Differential optimization in finite-dimensional spaces

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October 2016, 12(4): 1495-1505. doi: 10.3934/jimo.2016.12.1495

Differential optimization in finite-dimensional spaces

Xing Wang ^1, , Chang-Qi Tao ^2, and Guo-Ji Tang ^3,

1.	School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China
2.	School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China
3.	Department of Mathematics, Guangxi University for Nationalities, Nanning 530006, China

Received December 2014 Revised July 2015 Published January 2016

Abstract
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In this paper, a class of optimization problems coupled with differential equations in finite dimensional spaces are introduced and studied. An existence theorem of a Carathéodory weak solution of the differential optimization problem is established. Furthermore, when both the mapping and the constraint set in the optimization problem are perturbed by two different parameters, the stability analysis of the differential optimization problem is considered. Finally, an algorithm for solving the differential optimization problem is established.

Keywords: stability analysis, convergence analysis., Differential optimization, Carathéodory weak solution, algorithm.

Mathematics Subject Classification: 49J4.

Citation: Xing Wang, Chang-Qi Tao, Guo-Ji Tang. Differential optimization in finite-dimensional spaces. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1495-1505. doi: 10.3934/jimo.2016.12.1495

References:

[1]	F. Archetti and F. Schen, A survey on the global optimization problem: General theory and computational approaches, Annals of Operations Research, 1 (1984), 87-110. doi: 10.1007/BF01876141.
[2]	W. Behrman, An Efficient Gradient Flow Method for Unconstrained Optimization, PhD thesis, Stanford University, 1998.
[3]	J. F. Bonnans and A. Shapiro, Perturbation Analyisis of Optimization Problems, Springer-Verlag New York Inc., 2000. doi: 10.1007/978-1-4612-1394-9.
[4]	T. D. Chuong, N. Q. Huy and J. C. Yao, Stability of semi-infinite vector optimization problems under functional perturbations, Journal of Global Optimization, 45 (2009), 583-595. doi: 10.1007/s10898-008-9391-x.
[5]	W. R. Esposito and C. A. Floudas, Deterministic global optimization in nonlinear optimal control problems, Journal of Global Optimization, 17 (2000), 97-126. doi: 10.1023/A:1026578104213.
[6]	Z. G. Feng and K. F. C. Yiu, Manifold relaxations for integer programming, Journal of Industrial and Managemnt Optimization, 10 (2014), 557-566. doi: 10.3934/jimo.2014.10.557.
[7]	Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, Journal of Mathematical Analysis and Applications, 330 (2007), 352-363. doi: 10.1016/j.jmaa.2006.07.063.
[8]	X. Q. Hua and N. Yamashita, An inexact coordinate descent method for the weighted $l_1$-regularized convex optimization problem, Pacific Journal of Optimization, 9 (2013), 567-594.
[9]	N. Q. Huy and J. C. Yao, Semi-infinite optimization under convex function perturbations: Lipschitz stability, Journal of Optimization Theory and Application, 148 (2011), 237-256. doi: 10.1007/s10957-010-9753-7.
[10]	P. Q. Khanh, L. J. Lin and V. S. T. Long, On topological existence theorems and applications to optimization-related problems, Mathematical Method of Operations Research, 79 (2014), 253-272. doi: 10.1007/s00186-014-0462-0.
[11]	G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems, Journal of Global Optimization, 32 (2005), 597-612. doi: 10.1007/s10898-004-2696-5.
[12]	C. Y. Liu, Z. H. Gong and E. M. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pacific Journal of Optimization, 9 (2013), 595-612.
[13]	J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal investment-consumption problem with constraint, Journal of Industrial and Management Optimization, 9 (2013), 743-768. doi: 10.3934/jimo.2013.9.743.
[14]	J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints, Discrete and Continuous Dynamical Systems, 18 (2013), 1889-1907. doi: 10.3934/dcdsb.2013.18.1889.
[15]	Y. F. Liu, F. L. Wu and K. L. Teo, Conceptual study on applying optimal control theory for generator bidding in power markets, Automation of Electric Power Systems, 29 (2005), 1-6.
[16]	A. Nagurney, J. Pan and L. Zhao, Human migration networks, European Journal of Operational Research, 59 (1992), 262-274. doi: 10.1016/0377-2217(92)90140-5.
[17]	J. S. Pang and D. E. Stewart, Differential variational inequalities, Mathematical Programming Series A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.
[18]	I. Papamichail and C. S. Adjiman, A rigorous global optimization algorithm for problems with ordinary differential equations, Journal of Global Optimization, 24 (2002), 1-33. doi: 10.1023/A:1016259507911.
[19]	D. Preda and J. Noailles, Mixed integer programming for a special logic constrained optimal control problem, Mathematical Programming, 103 (2005), 309-333. doi: 10.1007/s10107-005-0584-5.
[20]	A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin and L. T. Biegler, Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Annals of Operations Research, 148 (2006), 251-270. doi: 10.1007/s10479-006-0086-8.
[21]	S. Sager, H. G. Bock and G. Reinelt, Direct methods with maximal lower bound for mixed-integer optimal control problems, Mathematical Programming, 118 (2009), 109-149. doi: 10.1007/s10107-007-0185-6.
[22]	A. B. Singer and P. I. Barton, Global optimization with nonlinear ordinary differential equations, Journal of Global Optimization, 34 (2006), 159-190. doi: 10.1007/s10898-005-7074-4.
[23]	A. B. Singer and P. I. Barton, Global solution of linear dynamic embedded optimization problems, Journal of Optimization Theory and Applications, 121 (2004), 613-646. doi: 10.1023/B:JOTA.0000037606.79050.a7.
[24]	S. Wang, X. Q. Yang and K. L. Teo, A unified gradient flow approach to constrained nonlinear optimization problems, Computational Optimization and Applications, 25 (2003), 251-268. doi: 10.1023/A:1022973608903.
[25]	L. Yang, Y. P. Chen and X. J. Tong, A note on local sensitivity analysis for parametric optimization problem, Pacific Journal of Optimization, 8 (2012), 185-195.
[26]	K. F. C. Yiu, W. Y. Yan, K. L. Teo and S. Y. Low, A new hybrid descent method with application to the optimal design of finite precision FIR filters, Optimization Methods and Software, 25 (2010), 725-735. doi: 10.1080/10556780903254104.
[27]	J. Zeng, S. J. Li, W. Y. Zhang and X. W. Xue, Stability results for convex vector-valued optimization problems, Positivity, 15 (2011), 441-453. doi: 10.1007/s11117-010-0093-5.
[28]	X. G. Zhou and B. Y. Cao, New global optimality conditions for cubic minimization subject to box or bivalent constraint, Pacific Journal of Optimization, 8 (2012), 631-647.
[29]	L. Zhu and F. Q. Xia, Scalarization method for Levitin-Polyak well-posedness of vectorial optimization problems, Mathematical Method of Operations Research, 76 (2012), 361-375. doi: 10.1007/s00186-012-0410-9.

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References:

[1]	F. Archetti and F. Schen, A survey on the global optimization problem: General theory and computational approaches, Annals of Operations Research, 1 (1984), 87-110. doi: 10.1007/BF01876141.
[2]	W. Behrman, An Efficient Gradient Flow Method for Unconstrained Optimization, PhD thesis, Stanford University, 1998.
[3]	J. F. Bonnans and A. Shapiro, Perturbation Analyisis of Optimization Problems, Springer-Verlag New York Inc., 2000. doi: 10.1007/978-1-4612-1394-9.
[4]	T. D. Chuong, N. Q. Huy and J. C. Yao, Stability of semi-infinite vector optimization problems under functional perturbations, Journal of Global Optimization, 45 (2009), 583-595. doi: 10.1007/s10898-008-9391-x.
[5]	W. R. Esposito and C. A. Floudas, Deterministic global optimization in nonlinear optimal control problems, Journal of Global Optimization, 17 (2000), 97-126. doi: 10.1023/A:1026578104213.
[6]	Z. G. Feng and K. F. C. Yiu, Manifold relaxations for integer programming, Journal of Industrial and Managemnt Optimization, 10 (2014), 557-566. doi: 10.3934/jimo.2014.10.557.
[7]	Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, Journal of Mathematical Analysis and Applications, 330 (2007), 352-363. doi: 10.1016/j.jmaa.2006.07.063.
[8]	X. Q. Hua and N. Yamashita, An inexact coordinate descent method for the weighted $l_1$-regularized convex optimization problem, Pacific Journal of Optimization, 9 (2013), 567-594.
[9]	N. Q. Huy and J. C. Yao, Semi-infinite optimization under convex function perturbations: Lipschitz stability, Journal of Optimization Theory and Application, 148 (2011), 237-256. doi: 10.1007/s10957-010-9753-7.
[10]	P. Q. Khanh, L. J. Lin and V. S. T. Long, On topological existence theorems and applications to optimization-related problems, Mathematical Method of Operations Research, 79 (2014), 253-272. doi: 10.1007/s00186-014-0462-0.
[11]	G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems, Journal of Global Optimization, 32 (2005), 597-612. doi: 10.1007/s10898-004-2696-5.
[12]	C. Y. Liu, Z. H. Gong and E. M. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pacific Journal of Optimization, 9 (2013), 595-612.
[13]	J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal investment-consumption problem with constraint, Journal of Industrial and Management Optimization, 9 (2013), 743-768. doi: 10.3934/jimo.2013.9.743.
[14]	J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints, Discrete and Continuous Dynamical Systems, 18 (2013), 1889-1907. doi: 10.3934/dcdsb.2013.18.1889.
[15]	Y. F. Liu, F. L. Wu and K. L. Teo, Conceptual study on applying optimal control theory for generator bidding in power markets, Automation of Electric Power Systems, 29 (2005), 1-6.
[16]	A. Nagurney, J. Pan and L. Zhao, Human migration networks, European Journal of Operational Research, 59 (1992), 262-274. doi: 10.1016/0377-2217(92)90140-5.
[17]	J. S. Pang and D. E. Stewart, Differential variational inequalities, Mathematical Programming Series A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.
[18]	I. Papamichail and C. S. Adjiman, A rigorous global optimization algorithm for problems with ordinary differential equations, Journal of Global Optimization, 24 (2002), 1-33. doi: 10.1023/A:1016259507911.
[19]	D. Preda and J. Noailles, Mixed integer programming for a special logic constrained optimal control problem, Mathematical Programming, 103 (2005), 309-333. doi: 10.1007/s10107-005-0584-5.
[20]	A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin and L. T. Biegler, Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Annals of Operations Research, 148 (2006), 251-270. doi: 10.1007/s10479-006-0086-8.
[21]	S. Sager, H. G. Bock and G. Reinelt, Direct methods with maximal lower bound for mixed-integer optimal control problems, Mathematical Programming, 118 (2009), 109-149. doi: 10.1007/s10107-007-0185-6.
[22]	A. B. Singer and P. I. Barton, Global optimization with nonlinear ordinary differential equations, Journal of Global Optimization, 34 (2006), 159-190. doi: 10.1007/s10898-005-7074-4.
[23]	A. B. Singer and P. I. Barton, Global solution of linear dynamic embedded optimization problems, Journal of Optimization Theory and Applications, 121 (2004), 613-646. doi: 10.1023/B:JOTA.0000037606.79050.a7.
[24]	S. Wang, X. Q. Yang and K. L. Teo, A unified gradient flow approach to constrained nonlinear optimization problems, Computational Optimization and Applications, 25 (2003), 251-268. doi: 10.1023/A:1022973608903.
[25]	L. Yang, Y. P. Chen and X. J. Tong, A note on local sensitivity analysis for parametric optimization problem, Pacific Journal of Optimization, 8 (2012), 185-195.
[26]	K. F. C. Yiu, W. Y. Yan, K. L. Teo and S. Y. Low, A new hybrid descent method with application to the optimal design of finite precision FIR filters, Optimization Methods and Software, 25 (2010), 725-735. doi: 10.1080/10556780903254104.
[27]	J. Zeng, S. J. Li, W. Y. Zhang and X. W. Xue, Stability results for convex vector-valued optimization problems, Positivity, 15 (2011), 441-453. doi: 10.1007/s11117-010-0093-5.
[28]	X. G. Zhou and B. Y. Cao, New global optimality conditions for cubic minimization subject to box or bivalent constraint, Pacific Journal of Optimization, 8 (2012), 631-647.
[29]	L. Zhu and F. Q. Xia, Scalarization method for Levitin-Polyak well-posedness of vectorial optimization problems, Mathematical Method of Operations Research, 76 (2012), 361-375. doi: 10.1007/s00186-012-0410-9.

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