# American Institute of Mathematical Sciences

September  2020, 16(5): 2267-2281. doi: 10.3934/jimo.2019053

## Differential equation method based on approximate augmented Lagrangian for nonlinear programming

 School of Mathematics, Zhejiang Ocean University, Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province, Zhoushan, Zhejiang 316022, China

* Corresponding author: Hongying Huang

Received  May 2018 Revised  February 2019 Published  September 2020 Early access  May 2019

Fund Project: The research is supported by the National Natural Science Foundation of China under Grant Nos.61673352 and 11771398

This paper analyzes the approximate augmented Lagrangian dynamical systems for constrained optimization. We formulate the differential systems based on first derivatives and second derivatives of the approximate augmented Lagrangian. The solution of the original optimization problems can be obtained at the equilibrium point of the differential equation systems, which lead the dynamic trajectory into the feasible region. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are analyzed, including the locally quadratic convergence rate of the discrete sequence for the second derivatives based differential system. The transient behavior of the differential equation systems is simulated and the validity of the approach is verified with numerical experiments.

Citation: Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2267-2281. doi: 10.3934/jimo.2019053
##### References:
 [1] K. J. Arrow and L. Hurwicz, Reduction of constrained maxima to saddle point problems, in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, (eds. J.Neyman), University of California Press, Berkeley, (1956), 1–20. [2] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, , in Computer Science and Applied Mathematics, Academic Press Inc, New York, 1982. [3] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, 1999. [4] Yu. G. Evtushenko, Numerical Optimization Techniques, Optimization Software, New York, 1985. doi: 10.1007/978-1-4612-5022-7. [5] Yu. G. Evtushenko and V. G. Zhadan, Barrier-projective methods for nonlinear programming, Comp.Math. Math. Phys., 34 (1994), 579-590. [6] Yu. G. Evtushenko and V. G. Zhandan, Stable barrier-projection and barrier-Newton methods in nonlinear programming, Comput. Optim. Appl., 3 (1994), 289-303.  doi: 10.1007/BF01299205. [7] Yu. G. Evtushenko and V. G. Zhandan, Stable barrier-projection and barrier-Newton methods for linear and nonlinear programming, in Algorithms for Continuous Optimization(eds.E. Spedicato), Kulwer Academic Publishers, 434 (1994), 255–285. [8] Yu. G. Evtushenko, Two numerical methods of solving nonlinear programming problems, Sov. Math. Dokl., 15 (1974), 420-423. [9] A. V. Fiacco and G. P. McCormick, Sequential Unconstrained Minimization Techniques, in Nonlinear Programming, John Wiely, New York, 1968. [10] M. Hestenes, Multiplier and gradient methods, J.Optim.Theor.Appl., 4 (1969), 303-320.  doi: 10.1007/BF00927673. [11] W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, in Lecture Notes Economics and Mathematical Systems, Springer-Verlag, Berlin Heidelberg-New York, 1981. doi: 10.1007/BF00934594. [12] X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs, Acta Mathematica Sinica (English Series), 22 (2006), 1283-1296.  doi: 10.1007/s10114-005-0702-6. [13] A. Ioffe, Necessary and sufficient conditions for a local minimum 3: Second-order conditions and augmented duality, SIAM J. Control Optim., 17 (1979), 266-288.  doi: 10.1137/0317021. [14] L. Jin, L. W. Zhang and X. T. Xiao, Two differential equation systems for equality-constrained optimization, Appl. Math. Comput., 190 (2007), 1030-1039.  doi: 10.1016/j.amc.2006.11.041. [15] L. Jin, L. W. Zhang and X. T. Xiao, Two differential equation systems for inequality constrained optimization, Appl. Math. Comput., 188 (2007), 1334-1343.  doi: 10.1016/j.amc.2006.10.085. [16] L. Jin, A stable differential equation approach for inequality constrained optimization problems, Appl. Math. Comput., 206 (2008), 186-192.  doi: 10.1016/j.amc.2008.09.007. [17] L. Jin, H. X. Yu and Z. S. Liu, Differential systems for constrained optimization via a nonlinear augmented Lagrangian, Appl. Math. Comput., 235 (2014), 482-491.  doi: 10.1016/j.amc.2014.02.097. [18] P. Q. Pan, New ODE methods for equality constrained optimization (1)-equations, J. Comput. Math., 10 (1992), 77-92. [19] M. J. D. Powell, A method for non-linear constraints in minimizations problems in optimization, in (eds. R. Fletcher), (1969), 283–298. [20] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math.Oper.Res., 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97. [21] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J.Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056. [22] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Control, 12 (1974), 268-285.  doi: 10.1137/0312021. [23] R. T. Rockafellar, Lagrange multipliers and optimality, SIAM Rev., 35 (1993), 183-238.  doi: 10.1137/1035044. [24] H. Yamadhita, A differential equation approach to nonlinear programming, Math. Prog., 18 (1980), 155-168.  doi: 10.1007/BF01588311. [25] L. W. Zhang, A modified version to the differential system of Evtushenko and Zhanda for solving nonlinear programming, in Numerical Linear Algebra and Optimization(eds.Ya-xiang Yuan), Science Press, (1999), 161–168. [26] L. W. Zhang, Q. Li and X. Zhang, Two differential systems for solving nonlinear programming problems, OR Trans., 4 (2000), 33-46.

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##### References:
 [1] K. J. Arrow and L. Hurwicz, Reduction of constrained maxima to saddle point problems, in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, (eds. J.Neyman), University of California Press, Berkeley, (1956), 1–20. [2] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, , in Computer Science and Applied Mathematics, Academic Press Inc, New York, 1982. [3] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, 1999. [4] Yu. G. Evtushenko, Numerical Optimization Techniques, Optimization Software, New York, 1985. doi: 10.1007/978-1-4612-5022-7. [5] Yu. G. Evtushenko and V. G. Zhadan, Barrier-projective methods for nonlinear programming, Comp.Math. Math. Phys., 34 (1994), 579-590. [6] Yu. G. Evtushenko and V. G. Zhandan, Stable barrier-projection and barrier-Newton methods in nonlinear programming, Comput. Optim. Appl., 3 (1994), 289-303.  doi: 10.1007/BF01299205. [7] Yu. G. Evtushenko and V. G. Zhandan, Stable barrier-projection and barrier-Newton methods for linear and nonlinear programming, in Algorithms for Continuous Optimization(eds.E. Spedicato), Kulwer Academic Publishers, 434 (1994), 255–285. [8] Yu. G. Evtushenko, Two numerical methods of solving nonlinear programming problems, Sov. Math. Dokl., 15 (1974), 420-423. [9] A. V. Fiacco and G. P. McCormick, Sequential Unconstrained Minimization Techniques, in Nonlinear Programming, John Wiely, New York, 1968. [10] M. Hestenes, Multiplier and gradient methods, J.Optim.Theor.Appl., 4 (1969), 303-320.  doi: 10.1007/BF00927673. [11] W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, in Lecture Notes Economics and Mathematical Systems, Springer-Verlag, Berlin Heidelberg-New York, 1981. doi: 10.1007/BF00934594. [12] X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs, Acta Mathematica Sinica (English Series), 22 (2006), 1283-1296.  doi: 10.1007/s10114-005-0702-6. [13] A. Ioffe, Necessary and sufficient conditions for a local minimum 3: Second-order conditions and augmented duality, SIAM J. Control Optim., 17 (1979), 266-288.  doi: 10.1137/0317021. [14] L. Jin, L. W. Zhang and X. T. Xiao, Two differential equation systems for equality-constrained optimization, Appl. Math. Comput., 190 (2007), 1030-1039.  doi: 10.1016/j.amc.2006.11.041. [15] L. Jin, L. W. Zhang and X. T. Xiao, Two differential equation systems for inequality constrained optimization, Appl. Math. Comput., 188 (2007), 1334-1343.  doi: 10.1016/j.amc.2006.10.085. [16] L. Jin, A stable differential equation approach for inequality constrained optimization problems, Appl. Math. Comput., 206 (2008), 186-192.  doi: 10.1016/j.amc.2008.09.007. [17] L. Jin, H. X. Yu and Z. S. Liu, Differential systems for constrained optimization via a nonlinear augmented Lagrangian, Appl. Math. Comput., 235 (2014), 482-491.  doi: 10.1016/j.amc.2014.02.097. [18] P. Q. Pan, New ODE methods for equality constrained optimization (1)-equations, J. Comput. Math., 10 (1992), 77-92. [19] M. J. D. Powell, A method for non-linear constraints in minimizations problems in optimization, in (eds. R. Fletcher), (1969), 283–298. [20] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math.Oper.Res., 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97. [21] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J.Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056. [22] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming, SIAM J. Control, 12 (1974), 268-285.  doi: 10.1137/0312021. [23] R. T. Rockafellar, Lagrange multipliers and optimality, SIAM Rev., 35 (1993), 183-238.  doi: 10.1137/1035044. [24] H. Yamadhita, A differential equation approach to nonlinear programming, Math. Prog., 18 (1980), 155-168.  doi: 10.1007/BF01588311. [25] L. W. Zhang, A modified version to the differential system of Evtushenko and Zhanda for solving nonlinear programming, in Numerical Linear Algebra and Optimization(eds.Ya-xiang Yuan), Science Press, (1999), 161–168. [26] L. W. Zhang, Q. Li and X. Zhang, Two differential systems for solving nonlinear programming problems, OR Trans., 4 (2000), 33-46.
Performances of the variable $x$ and $z$ in Problem 71
Performances of the variable $x$ and $z$ in Problem 53
Performances of the variable $x$ and $z$ in Problem 100
Performances of the variable $x$ and $z$ in Problem 113
Performances of the variable x and z in Problem 100
Performances of the cost function and the objective function in Problem 100
Performances of the variable x and z in Problem 113
Performances of the cost function and the objective function in Problem 113
numerical results
 Test n p q IT $S(z)$ $f(x^*)$ $F(x^*)$ P.71 4 10 1 349 8.125604 $\times10^{-10}$ 17.014 17.0140173 P.53 5 13 3 127 1.175666 $\times10^{-11}$ 4.0930 4.093023 P.100 7 4 0 967 3.829630$\times10^{-12}$ 678.6796 680.6300573 P.113 10 8 0 991 2.452665$\times10^{-12}$ 24.3062 24.306291
 Test n p q IT $S(z)$ $f(x^*)$ $F(x^*)$ P.71 4 10 1 349 8.125604 $\times10^{-10}$ 17.014 17.0140173 P.53 5 13 3 127 1.175666 $\times10^{-11}$ 4.0930 4.093023 P.100 7 4 0 967 3.829630$\times10^{-12}$ 678.6796 680.6300573 P.113 10 8 0 991 2.452665$\times10^{-12}$ 24.3062 24.306291
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