American Institute of Mathematical Sciences

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January  2020, 16(1): 1-9. doi: 10.3934/jimo.2018136

An accelerated augmented Lagrangian method for multi-criteria optimization problem

Xueyong Wang 1, , Yiju Wang and  Gang Wang

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  January 2020 Early access  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 $O(1/k)$.

Keywords: Iteration-complexity, augmented Lagrangian method, mulit-criteria optimization problem.
Mathematics Subject Classification: 15A18, 15A69, 65F15, 65F10.
Citation: Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 1-9. doi: 10.3934/jimo.2018136
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Analysis, Berlin: Spring, 2005.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Al., 4 (1969), 303-320.  doi: 10.1007/BF00927673.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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[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.  Google Scholar

[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.  Google Scholar

[21]

D. T. Luc, Scalarization in vector optimization problem, J. Optim. Theory Al., 55 (1987), 85-102.  doi: 10.1007/BF00939046.  Google Scholar

[22]

K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, 1999.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. Sawaragi, H. Tanino and T. Tanino, Theory of Multiobjective Optimization, Orlando, FL: Academic Press, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Analysis, Berlin: Spring, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Al., 4 (1969), 303-320.  doi: 10.1007/BF00927673.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. T. Luc, Scalarization in vector optimization problem, J. Optim. Theory Al., 55 (1987), 85-102.  doi: 10.1007/BF00939046.  Google Scholar

[22]

K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, 1999.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. Sawaragi, H. Tanino and T. Tanino, Theory of Multiobjective Optimization, Orlando, FL: Academic Press, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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