一类多参数不确定切换系统的分布鲁棒最优控制
Distributionally Robust Optimal Control of a Class of Switched Systems with Multiple Uncertain Parameters
摘要: 分布鲁棒优化结合了传统随机规划和鲁棒优化的优点，成为了近些年来解决含有不确定信息的优化问题的研究热点。结合分布鲁棒优化的特点，本文考虑了不确定参数的分布是未知，但一阶矩和二阶矩已知的切换系统分布鲁棒最优控制问题。该问题本质上是一个极小极大最优控制问题。本文利用控制参数化将原问题近似为有限维的参量最优控制问题，并将内层问题和外层问题分别求解。内层用非线性规划方法进行求解，外层则通过构造竞争粒子群算法进行求解。并以一类微生物批式流加发酵分布鲁棒最优控制问题作为实例验证了算法的有效性。
Abstract: Distributionally robust programming has become a hotspot to solve optimization problems with uncertain information in recent years, which combines the advantages of traditional stochastic programming with robust optimization. Motivated by the feature of distributionally robust pro-gramming, this paper considers a distributionally robust optimal control problem of switched sys-tems with uncertain parameters, in which the exact distributions of the uncertain parameters are unknown but the information of the first-order moment and the second-order moment are known. The problem is a minimax optimal control problem in essential, which is approximated by a fi-nite-dimensional parameter optimization problem via the control parametrization method. Then, we solve inner subproblem and outer subproblem separately, where the inner subproblem is solved by the nonlinear programming technique and the outer subproblem is solved by a competi-tive particle swarm optimization algorithm. Finally, the effectiveness of algorithm is verified by a distributionally robust optimal control problem of a microbialfed-batch process.
文章引用：王佳, 张政颖. 一类多参数不确定切换系统的分布鲁棒最优控制[J]. 应用数学进展, 2022, 11(12): 9072-9080. https://doi.org/10.12677/AAM.2022.1112957

参考文献

[1] Heitsch, H. and Römisch, W. (2003) Scenario Reduction Algorithms in Stochastic Programming. Computational Opti-mization and Applications, 24, 187-206. [Google Scholar] [CrossRef] 
[2] Shapiro, A. and Homem-de-Mello, T. (2000) On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Sto-chastic Programs. SIAM Journal on Optimization, 11, 70-86. [Google Scholar] [CrossRef] 
[3] Chen, M., Mehrotra, S. and Papp, D. (2015) Scenario Genera-tion for Stochastic Optimization Problems via the Sparse Grid Method. Computational Optimization and Applications, 62, 669-692. [Google Scholar] [CrossRef] 
[4] Scarf, H. (1958) A Min-Max Solution of an Inven-tory Problem. Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, 201-209.
[5] Bertsimas, D. and Popescu, I. (2005) Optimal Inequalities in Probability Theory: A Convex Optimization Approach. SIAM Journal on Optimization, 15, 780-804. [Google Scholar] [CrossRef] 
[6] Bertsimas, D., Natarajan, K. and Teo, C.P. (2004) Probabilistic Combinatorial Optimization: Moments, Semidefinite Programming, and Asymptotic Bounds. SIAM Journal on Optimi-zation, 15, 185-209. [Google Scholar] [CrossRef] 
[7] Bertsimas, D., Natarajan, K. and Teo, C.P. (2006) Persistence in Discrete Optimization under Data Uncertainty. Mathematical Programming, 108, 251-274. [Google Scholar] [CrossRef] 
[8] Delage, E. and Ye, Y. (2010) Distributionally Robust Optimiza-tion under Moment Uncertainty with Application to Data- Driven Problems. Operations Research, 58, 595-612. [Google Scholar] [CrossRef] 
[9] Wiesemann, W., Kuhn, D. and Sim, M. (2014) Distributionally Ro-bust Convex Optimization. Operations Research, 62, 1358-1376. [Google Scholar] [CrossRef] 
[10] Gibbs, A.L. and Su, F.E. (2002) On Choosing and Bounding Proba-bility Metrics. International Statistical Review, 70, 419-435. [Google Scholar] [CrossRef] 
[11] Mohajerin Esfahani, P. and Kuhn, D. (2018) Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformula-tions. Mathematical Programming, 171, 115-166. [Google Scholar] [CrossRef] 
[12] Ye, J., Wang, L., Wu, C., Sun, J., Teo, K.L. and Wang, X. (2016) A Robust Optimal Control Problem with Moment Constraints on Distribution: Theoretical Analysis and an Algorithm. arXiv:1611.07708.
[13] Wang, J., Xu, H., Teo, K.L., Sun, J. and Ye, J. (2020) Mixed-Integer Minimax Dynamic Opti-mization for Structure Identification of Glycerol Metabolic Network. Applied Mathematical Modelling, 82, 503-520. [Google Scholar] [CrossRef] 

为你推荐