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July  2019, 15(3): 1241-1261. doi: 10.3934/jimo.2018094

A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

2. 

School of Mathematics, Tianjin University, Tianjin, 300072, China

3. 

School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China

* Corresponding author: Mengwei Xu

Received  July 2017 Revised  October 2017 Published  July 2018

Fund Project: The second author is supported by NSFC grant 11601376. The third author is supported by NSFC grant 11601389, the Doctoral Foundation of Tianjin Normal University grant 52XB1513 and and 2017- Outstanding Young Innovation Team Cultivation Program of Tianjin Normal University grant 135202TD1703. The fourth author is supported by NSFC grant 11571059 and 11731013.

In this paper, we consider a class of nonsmooth and nonconvex optimization problem with an abstract constraint. We propose an augmented Lagrangian method for solving the problem and construct global convergence under a weakly nonsmooth Mangasarian-Fromovitz constraint qualification. We show that any accumulation point of the iteration sequence generated by the algorithm is a feasible point which satisfies the first order necessary optimality condition provided that the penalty parameters are bounded and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. Numerical experiments show that the algorithm is efficient for obtaining stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the nonsmooth Mangasarian-Fromovitz constraint qualification.

Citation: Qingsong Duan, Mengwei Xu, Yue Lu, Liwei Zhang. A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1241-1261. doi: 10.3934/jimo.2018094
References:
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R. AndreaniE. G. BirginJ. M. Martínez and M. L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints, SIAM J. Optim., 18 (2007), 1286-1309.  doi: 10.1137/060654797.  Google Scholar

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R. AndreaniE. G. BirginJ. M. Martínez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Program., Ser. B, 111 (2008), 5-32.  doi: 10.1007/s10107-006-0077-1.  Google Scholar

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R. AndreaniG. Haeser and M. L. Schuverdt, A relaxed constant positive linear dependence constraint qualification and applications, Math. Program., 135 (2012), 255-273.  doi: 10.1007/s10107-011-0456-0.  Google Scholar

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J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publications, Dordrecht, Netherlands, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar

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D. P. Bertsekas, Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York, 1982.  Google Scholar

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W. Bian and X. Chen, Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 545-556.  doi: 10.1109/TNNLS.2013.2278427.  Google Scholar

[8]

E. G. BirginD. Fernández and J. M. Martínez, The boundedness of penalty parameters in an Augmented Lagrangian method with lower level constraints, Optim. Methods Soft., 27 (2012), 1001-1024.  doi: 10.1080/10556788.2011.556634.  Google Scholar

[9]

J. V. Burke and T. Hoheisel, Epi-convergent smoothing with applications to convex composite functions, SIAM J. Optim., 23 (2013), 1457-1479.  doi: 10.1137/120889812.  Google Scholar

[10]

J. V. BurkeT. Hoheisel and C. Kanzow, Gradient consistency for integral-convolution smoothing functions, Set-Valued Var. Anal., 21 (2013), 359-376.  doi: 10.1007/s11228-013-0235-6.  Google Scholar

[11]

B. Chen and X. Chen, A global and local superlinear continuation-smoothing method for $P_0$ and $R_0$ NCP or monotone NCP, SIAM J. Optim., 9 (1999), 624-645.  doi: 10.1137/S1052623497321109.  Google Scholar

[12]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

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B. Chen and P. T. Harker, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), 1168-1190.  doi: 10.1137/0614081.  Google Scholar

[14]

X. ChenL. GuoZ. Lu and J. J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM J. Numer. Anal., 55 (2017), 168-193.  doi: 10.1137/15M1052834.  Google Scholar

[15]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Math. Program., 71 (1995), 51-70.  doi: 10.1007/BF01592244.  Google Scholar

[16]

X. ChenR. S. Womersley and J. J. Ye, Minimizing the condition number of a gram matrix, SIAM J. Optim., 21 (2011), 127-148.  doi: 10.1137/100786022.  Google Scholar

[17]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.  Google Scholar

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F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[19]

A. R. ConnN. I. M. Gould and Ph. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bound, SIAM J. Numer. Anal., 28 (1991), 545-572.  doi: 10.1137/0728030.  Google Scholar

[20]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust Region Methods, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[21]

F. E. CurtisH. Jiang and D. P. Robinson, An adaptive augmented Lagrangian method for large-scale constrained optimization, Math. Program., 152 (2015), 201-245.  doi: 10.1007/s10107-014-0784-y.  Google Scholar

[22]

F. E. Curtis and M. L. Overton, A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization, SIAM J. Optim., 22 (2012), 474-500.  doi: 10.1137/090780201.  Google Scholar

[23]

S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, 2002.  Google Scholar

[24]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optim., 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.  Google Scholar

[25]

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

[26]

C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), 851-868.  doi: 10.1137/S0895479894273134.  Google Scholar

[27]

LANCELOT, http://www.cse.scitech.ac.uk/nag/lancelot/lancelot.shtml. Google Scholar

[28]

G. H. LinM. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Math. Program., series A, 144 (2014), 277-305.  doi: 10.1007/s10107-013-0633-4.  Google Scholar

[29]

Z. Lu and Y. Zhang, An augmented Lagrangian approach for sparse principal component analysis, Math. Program. series A, 135 (2012), 149-193.  doi: 10.1007/s10107-011-0452-4.  Google Scholar

[30]

J. Mirrlees, The theory of moral hazard and unobservable behaviour: Part Ⅰ, Rev. Econ. Stud., 66 (1999), 3-22.  doi: 10.1093/acprof:oso/9780198295211.003.0020.  Google Scholar

[31]

A. Mitsos and P. Barton, A Test Set for Bilevel Programs, Technical Report, Massachusetts Institute of Technology, 2006. Google Scholar

[32]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. 1: Basic Theory, Vol. 2: Applications, Springer, 2006.  Google Scholar

[33]

Y. Nesterov, Smoothing minimization of nonsmooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[34]

J. V. Outrata, On the numerical solution of a class of Stackelberg problems, Z. Oper. Res., 34 (1990), 255-277.  doi: 10.1007/BF01416737.  Google Scholar

[35]

M. J. D. Powell, A method for nonlinear constraints in minimization problems, in Optimization(eds. R. Fletcher), 283-298, London and New York, 1969. Academic Press  Google Scholar

[36]

R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970.  Google Scholar

[37]

R. T. Rockfellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. Program., 5 (1973), 354-373.  doi: 10.1007/BF01580138.  Google Scholar

[38]

R. T. Rockfellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Con., 12 (1974), 268-285.  doi: 10.1137/0312021.  Google Scholar

[39]

R. T. Rockfellar, Monotone operators and the proximal point algorithm, SIAM J. Con. Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[40]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[41]

K. Shimizu, Y. Ishizuka and J. F. Bard, Nondifferentiable and Two-Level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4615-6305-1.  Google Scholar

[42]

S. Smale, Algorithms for solving equations, In: Proceedings of the International Congress of Mathematicians, Berkeley, CA., (1986), 172-195  Google Scholar

[43]

L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review, J. Global Optim., 5 (1994), 291-306.  doi: 10.1007/BF01096458.  Google Scholar

[44]

M. Xu and J. J. Ye, A smoothing augmented Lagrangian method for solving simple bilevel programs, Compu. Optim. App., 59 (2014), 353-377.  doi: 10.1007/s10589-013-9627-7.  Google Scholar

[45]

M. XuJ. J. Ye and L. Zhang, Smoothing sequential quadratic programming method for solving nonconvex, nonsmooth constrained optimization problems, SIAM J. Optim., 25 (2015), 1388-1410.  doi: 10.1137/140971580.  Google Scholar

[46]

M. XuJ. J. Ye and L. Zhang, Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems, J. Glob. Optim., 62 (2015), 675-694.  doi: 10.1007/s10898-014-0242-7.  Google Scholar

[47]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optim., 33 (1995), 9-27.  doi: 10.1080/02331939508844060.  Google Scholar

[48]

J. J. Ye and D. L. Zhu, A note on optimality conditions for bilevel programming problems, Optim., 39 (1997), 361-366.  doi: 10.1080/02331939708844290.  Google Scholar

[49]

J. J. Ye and D. L. Zhu, New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach, SIAM J. Optim., 20 (2010), 1885-1905.  doi: 10.1137/080725088.  Google Scholar

show all references

References:
[1]

ALGENCAN, http://www.ime.usp.br/$\sim$egbirgin/tango/. Google Scholar

[2]

R. AndreaniE. G. BirginJ. M. Martínez and M. L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints, SIAM J. Optim., 18 (2007), 1286-1309.  doi: 10.1137/060654797.  Google Scholar

[3]

R. AndreaniE. G. BirginJ. M. Martínez and M. L. Schuverdt, Augmented Lagrangian methods under the constant positive linear dependence constraint qualification, Math. Program., Ser. B, 111 (2008), 5-32.  doi: 10.1007/s10107-006-0077-1.  Google Scholar

[4]

R. AndreaniG. Haeser and M. L. Schuverdt, A relaxed constant positive linear dependence constraint qualification and applications, Math. Program., 135 (2012), 255-273.  doi: 10.1007/s10107-011-0456-0.  Google Scholar

[5]

J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publications, Dordrecht, Netherlands, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar

[6]

D. P. Bertsekas, Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York, 1982.  Google Scholar

[7]

W. Bian and X. Chen, Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 545-556.  doi: 10.1109/TNNLS.2013.2278427.  Google Scholar

[8]

E. G. BirginD. Fernández and J. M. Martínez, The boundedness of penalty parameters in an Augmented Lagrangian method with lower level constraints, Optim. Methods Soft., 27 (2012), 1001-1024.  doi: 10.1080/10556788.2011.556634.  Google Scholar

[9]

J. V. Burke and T. Hoheisel, Epi-convergent smoothing with applications to convex composite functions, SIAM J. Optim., 23 (2013), 1457-1479.  doi: 10.1137/120889812.  Google Scholar

[10]

J. V. BurkeT. Hoheisel and C. Kanzow, Gradient consistency for integral-convolution smoothing functions, Set-Valued Var. Anal., 21 (2013), 359-376.  doi: 10.1007/s11228-013-0235-6.  Google Scholar

[11]

B. Chen and X. Chen, A global and local superlinear continuation-smoothing method for $P_0$ and $R_0$ NCP or monotone NCP, SIAM J. Optim., 9 (1999), 624-645.  doi: 10.1137/S1052623497321109.  Google Scholar

[12]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[13]

B. Chen and P. T. Harker, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), 1168-1190.  doi: 10.1137/0614081.  Google Scholar

[14]

X. ChenL. GuoZ. Lu and J. J. Ye, An augmented Lagrangian method for non-Lipschitz nonconvex programming, SIAM J. Numer. Anal., 55 (2017), 168-193.  doi: 10.1137/15M1052834.  Google Scholar

[15]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Math. Program., 71 (1995), 51-70.  doi: 10.1007/BF01592244.  Google Scholar

[16]

X. ChenR. S. Womersley and J. J. Ye, Minimizing the condition number of a gram matrix, SIAM J. Optim., 21 (2011), 127-148.  doi: 10.1137/100786022.  Google Scholar

[17]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.  Google Scholar

[18]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[19]

A. R. ConnN. I. M. Gould and Ph. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bound, SIAM J. Numer. Anal., 28 (1991), 545-572.  doi: 10.1137/0728030.  Google Scholar

[20]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust Region Methods, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[21]

F. E. CurtisH. Jiang and D. P. Robinson, An adaptive augmented Lagrangian method for large-scale constrained optimization, Math. Program., 152 (2015), 201-245.  doi: 10.1007/s10107-014-0784-y.  Google Scholar

[22]

F. E. Curtis and M. L. Overton, A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization, SIAM J. Optim., 22 (2012), 474-500.  doi: 10.1137/090780201.  Google Scholar

[23]

S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, 2002.  Google Scholar

[24]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optim., 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.  Google Scholar

[25]

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

[26]

C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), 851-868.  doi: 10.1137/S0895479894273134.  Google Scholar

[27]

LANCELOT, http://www.cse.scitech.ac.uk/nag/lancelot/lancelot.shtml. Google Scholar

[28]

G. H. LinM. Xu and J. J. Ye, On solving simple bilevel programs with a nonconvex lower level program, Math. Program., series A, 144 (2014), 277-305.  doi: 10.1007/s10107-013-0633-4.  Google Scholar

[29]

Z. Lu and Y. Zhang, An augmented Lagrangian approach for sparse principal component analysis, Math. Program. series A, 135 (2012), 149-193.  doi: 10.1007/s10107-011-0452-4.  Google Scholar

[30]

J. Mirrlees, The theory of moral hazard and unobservable behaviour: Part Ⅰ, Rev. Econ. Stud., 66 (1999), 3-22.  doi: 10.1093/acprof:oso/9780198295211.003.0020.  Google Scholar

[31]

A. Mitsos and P. Barton, A Test Set for Bilevel Programs, Technical Report, Massachusetts Institute of Technology, 2006. Google Scholar

[32]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. 1: Basic Theory, Vol. 2: Applications, Springer, 2006.  Google Scholar

[33]

Y. Nesterov, Smoothing minimization of nonsmooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[34]

J. V. Outrata, On the numerical solution of a class of Stackelberg problems, Z. Oper. Res., 34 (1990), 255-277.  doi: 10.1007/BF01416737.  Google Scholar

[35]

M. J. D. Powell, A method for nonlinear constraints in minimization problems, in Optimization(eds. R. Fletcher), 283-298, London and New York, 1969. Academic Press  Google Scholar

[36]

R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970.  Google Scholar

[37]

R. T. Rockfellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. Program., 5 (1973), 354-373.  doi: 10.1007/BF01580138.  Google Scholar

[38]

R. T. Rockfellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Con., 12 (1974), 268-285.  doi: 10.1137/0312021.  Google Scholar

[39]

R. T. Rockfellar, Monotone operators and the proximal point algorithm, SIAM J. Con. Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[40]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[41]

K. Shimizu, Y. Ishizuka and J. F. Bard, Nondifferentiable and Two-Level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4615-6305-1.  Google Scholar

[42]

S. Smale, Algorithms for solving equations, In: Proceedings of the International Congress of Mathematicians, Berkeley, CA., (1986), 172-195  Google Scholar

[43]

L. N. Vicente and P. H. Calamai, Bilevel and multilevel programming: A bibliography review, J. Global Optim., 5 (1994), 291-306.  doi: 10.1007/BF01096458.  Google Scholar

[44]

M. Xu and J. J. Ye, A smoothing augmented Lagrangian method for solving simple bilevel programs, Compu. Optim. App., 59 (2014), 353-377.  doi: 10.1007/s10589-013-9627-7.  Google Scholar

[45]

M. XuJ. J. Ye and L. Zhang, Smoothing sequential quadratic programming method for solving nonconvex, nonsmooth constrained optimization problems, SIAM J. Optim., 25 (2015), 1388-1410.  doi: 10.1137/140971580.  Google Scholar

[46]

M. XuJ. J. Ye and L. Zhang, Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems, J. Glob. Optim., 62 (2015), 675-694.  doi: 10.1007/s10898-014-0242-7.  Google Scholar

[47]

J. J. Ye and D. L. Zhu, Optimality conditions for bilevel programming problems, Optim., 33 (1995), 9-27.  doi: 10.1080/02331939508844060.  Google Scholar

[48]

J. J. Ye and D. L. Zhu, A note on optimality conditions for bilevel programming problems, Optim., 39 (1997), 361-366.  doi: 10.1080/02331939708844290.  Google Scholar

[49]

J. J. Ye and D. L. Zhu, New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach, SIAM J. Optim., 20 (2010), 1885-1905.  doi: 10.1137/080725088.  Google Scholar

Table 1.  Mirrlees' problem
(x*; y*) d(x*; y*)
Algorithm 3.1 (1, 0.957504) 5.73e-006
SQP algorithm (1.000002, 0.957598) 9.79e-005
SAL algorithm (1.000905, 0.957459) 9.06e-004
(x*; y*) d(x*; y*)
Algorithm 3.1 (1, 0.957504) 5.73e-006
SQP algorithm (1.000002, 0.957598) 9.79e-005
SAL algorithm (1.000905, 0.957459) 9.06e-004
Table 2.  Example 4.4
(x*; y*) d(x*; y*)
Algorithm 3.1 (0.500003, 0.500003) 4.08e-006
SQP algorithm (0.499996, 0.499996) 5.85e-006
SAL algorithm (0.500000, 0.499995) 2.89e-005
(x*; y*) d(x*; y*)
Algorithm 3.1 (0.500003, 0.500003) 4.08e-006
SQP algorithm (0.499996, 0.499996) 5.85e-006
SAL algorithm (0.500000, 0.499995) 2.89e-005
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