Solving convex uncertain PDE-constrained multi-dimensional fractional control problems via a new approach. (English) Zbl 1540.49033
Summary: In this paper, the class of uncertain multi-dimensional fractional control problems with the first-order PDE constraints is investigated. The robust approach and the parametric method are applied for solving such control problems. Then, robust optimality is analyzed for the considered PDE-constrained multi-dimensional fractional control problem with uncertainty. Further, the exact absolute penalty function method is used for solving control problems created in both the aforementioned approaches. Then, under appropriate convexity hypotheses, exactness of the penalization of this exact penalty function method is investigated in the case when it is used for solving the considered control problem with uncertainty. Further, an algorithm based on the used method is presented, the main goal of which is to illustrate the precise steps to solve the unconstrained multi-dimensional non-fractional control problem with uncertainty associated with the constrained fractional control problem.
MSC:
| 49M37 | Numerical methods based on nonlinear programming |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 90C17 | Robustness in mathematical programming |
| 90C25 | Convex programming |
| 90C32 | Fractional programming |
| 90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |
Keywords:
exact absolute penalty function method; parametric method; PDE-constrained multi-dimensional fractional control problem with uncertainty; robust optimization; uncertainty modelingReferences:
| [1] | Lara, P.; Stancu-Minasian, I., Fractional programming: a tool for the assessment of sustainability, Agric Syst, 62, 131-141, 1999 · doi:10.1016/S0308-521X(99)00062-1 |
| [2] | Pitea, A.; Udriste, C.; Mititelu, S., New type dualities in PDI and PDE constrained optimization problems, J Adv Math Stud, 2, 81-91, 2009 · Zbl 1177.49054 |
| [3] | Stancu, AM, Mathematical programming with type-I functions, 2013, Bucharest: Matrix Rom, Bucharest |
| [4] | Stancu-Minasian, M., Fractional Programming, 1997, Theory, Methods and Applications: Academic Publishers, Dordrecht, Theory, Methods and Applications · Zbl 0899.90155 · doi:10.1007/978-94-009-0035-6 |
| [5] | Stancu-Minasian, IM, A seventh bibliography of fractional programming, Adv Model Optim, 15, 309-386, 2013 · Zbl 1413.90282 |
| [6] | Stancu-Minasian, IM, An eighth bibliography of fractional programming, Optimization, 3, 439-470, 2017 · Zbl 1365.90249 · doi:10.1080/02331934.2016.1276179 |
| [7] | Stancu-Minasian, IM, A ninth bibliography of fractional programming, Optimization, 11, 2123-2167, 2019 · Zbl 1431.90156 |
| [8] | Treanţă, S.; Agarwal, D.; Sachdev, G., Robust efficiency conditions in multiple-objective fractional variational control problems, Fractal Fract, 7, 18, 2022 · doi:10.3390/fractalfract7010018 |
| [9] | Ben-Tal, A.; Nemirovski, A., Robust convex optimization, Math Oper Res, 23, 769-805, 1998 · Zbl 0977.90052 · doi:10.1287/moor.23.4.769 |
| [10] | Baranwal, A.; Jayswal, A.; Kardam, P., Robust duality for the uncertain multitime control optimization problems, Int J Robust Nonlinear Control, 32, 5837-5847, 2022 · Zbl 1530.49032 · doi:10.1002/rnc.6113 |
| [11] | Jayswal, A.; Baranwal, A., Robust approach for uncertain multi-dimensional fractional control optimization problems, Bull Malays Math Sci Soc, 46, 1-17, 2023 · Zbl 1522.90221 · doi:10.1007/s40840-023-01469-3 |
| [12] | Minh, VT; Afzulpurkar, N., Robust model predictive control for input saturated and softened state constraints, Asian J Control, 7, 319-325, 2005 · doi:10.1111/j.1934-6093.2005.tb00241.x |
| [13] | Kim, MH; Kim, GS, Optimality conditions and duality in fractional robust optimization problems, East Asian Math J, 31, 345-349, 2015 · Zbl 1327.90288 · doi:10.7858/eamj.2015.025 |
| [14] | Antczak, T., Parametric approach for approximate efficiency of robust multiobjective fractional programming problems, Math Methods Appl Sci, 44, 11211-11230, 2021 · Zbl 1476.90320 · doi:10.1002/mma.7482 |
| [15] | Jayswal, A.; Baranwal, A.; Jiménez, MA, \(G\)-penalty approach for multi-dimensional control optimization problem with non-linear dynamical system, Int J Control, 96, 1165-1176, 2022 · Zbl 1519.93118 · doi:10.1080/00207179.2022.2032833 |
| [16] | Antczak, T.; Treanţă, S., Solving invex multitime control problems with first-order PDE constraints via the absolute value exact penalty method, Optim Control Appl Methods, 44, 6, 3379-3395, 2023 · Zbl 1531.93181 · doi:10.1002/oca.3043 |
| [17] | Antczak, T., Exact penalty functions method for mathematical programming problems involving invex functions, Eur J Oper Res, 198, 29-36, 2009 · Zbl 1163.90792 · doi:10.1016/j.ejor.2008.07.031 |
| [18] | Antczak, T., The vector exact \(l_1\) penalty method for nondifferentiable convex multiobjective programming problems, Appl Math Comput, 218, 9095-9106, 2012 · Zbl 1245.90113 |
| [19] | Pitea, A.; Postolache, M., Minimization of vectors of curvilinear functionals on the second order jet bundle: necessary conditions, Optim Lett, 6, 459-470, 2012 · Zbl 1280.90109 · doi:10.1007/s11590-010-0272-0 |
| [20] | Pitea, A., Multiobjective optimization problems on jet bundles, Front Phys, 10, 2022 · doi:10.3389/fphy.2022.875847 |
| [21] | Dinkelbach, W., On nonlinear fractional programming, Manage Sci, 13, 492-498, 1967 · Zbl 0152.18402 · doi:10.1287/mnsc.13.7.492 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
