doi: 10.3934/jimo.2018136

An accelerated augmented Lagrangian method for multi-criteria optimization problem

School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China

1This research was done during his postdoctoral period in Qufu Normal Univeristy

Received  December 2016 Revised  November 2017 Published  September 2018

Fund Project: This work was supported by the Natural Science Foundation of China (11671228, 11801309), Shandong Provincial Natural Science Foundation (ZR2016AM10), and Science & Technology Planning Project of Qufu Normal University (XKJ201623)

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 $O(1/k)$.

Citation: Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018136
References:
[1]

C. J. Y. BelloP. 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. ChenS. 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. CheY. 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. ChenL. 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. GaoD. 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. LertworawanichM. Kuwahara and M. MIska, A new multiobjective signal optimization for oversaturated networks, IEEE Trans. Intel. Trans. Systems, 12 (2011), 967-976.

[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. LiuB. 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. QuM. 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. QuM. GohY. 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. QuB. 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. SunL. 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. SunY. 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. WangH. 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. WangK. 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. WangX. 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. WangL. 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. BelloP. 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. ChenS. 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. CheY. 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. ChenL. 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. GaoD. 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. LertworawanichM. Kuwahara and M. MIska, A new multiobjective signal optimization for oversaturated networks, IEEE Trans. Intel. Trans. Systems, 12 (2011), 967-976.

[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. LiuB. 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. QuM. 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. QuM. GohY. 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. QuB. 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. SunL. 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. SunY. 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. WangH. 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. WangK. 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. WangX. 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. WangL. 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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