Modified ADMM algorithm for solving proximal bound formulation of multi-delay optimal control problem with bounded control

Document Type : Research Article

Author

Department of Mathematical Sciences, The Federal University of technology Akure, Ondo-State, Nigeria.

Abstract

 This study presents an algorithm for solving optimal control problems with the objective function of the Lagrange-type and multiple delays on both the state and control variables of the constraints, with bounds on the control variable. The full discretization of the objective functional and the multiple delay constraints is carried out by using the Simpson numerical scheme. The discrete recurrence relations generated from the discretization of both the objective functional and constraints are used to develop the matrix operators, which satisfy the basic spectral properties. The primal-dual residuals of the algorithm are derived in order to ascertain the rate of convergence of the algorithm, which performs faster when relaxed with an accelerator variant in the sense of Nesterov. The direct numerical approach for handling the multi-delay control problem is observed to obtain an accurate result at a faster rate of convergence when over-relaxed with an accelerator variant. This research problem is limited to linear constraints and objective functional of the Lagrange-type and can address real-life models with multiple delays as applicable to quadratic optimization of intensity modulated radiation theory planning. The novelty of this research paper lies in the method of discretization and its adaptation to handle linearly and proximal bound-constrained program formulated from the multiple delay optimal control problems.

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1. Bashier, E.B.M. and Patidar, K.C. Optimal control of an epidemiological model with multiple time delays, Appl. Math. Comput. 292 (2017), 47–56.
2. Boley, D.
Local linear convergence of the alternating direction method of multipliers on quadratic or linear programs, SIAM J. Optim. 23(4) (2013), 2183–2207.
3. Boyd, S., Parikh, N., Chu, E. and Eckstein, J.
Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3(1) (2011), 11–22.
4. Boyd, S. and Vandenberghe, L.
Convex Optimization, Cambridge University Press, NY, 2004.
5. Carreira-Perpi
n˜a´n, M.A. An ADMM algorithm for solving a proximal bound-constrained quadratic program, arXive:1412.8493[math.OC], 2014.
6. Combettes, P. and Pesquet, J.
Proximal splitting methods in signal processing, Fixed-point algorithms for inverse problems in science and engineering, 185–212, Springer Optim. Appl., 49, Springer, New York, 2011.
7. Eckstein, J. and Ferris, M.
Operator-splitting methods for monotone affne variational inequalities, with a parallel application to optimal control, INFORMS J. Comput. 10(2) (1998), 218–235.
8. Ghadimi, E., Teixeira, A., Shames, I. and Johansson, M.
Optimal Parameter Selection for the Alternating Direction Method of Multipliers (ADMM): Quadratic Problems, Linnaeus center, Elect. Eng., Royal Inst. of Technology, Sidney, 2014.
9. Gollman, L., Kern, D. and Maurer, H.
Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Appl. Methods 30(4) (2009), 341–365.
10. Gollman, L. and Maurer, H. Theory and Applications of Optimal Control Problems with Multiple Delays, J. Ind. Manag. Optim. 10 (2014), 413–441.
11. Guinn, T.
Reduction of delayed optimal control problems to nondelayed problems, J. Optim. Theory Appl. 18 (1976), 371–377.
12. He, B., Hong, X. and Yuan, X.
On the proximal Jacobian decomposition of ALM for multiple-block separable convex minimization problems and its relationship to ADMM, J. Sci. Comput. 66 (2016), 1204–1217.
13. He, B. and Yuan, X.
Block-wise alternating direction method of multipliers for multiple-block convex programming and beyond, SSMAI J. Comput. Math. 691 (2015), 145–174.
14. Laarabi, H., Abta, A. and Hattaf, K.
Optimal control of a delayed SIRS epidemic model with vaccination and treatment, Acta Biotheor, 63 (2015), 87–97.
15. Moreau, J.J.
Fonctions convexes duales et points proximaux dans un espace hilbertien, (French) C. R. Acad. Sci. Paris 255 (1962), 2897–2899.
16. Nesterov, Y.E.
A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543–547.
17. Olotu, O. and Dawodu, K.A.
On the discretized algorithm for optimal proportional control constrained by delay differential equation, Journal of Mathematical Theory and Modelling by IISTE, 3(8) (2013), 157–169.
18. Olotu, O. and Dawodu, K.A.
Numerical solution to one-dimensional multi-delay system using the modified alternating direction method of multipliers, Journal of the Nigerian Association of Mathematical Physics, Nigeria. 50 (2019), 107–116.
19. Olotu, O., Dawodu, K.A. and Yusuf, T.T.
Algorithm for solving multi-delay optimal control problems using modified alternating direction method of multipliers, J. Appl. Computat. Math. 8 (3) (2019), 1000445.
20. Pasquale, L.D. and Gerardo, T.
Quadratic programming with bound constraints, Encyclopedia of Optimization, USA, 2009.
21. Rihan, F.A. and Anwar, M.N.
Qualitative analysis of delayed SIR epidemic model with a saturated incidence rate, Int. J. Differ. Equ. 2012, Art. ID 408637, 13 pp.
22. Rockafellar, R.T.
Monotone operators and the proximal point algorithm, SIAM J. Control and Optim., 14 (5), (1976), 877–898.
23. Sun, D., Toh, K.C. and Yang, L. A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-type of constraints, SIAM J. Optim., 25(2) (2015) 882–915.
24. Voglis, C. and Lagaris, I.E.
Boxcqp: An algorithm for bound constrained convex quadratic problems 1st IC-SCCE, Athens, 2004.
25. Wu, D. and Shang, Y.
Complete solutions to general box-constrained global optimization problems, J. Appl. Math. 2011, Art. ID 478608, 17 pp.
26. Xinmin, L., Andrew, H.B., Zachary, G. and Rodney, D.
Constrained quadratic optimization for radiation treatment planning, American Control Conference (ACC), Boston, USA, 2016.
27. Xue, Y. and Duan, X.
Dynamic analysis of an SIR epidemic model with nonlinear incidence rate and double delays, Intl. J. Inf. Syst. Sci. Computer Inf. 7 (2011), 92–102.
28. Yao, Y., Zhu, X., Dong, H., Wu, S., Wu, H., Carol Tong, L. and Zhou, X.
ADMM-based problem decomposition scheme for vehicle routing problem with time windows, Transp. Res. B: Meth. 129, (2019), 156–174.
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