Two methods based on second-order dynamical systems for solving a special class of nonlinear optimization problems

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Abstract

Symbolic functions, as an important class of nonsmooth functions, play a key role in many fields such as system control and optimization theory. In this study, the special case of nonlinear optimization problems with sign function constraints is discussed in depth. Based on the smooth approximation theory, the study first adopts an approximate substitution method to smooth the nonsmooth constraints and then transforms them into differentiable optimization problems, thus effectively avoiding the numerical computational difficulties caused by the sign function. Second, the constrained optimization problem is transformed into an unconstrained optimization problem by constructing an exact penalty function. Third, two forms of second-order dynamical systems are established to solve the problem, and a rigorous theoretical analysis of the stability of these systems is conducted. Finally, the convergence and computational efficiency of the proposed method are verified by numerical simulation experiments of the system, and the simulation results fully prove the effectiveness and practicality of the algorithm.

Keywords

symbolic functions / nonlinear optimization problems / smooth approximation / penalty function method / second-order dynamical systems

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References

[1] A. Bensoussan, J. Lions, Optimal impulse and continuous control: Method of non linear quasi-variational inequalities, Trudy Mat. Inst. Steklov, 134 (1975), 5-22.

[2] W. Gong, Y. Ho, Smoothed (conditional) perturbation analysis of discrete event dynamical systems, IEEE Trans. Autom. Control, 32 (1987), 858-866. http://doi.org/10.1109/TAC.1987.1104464

[3] R. Rubinstein, How to optimize discrete-event systems from a single sample path by the score function method, Ann. Oper. Res., 27 (1990), 175-212. https://doi.org/10.1007/BF02055195

[4] J. Zhang, L. Zhao, A class of optimization problems with discontinuous constraints and applications, J. Xiamen Univ. Nat. Sci., 49 (2010), 16-19.

[5] A. M. Gupal, V. I. Norkin, Algorithm for the minimization of discontinuous functions, Cybernetics, 13 (1977), 220-223. http://doi.org/10.1007/BF01073313

[6] Y. Ermoliev, V. Norkin, On constrained discontinuous optimization, in Stochastic Programming Methods and Technical Applications, 458 (1998), 128-144. https://doi.org/10.1007/978-3-642-45767-87

[7] V. Y. Katkovnik, O. Y. Kul’chitskii, Convergence of one class of random search algorithms, Autom. Remote Control, 33 (1972), 1321-1326.

[8] J. Warga, Necessary conditions without differentiability assumptions in optimal control, J. Differ. Equations, 18 (1975), 41-62. http://doi.org/10.1016/0022-0396(75)90080-7

[9] V. D. Batukhtin, An approach to the solution of discontinuous extremal problems, J. Comput. Syst. Sci. Int., 33 (1995), 30-38.

[10] Y. M. Ermoliev, V. I. Norkin, R. J. Wets, The minimization of semicontinuous functions: molliffer subgradients, SIAM J. Control Optim., 33 (1995), 149-167. http://doi.org/10.1137/S0363012992238369

[11] H. T. Jongen, O. Stein, Smoothing by molliffers. Part I: Semi-infinite optimization, J. Glob. Optim., 41 (2008), 319-334. http://doi.org/10.1007/s10898-007-9232-3

[12] H. T. Jongen, O. Stein, Smoothing by molliffers. Part II: Nonlinear optimization, J. Glob. Optim., 41 (2008), 335-350. http://doi.org/10.1007/s10898-007-9231-4

[13] Y. Cui, J. Liu, J. Pang, The minimization of piecewise functions: Pseudo stationarity, preprint, arXiv:2305.14798.

[14] B. W. Kort, D. P. Bertsekas, A new penalty function method for constrained minimization, in Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes, (1972), 162-166. http://doi.org/10.1109/CDC.1972.268971

[15] Ö. Yeniay, Penalty function methods for constrained optimization with genetic algorithms, Math. Comput. Appl., 10 (2005), 45-56. http://doi.org/10.3390/mca10010045

[16] Q. Wu, A Bundle Methods for A Class of Nonconvex and Nonsmooth Optimization Problems and Applications, Ph.D. Thesis, Dalian University of Technology in Dalian, 2022. http://doi.org/10.26991/d.cnki.gdllu.2021.003891

[17] J. Sun, W. Fu, J. H. Alcantara, J. Chen, A neural network based on the metric projector for solving soccvi problem, IEEE Trans. Neural Networks Learn. Syst., 32 (2021), 2886-2900. http://doi.org/10.1109/TNNLS.2020.3008661

[18] H. Attouch, J. Peypouquet, P. Redont, Fast convex optimization via inertial dynamics with hessian driven damping, J. Differ. Equations, 261 (2016), 5734-5783. http://doi.org/10.1016/j.jde.2016.08.020

[19] H. Attouch, Z. Chbani, H. Riahi, Fast convex optimization via time scaling of damped inertial gradient dynamics, 2019. Available from: https://hal.science/hal-02138954.

[20] A. Cabot, H. Engler, S. Gadat, On the long time behavior of second order differential equations with asymptotically small dissipation, Trans. Am. Math. Soc., 361 (2009), 5983-6017. http://doi.org/10.1090/S0002-9947-09-04785-0

[21] A. Cabot, P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differ. Equations, 252 (2012), 294-322. http://doi.org/10.1016/j.jde.2011.09.012