multicriteria optimization, Math. Methods Oper. Res., 51 (2000), 479-494.
doi: 10.1007/s001860000043. |
[12] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time delay systems with delayed impulse effects, Appl. Math. Comput., 268 (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
[13] |
M. R. Hestenes,
Multiplier and gradient methods, J. Optim. Theory Al., 4 (1969), 303-320.
doi: 10.1007/BF00927673. |
[14] |
P. Lertworawanich, M. Kuwahara and M. MIska, A new multiobjective signal optimization for oversaturated networks, IEEE Trans. Intel. Trans. Systems, 12 (2011), 967-976. Google Scholar |
[15] |
S. Lian and L. Zhang,
A simple smooth exact penalty function for smooth optimization problem, J.Syst.Sci.Complex., 25 (2012), 521-528.
doi: 10.1007/s11424-012-9226-1. |
[16] |
S. Lian,
Smoothing approximation to l1 exact penalty function for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.
doi: 10.1016/j.amc.2012.09.042. |
[17] |
S. Lian and Y. Duan, Smoothing of the lower-order exact penalty function for inequality constrained optimization, J. Inequal. Appl., 2016 (2016), Paper No. 185, 12 pp.
doi: 10.1186/s13660-016-1126-9. |
[18] |
B. Liu, B. Qu and N. Zheng,
A successive projection algorithm for solving the multiple-sets split feasibility problem, Numer. Funct. Anal. Optim., 35 (2014), 1459-1466.
doi: 10.1080/01630563.2014.895755. |
[19] |
W. Liu and C. Wang,
A smoothing Levenberg-Marquardt method for generalized semi-infinite programming, Comput. Appl. Math., 32 (2013), 89-105.
doi: 10.1007/s40314-013-0013-y. |
[20] |
F. Lu and C. R. Chen,
Newton-like methods for solving vector optimization problems, Al. Anal., 93 (2014), 1567-1586.
doi: 10.1080/00036811.2013.839781. |
[21] |
D. T. Luc,
Scalarization in vector optimization problem, J. Optim. Theory Al., 55 (1987), 85-102.
doi: 10.1007/BF00939046. |
[22] |
K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, 1999. |
[23] |
Y. Y. E. Nesterov,
A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.
|
[24] |
S. J. Qu, M. Goh and F. T. S. Chan,
Quasi-Newton methods for solving multiobjective optimization, Oper. Res. Lett., 39 (2011), 397-399.
doi: 10.1016/j.orl.2011.07.008. |
[25] |
S. J. Qu, M. Goh, Y. Ji and R. D. Souza,
A new algorithm for linearly constrained c-convex vector optimization with a suly chain network risk alication, Euro. J Oper. Research, 247 (2015), 359-365.
doi: 10.1016/j.ejor.2015.06.016. |
[26] |
B. Qu, B. Liu and N. Zheng,
On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218-223.
doi: 10.1016/j.amc.2015.04.056. |
[27] |
B. Qu and H. Chang,
Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem, Numer. Funct. Anal. Optim., 38 (2017), 1614-1623.
doi: 10.1080/01630563.2017.1369109. |
[28] |
R. T. Rockafellar,
Augmented lagrangians and alications of the proximal point algorithm in optimizing convex programming, Math. Oper. Res., 1 (1976), 97-116.
doi: 10.1287/moor.1.2.97. |
[29] |
Y. Sawaragi, H. Tanino and T. Tanino, Theory of Multiobjective Optimization, Orlando, FL: Academic Press, 1985. |
[30] |
Y. Sun, L. S. Liu and Y. H. Wu,
The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrodinger equations on infinite domains, J. Comput. Appl. Math., 321 (2017), 478-486.
doi: 10.1016/j.cam.2017.02.036. |
[31] |
M. Sun, Y. Wang and J. Liu,
Generalized Peaceman-Rachford splitting method for multipleblock separable convex programming with applications to robust PCA, Calcolo, 54 (2017), 77-94.
doi: 10.1007/s10092-016-0177-0. |
[32] |
G. Wang and H. T. Che,
Generalized strict feasibility and solvability for generalized vector equilibrium problem with set-valued map in reflexive Banach spaces, J. Inequal. Appl., 2012 (2012), 1-11.
doi: 10.1186/1029-242X-2012-66. |
[33] |
G. Wang, Existence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spaces, J. Inequal. Appl., 2015 (2015), 14pp.
doi: 10.1186/s13660-015-0760-y. |
[34] |
X. Wang, Alternating proximal penalization algorithm for the modified multiple-set split feasibility problems, J. Inequal. Appl., 2018 (2018), Paper No. 48, 8 pp.
doi: 10.1186/s13660-018-1641-y. |
[35] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China, 313 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[36] |
Y. Wang, K. L. Zhang and H. C. Sun,
Criteria for strong H-tensors, Front. Math. China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[37] |
B. Wang, X. Wu and F. Meng,
Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations, J. Comput. Appl. Math., 313 (2017), 185-201.
doi: 10.1016/j.cam.2016.09.017. |
[38] |
Y. Wang and L. S. Liu, Uniqueness and existence of positive solutions for the fractional integro differential equation, Bound. Value Probl., 12 (2017), 2017.
doi: 10.1186/s13661-016-0741-1. |
[39] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[40] |
Y. Wang and J. Jiang, Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian, Adv. Difference Equations, 2017 (2017), Paper No. 337, 19 pp.
doi: 10.1186/s13662-017-1385-x. |
[41] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Anal. Appl., 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[42] |
H. Zhang and Y. Wang,
A new CQ method for solving split feasibility problem, Front. Math. China, 5 (2010), 37-46.
doi: 10.1007/s11464-009-0047-z. |
[43] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10 pp.
doi: 10.1002/nla.2134. |
show all references
References:
[1] |
C. J. Y. Bello, P. L. R. Lucambio and J. G. Melo,
Convergence of the projected gradient method for quasiconvex multiobjective optimization, Nonlinear Anal., 74 (2011), 5268-5273.
doi: 10.1016/j.na.2011.04.067. |
[2] |
A. Chinchuluun and P. M. Pardalos,
A survey of recent developments in multiobjective optimization, Ann.Oper.Res., 154 (2007), 29-50.
doi: 10.1007/s10479-007-0186-0. |
[3] |
C. R. Chen, S. J. Li and X. Q. Yang,
Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[4] |
G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Analysis, Berlin: Spring, 2005. |
[5] |
H. Chen and Y. Wang,
A Family of higher-order convergent iterative methods for computing the Moore-Penrose inverse, Appl. Math. Comput., 218 (2011), 4012-4016.
doi: 10.1016/j.amc.2011.05.066. |
[6] |
H. Che, Y. Wang and M. Li,
A smoothing inexact Newton method for P-0 nonlinear complementarity problem, Front. Math. China, 7 (2012), 1043-1058.
doi: 10.1007/s11464-012-0245-y. |
[7] |
H. Chen, Y. Wang and G. Wang, Strong convergence of extragradient method for generalized variational inequalities in Hilbert space, J. Inequal. Appl., 2014 (2014), 11pp.
doi: 10.1186/1029-242X-2014-223. |
[8] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[9] |
H. Chen, Y. Wang and Y. Xu, An alternative extragradient projection method for quasi-equilibrium problems, J. Inequal. Appl., 26 (2018), 15pp.
doi: 10.1186/s13660-018-1619-9. |
[10] |
D. Feng, M. Sun and X. Wang, A family of conjugate gradient methods for large-scale nonlinear equation, J. Inequal. Appl., 236 (2017), 8pp.
doi: 10.1186/s13660-017-1510-0. |
[11] |
J. Fliege and B. F. Svaiter,
Steepest descent methods for multicriteria optimization, Math. Methods Oper. Res., 51 (2000), 479-494.
doi: 10.1007/s001860000043. |
[12] |
L. Gao, D. Wang and G. Wang,
Further results on exponential stability for impulsive switched nonlinear time delay systems with delayed impulse effects, Appl. Math. Comput., 268 (2015), 186-200.
doi: 10.1016/j.amc.2015.06.023. |
[13] |
M. R. Hestenes,
Multiplier and gradient methods, J. Optim. Theory Al., 4 (1969), 303-320.
doi: 10.1007/BF00927673. |
[14] |
P. Lertworawanich, M. Kuwahara and M. MIska, A new multiobjective signal optimization for oversaturated networks, IEEE Trans. Intel. Trans. Systems, 12 (2011), 967-976. Google Scholar |
[15] |
S. Lian and L. Zhang,
A simple smooth exact penalty function for smooth optimization problem, J.Syst.Sci.Complex., 25 (2012), 521-528.
doi: 10.1007/s11424-012-9226-1. |
[16] |
S. Lian,
Smoothing approximation to l1 exact penalty function for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.
doi: 10.1016/j.amc.2012.09.042. |
[17] |
S. Lian and Y. Duan, Smoothing of the lower-order exact penalty function for inequality constrained optimization, J. Inequal. Appl., 2016 (2016), Paper No. 185, 12 pp.
doi: 10.1186/s13660-016-1126-9. |
[18] |
B. Liu, B. Qu and N. Zheng,
A successive projection algorithm for solving the multiple-sets split feasibility problem, Numer. Funct. Anal. Optim., 35 (2014), 1459-1466.
doi: 10.1080/01630563.2014.895755. |
[19] |
W. Liu and C. Wang,
A smoothing Levenberg-Marquardt method for generalized semi-infinite programming, Comput. Appl. Math., 32 (2013), 89-105.
doi: 10.1007/s40314-013-0013-y. |
[20] |
F. Lu and C. R. Chen,
Newton-like methods for solving vector optimization problems, Al. Anal., 93 (2014), 1567-1586.
doi: 10.1080/00036811.2013.839781. |
[21] |
D. T. Luc,
Scalarization in vector optimization problem, J. Optim. Theory Al., 55 (1987), 85-102.
doi: 10.1007/BF00939046. |
[22] |
K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, 1999. |
[23] |
Y. Y. E. Nesterov,
A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.
|
[24] |
S. J. Qu, M. Goh and F. T. S. Chan,
Quasi-Newton methods for solving multiobjective optimization, Oper. Res. Lett., 39 (2011), 397-399.
doi: 10.1016/j.orl.2011.07.008. |
[25] |
S. J. Qu, M. Goh, Y. Ji and R. D. Souza,
A new algorithm for linearly constrained c-convex vector optimization with a suly chain network risk alication, Euro. J Oper. Research, 247 (2015), 359-365.
doi: 10.1016/j.ejor.2015.06.016. |
[26] |
B. Qu, B. Liu and N. Zheng,
On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218-223.
doi: 10.1016/j.amc.2015.04.056. |
[27] |
B. Qu and H. Chang,
Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem, Numer. Funct. Anal. Optim., 38 (2017), 1614-1623.
doi: 10.1080/01630563.2017.1369109. |
[28] |
R. T. Rockafellar,
Augmented lagrangians and alications of the proximal point algorithm in optimizing convex programming, Math. Oper. Res., 1 (1976), 97-116.
doi: 10.1287/moor.1.2.97. |
[29] |
Y. Sawaragi, H. Tanino and T. Tanino, Theory of Multiobjective Optimization, Orlando, FL: Academic Press, 1985. |
[30] |
Y. Sun, L. S. Liu and Y. H. Wu,
The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrodinger equations on infinite domains, J. Comput. Appl. Math., 321 (2017), 478-486.
doi: 10.1016/j.cam.2017.02.036. |
[31] |
M. Sun, Y. Wang and J. Liu,
Generalized Peaceman-Rachford splitting method for multipleblock separable convex programming with applications to robust PCA, Calcolo, 54 (2017), 77-94.
doi: 10.1007/s10092-016-0177-0. |
[32] |
G. Wang and H. T. Che,
Generalized strict feasibility and solvability for generalized vector equilibrium problem with set-valued map in reflexive Banach spaces, J. Inequal. Appl., 2012 (2012), 1-11.
doi: 10.1186/1029-242X-2012-66. |
[33] |
G. Wang, Existence-stability theorems for strong vector set-valued equilibrium problems in reflexive Banach spaces, J. Inequal. Appl., 2015 (2015), 14pp.
doi: 10.1186/s13660-015-0760-y. |
[34] |
X. Wang, Alternating proximal penalization algorithm for the modified multiple-set split feasibility problems, J. Inequal. Appl., 2018 (2018), Paper No. 48, 8 pp.
doi: 10.1186/s13660-018-1641-y. |
[35] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Front. Math. China, 313 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[36] |
Y. Wang, K. L. Zhang and H. C. Sun,
Criteria for strong H-tensors, Front. Math. China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[37] |
B. Wang, X. Wu and F. Meng,
Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations, J. Comput. Appl. Math., 313 (2017), 185-201.
doi: 10.1016/j.cam.2016.09.017. |
[38] |
Y. Wang and L. S. Liu, Uniqueness and existence of positive solutions for the fractional integro differential equation, Bound. Value Probl., 12 (2017), 2017.
doi: 10.1186/s13661-016-0741-1. |
[39] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra Appl., 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[40] |
Y. Wang and J. Jiang, Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian, Adv. Difference Equations, 2017 (2017), Paper No. 337, 19 pp.
doi: 10.1186/s13662-017-1385-x. |
[41] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Anal. Appl., 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[42] |
H. Zhang and Y. Wang,
A new CQ method for solving split feasibility problem, Front. Math. China, 5 (2010), 37-46.
doi: 10.1007/s11464-009-0047-z. |
[43] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10 pp.
doi: 10.1002/nla.2134. |
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