Abstract
The solution of nonlinear optimization is usually encountered in many fields of scientific researches and engineering applications, which spawns a large number of corresponding algorithms to cope with it. Besides, with developments of modern cybernetics technology, it imperatively requires some advanced numerical algorithms to solve online dynamic nonlinear optimization (ODNO). Nevertheless, the major existing algorithms are limited to the static nonlinear optimization models, few works considering the dynamic ones, let alone tolerating noise. For the abovementioned reasons, this paper proposes a modified Newton integration (MNI) algorithm for ODNO with strong robustness and high-accuracy computing solution, which can effectively suppress the influence caused by noise components. In addition, the correlative theoretical analyses and mathematical proofs on convergence and robustness of the MNI algorithm are carried out, which indicates that computing solutions of the proposed MNI algorithm can globally converge to relative small value in the presence of various noise or zero noise conditions. Finally, to illustrate the advantages and feasibilities of the proposed MNI algorithm for ODNO problems, four numerical simulation examples and an application to robot manipulator motion generation are performed.
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References
Courvoisier, Y., Gander, M.J.: Optimization of Schwarz waveform relaxation over short time windows. Numer. Algor. 64(2), 221–243 (2013)
Jin, L., Zhang, Y.: Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization. Numer. Algor. 73(1), 115–140 (2016)
Qiu, B., Zhang, Y., Guo, J., Yang, Z., Li, X.: New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern. Numer. Algor. 81(3), 1043–1065 (2019)
Amini, K., Ahookhosh, M., Nosratipour, H.: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algor. 66(1), 49–78 (2014)
Xiao, X., Xiong, N.N., Lai, J., Wang, C., Sun, Z., Yan, J.: A local consensus index scheme for random-valued impulse noise detection systems. IEEE Trans. Syst. Man Cybern. Syst. In Press with https://doi.org/10.1109/TSMC.2019.2925886
Gammell, J.D., Barfoot, T.D., Srinivasa, S.S.: Informed sampling for asymptotically optimal path planning. IEEE Trans. Robot. 34(4), 966–984 (2018)
Xie, Z., Jin, L., Du, X., Xiao, X., Li, H., Li, S.: On generalized RMP scheme for redundant robot manipulators aided with dynamic neural networks and nonconvex bound constraints. IEEE Trans. Ind. Informat. 15(9), 5172–5181 (2019)
Zhang, Y., Li, S., Kadry, S., Liao, B.: Recurrent neural network for kinematic control of redundant manipulators with periodic input disturbance and physical constraints. IEEE Trans. Cybern. 49(12), 4194–4205 (2018)
Qi, Y., Jin, L., Wang, Y., Xiao, L., Zhang, J.: Complex-valued discrete-time neural dynamics for perturbed time-dependent complex quadratic programming with applications. IEEE Trans. Neural Netw. Learn. Syst. In Press with https://doi.org/10.1109/TNNLS.2019.2944992
Li, X., Rui, H., Chen, S.: A fully conservative block-centered finite difference method for simulating Darcy-Forchheimer compressible wormhole propagation. Numer. Algor. 82(2), 451–478 (2019)
Kumar, M., Aggarwal, A., Rawat, T., Parthasarathy, H.: Optimal nonlinear system identification using fractional delay second-order volterra system. IEEE/CAA J. Autom. Sinica. In Press with https://doi.org/10.1109/JAS.2016.7510184
Mustafa, A., Dhar, N.K., Verma, N.K.: Event-triggered sliding mode control for trajectory tracking of nonlinear systems. IEEE/CAA J. Autom. Sinica 7(1), 307–314 (2020)
Sandy, T., Stadelmann, L., Kerscher, S., Buchli, J.: Confusion: sensor fusion for complex robotic systems using nonlinear optimization. IEEE Robot Autom. Lett. 4(2), 1093–1100 (2019)
Mahmoodabadi, M.J., Mostaghim, S.A.: Stability of nonlinear systems using optimal fuzzy controllers and its simulation by Java programming. IEEE/CAA J. Autom. Sinica 6(6), 1519–1527 (2019)
Li, X., Shi, J., Dong, X., Yu, J.: A new conjugate gradient method based on quasi-newton equation for unconstrained optimization. Comput. Appl. Math. 350, 372–379 (2019)
Dehghani, R., Mahdavi-Amiri, N.: Scaled nonlinear conjugate gradient methods for nonlinear least squares problems. Numer. Algor. 82(1), 1–20 (2019)
Babaie-Kafaki, S., Ghanbari, R.: A modified scaled conjugate gradient method with global convergence for nonconvex functions. Bull. Belg. Math. Soc. Simon Stevin 21(3), 465–477 (2014)
Sun, Z., Li, H., Wang, J., Tian, Y.: Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization. Int. J. Comput. Math. 95(10), 2082–2099 (2018)
Raudys, S., Duin, R.P.W.: Expected classification error of the fisher linear classifier with pseudo-inverse covariance matrix. Pattern Recogn. Lett. 19(5–6), 385–392 (1998)
Qi, Y., Jin, L., Li, H., Li, Y., Liu, M.: Discrete computational neural dynamics models for solving time-dependent Sylvester equations with applications to robotics and MIMO systems. IEEE Trans. Ind. Informat. 16(10), 6231–6241 (2020)
Kong, L., He, W., Yang, C., Li, Z., Sun, C.: Adaptive fuzzy control for coordinated multiple robots with constraint using impedance learning. IEEE Trans Cybern. 49(8), 3052–3063 (2019)
Zhang, Y., Li, H., Sun, J., He, W.: Cooperative adaptive event-triggered control for multiagent systems with actuator failures. IEEE Trans. Syst. Man Cybern. Syst. 49(9), 1759–1768 (2018)
Yang, C., Chen, C., He, W., Cui, R., Li, Z.: Robot learning system based on adaptive neural control and dynamic movement primitives. IEEE Trans. Neural Netw. Learn. Syst. 30(3), 777–787 (2018)
Liao, S., Liu, J., Xiao, X., Fu, D., Wang, G., Jin, L.: Modified gradient neural networks for solving the time-varying Sylvester equation with adaptive coefficients and elimination of matrix inversion. Neurocomputing 379, 1–11 (2020)
Zerari, N., Chemachema, M., Essounbouli, N.: Neural network based adaptive tracking control for a class of pure feedback nonlinear systems with input saturation. IEEE/CAA J. Autom. Sinica 6(1), 278–290 (2019)
Myung, H., Kim, J.H.: Time-varying two-phase optimization and its application to neural-network learning. IEEE Trans. Neural Netw. 8(6), 1293–1300 (1997)
Jin, L., Zhang, Y.: Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation. IEEE Trans. Neural Netw. Learn. Syst. 26(7), 1525–1531 (2014)
Li, J., Mao, M., Uhlig, F., Zhang, Y.: Z-type neural-dynamics for time-varying nonlinear optimization under a linear equality constraint with robot application. J. Comput. Appl. Math. 327, 155–166 (2018)
Zhang, Y., He, L., Hu, C., Guo, J., Li, J., Shi, Y.: General four-step discrete-time zeroing and derivative dynamics applied to time-varying nonlinear optimization. J. Comput. Appl. Math. 347, 314–329 (2019)
Guo, D., Zhang, Y.: Neural dynamics and newton-raphson iteration for nonlinear optimization. J. Comput. Nonlinear Dyn. 9(2), 012016 (2014)
Guo, D., Yan, L., Nie, Z.: Design, analysis, and representation of novel five-step DTZD algorithm for time-varying nonlinear optimization. IEEE Trans. Neural Netw. Learn. Syst. 29(9), 4248–4260 (2017)
Zhang, Y., Li, Z., Guo, D., Ke, Z., Chen, P.: Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer. Algor. 64 (4), 721–740 (2013)
Wei, L., Jin, L., Yang, C., Chen, K., Li, W.: New noise-tolerant neural algorithms for future dynamic nonlinear optimization with estimation on hessian matrix inversion. IEEE Trans. Syst. Man Cybern. Syst. In Press with https://doi.org/10.1109/TSMC.2019.2916892
Jin, L., Zhang, Y.: Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization. Numer. Algor. 73(1), 115–140 (2016)
Guo, D., Lin, X., Su, Z., Sun, S., Huang, Z.: Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization. Numer. Algor. 77(1), 23–36 (2018)
Lu, H., Jin, L., Luo, X., Liao, B., Guo, D., Xiao, L.: RNN For solving perturbed time-varying underdetermined linear system with double bound limits on residual errors and state variables. IEEE Trans. Ind. Informat. 15(11), 5931–5942 (2019)
Martínez, J.M., Prudente, L.F.: Handling infeasibility in a large-scale nonlinear optimization algorithm. Numer. Algor. 60(2), 263–277 (2012)
Jin, L., Yan, J., Du, X., Xiao, X., Fu, D.: RNN for solving time-variant generalized Sylvester equation with applications to robots and acoustic source localization. IEEE Trans. Ind. Informat. 16(10), 6359–6369 (2020)
Yang, C., Wang, X., Li, Z., Li, Y., Su, C.Y.: Teleoperation control based on combination of wave variable and neural networks. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 2125–2136 (2017)
Zhang, J., Jin, L., Cheng, L.: RNN for perturbed manipulability optimization of manipulators based on a distributed scheme: a game-theoretic perspective. IEEE Trans. Neural Netw. Learn. Syst. In Press with https://doi.org/10.1109/TNNLS.2020.2963998
Xiao, L., Zhang, Y.: Dynamic design, numerical solution and effective verification of acceleration-level obstacle-avoidance scheme for robot manipulators. Int. J. Syst. Sci. 47(4), 932–945 (2016)
Funding
This work is supported by the Fund of Southern Marine Science and Engineering Guangdong Laboratory of Zhanjiang, China under Grant ZJW-2019-08, by the Key Projects of the Guangdong Education Department under Grant 2019KZDXM019, in part by the High-Level Marine Discipline Team Project of Guangdong Ocean University under Grant 002026002009, by the Guangdong Graduate Academic Forum Project under Grant 230420003, by the “First Class” Discipline Construction Platform Project in 2019 of Guangdong Ocean University under Grant 231419026, by the Innovation and Strength Project in Guangdong Province, China (Natural Science) under Grant 230419065, by the Key Lab of Digital Signal and Image Processing of Guangdong Province, China under Grant 2019GDDSIPL-01, by the Industry-University-Research Cooperation Education Project of Ministry of Education under Grant 201801328005, by the Guangdong Graduate Education Innovation Project, Graduate Summer School under Grant 2020SQXX19, by the Guangdong Graduate Education Innovation Project, Graduate Academic Forum under Grant 2020XSLT27, by the Doctoral Initiating Project of Guangdong Ocean University under Grant E13428, and also by the Special Project in Key Fields of Universities in Department of Education of Guangdong Province, China under Grant 2019033.
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Huang, H., Fu, D., Wang, G. et al. Modified Newton integration algorithm with noise suppression for online dynamic nonlinear optimization. Numer Algor 87, 575–599 (2021). https://doi.org/10.1007/s11075-020-00979-6
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DOI: https://doi.org/10.1007/s11075-020-00979-6