# American Institute of Mathematical Sciences

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

## Differential optimization in finite-dimensional spaces

 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

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.
Citation: Xing Wang, Chang-Qi Tao, Guo-Ji Tang. Differential optimization in finite-dimensional spaces. Journal of Industrial & 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. doi: 10.1007/BF01876141. Google Scholar [2] W. Behrman, An Efficient Gradient Flow Method for Unconstrained Optimization,, PhD thesis, (1998). Google Scholar [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. Google Scholar [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. doi: 10.1007/s10898-008-9391-x. Google Scholar [5] W. R. Esposito and C. A. Floudas, Deterministic global optimization in nonlinear optimal control problems,, Journal of Global Optimization, 17 (2000), 97. doi: 10.1023/A:1026578104213. Google Scholar [6] Z. G. Feng and K. F. C. Yiu, Manifold relaxations for integer programming,, Journal of Industrial and Managemnt Optimization, 10 (2014), 557. doi: 10.3934/jimo.2014.10.557. Google Scholar [7] Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces,, Journal of Mathematical Analysis and Applications, 330 (2007), 352. doi: 10.1016/j.jmaa.2006.07.063. Google Scholar [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. Google Scholar [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. doi: 10.1007/s10957-010-9753-7. Google Scholar [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. doi: 10.1007/s00186-014-0462-0. Google Scholar [11] G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems,, Journal of Global Optimization, 32 (2005), 597. doi: 10.1007/s10898-004-2696-5. Google Scholar [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. Google Scholar [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. doi: 10.3934/jimo.2013.9.743. Google Scholar [14] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints,, Discrete and Continuous Dynamical Systems, 18 (2013), 1889. doi: 10.3934/dcdsb.2013.18.1889. Google Scholar [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. Google Scholar [16] A. Nagurney, J. Pan and L. Zhao, Human migration networks,, European Journal of Operational Research, 59 (1992), 262. doi: 10.1016/0377-2217(92)90140-5. Google Scholar [17] J. S. Pang and D. E. Stewart, Differential variational inequalities,, Mathematical Programming Series A, 113 (2008), 345. doi: 10.1007/s10107-006-0052-x. Google Scholar [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. doi: 10.1023/A:1016259507911. Google Scholar [19] D. Preda and J. Noailles, Mixed integer programming for a special logic constrained optimal control problem,, Mathematical Programming, 103 (2005), 309. doi: 10.1007/s10107-005-0584-5. Google Scholar [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. doi: 10.1007/s10479-006-0086-8. Google Scholar [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. doi: 10.1007/s10107-007-0185-6. Google Scholar [22] A. B. Singer and P. I. Barton, Global optimization with nonlinear ordinary differential equations,, Journal of Global Optimization, 34 (2006), 159. doi: 10.1007/s10898-005-7074-4. Google Scholar [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. doi: 10.1023/B:JOTA.0000037606.79050.a7. Google Scholar [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. doi: 10.1023/A:1022973608903. Google Scholar [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. Google Scholar [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. doi: 10.1080/10556780903254104. Google Scholar [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. doi: 10.1007/s11117-010-0093-5. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00186-012-0410-9. Google Scholar

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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. doi: 10.1007/BF01876141. Google Scholar [2] W. Behrman, An Efficient Gradient Flow Method for Unconstrained Optimization,, PhD thesis, (1998). Google Scholar [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. Google Scholar [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. doi: 10.1007/s10898-008-9391-x. Google Scholar [5] W. R. Esposito and C. A. Floudas, Deterministic global optimization in nonlinear optimal control problems,, Journal of Global Optimization, 17 (2000), 97. doi: 10.1023/A:1026578104213. Google Scholar [6] Z. G. Feng and K. F. C. Yiu, Manifold relaxations for integer programming,, Journal of Industrial and Managemnt Optimization, 10 (2014), 557. doi: 10.3934/jimo.2014.10.557. Google Scholar [7] Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces,, Journal of Mathematical Analysis and Applications, 330 (2007), 352. doi: 10.1016/j.jmaa.2006.07.063. Google Scholar [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. Google Scholar [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. doi: 10.1007/s10957-010-9753-7. Google Scholar [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. doi: 10.1007/s00186-014-0462-0. Google Scholar [11] G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems,, Journal of Global Optimization, 32 (2005), 597. doi: 10.1007/s10898-004-2696-5. Google Scholar [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. Google Scholar [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. doi: 10.3934/jimo.2013.9.743. Google Scholar [14] J. Z. Liu and K. F. C. Yiu, Optimal stochastic differential games with var constraints,, Discrete and Continuous Dynamical Systems, 18 (2013), 1889. doi: 10.3934/dcdsb.2013.18.1889. Google Scholar [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. Google Scholar [16] A. Nagurney, J. Pan and L. Zhao, Human migration networks,, European Journal of Operational Research, 59 (1992), 262. doi: 10.1016/0377-2217(92)90140-5. Google Scholar [17] J. S. Pang and D. E. Stewart, Differential variational inequalities,, Mathematical Programming Series A, 113 (2008), 345. doi: 10.1007/s10107-006-0052-x. Google Scholar [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. doi: 10.1023/A:1016259507911. Google Scholar [19] D. Preda and J. Noailles, Mixed integer programming for a special logic constrained optimal control problem,, Mathematical Programming, 103 (2005), 309. doi: 10.1007/s10107-005-0584-5. Google Scholar [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. doi: 10.1007/s10479-006-0086-8. Google Scholar [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. doi: 10.1007/s10107-007-0185-6. Google Scholar [22] A. B. Singer and P. I. Barton, Global optimization with nonlinear ordinary differential equations,, Journal of Global Optimization, 34 (2006), 159. doi: 10.1007/s10898-005-7074-4. Google Scholar [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. doi: 10.1023/B:JOTA.0000037606.79050.a7. Google Scholar [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. doi: 10.1023/A:1022973608903. Google Scholar [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. Google Scholar [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. doi: 10.1080/10556780903254104. Google Scholar [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. doi: 10.1007/s11117-010-0093-5. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00186-012-0410-9. Google Scholar
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