Abstract
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.
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References
Lara P, Stancu-Minasian I (1999) Fractional programming: a tool for the assessment of sustainability. Agric Syst 62:131–141
Pitea A, Udriste C, Mititelu S (2009) New type dualities in PDI and PDE constrained optimization problems. J Adv Math Stud 2:81–91
Stancu AM (2013) Mathematical programming with type-I functions. Matrix Rom, Bucharest
Stancu-Minasian M (1997) Fractional Programming, vol 426. Academic Publishers, Dordrecht, Theory, Methods and Applications
Stancu-Minasian IM (2013) A seventh bibliography of fractional programming. Adv Model Optim 15:309–386
Stancu-Minasian IM (2017) An eighth bibliography of fractional programming. Optimization 3:439–470
Stancu-Minasian IM (2019) A ninth bibliography of fractional programming. Optimization 11:2123–2167
Treanţă S, Agarwal D, Sachdev G (2022) Robust efficiency conditions in multiple-objective fractional variational control problems. Fractal Fract 7:18
Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23:769–805
Baranwal A, Jayswal A, Kardam P (2022) Robust duality for the uncertain multitime control optimization problems. Int J Robust Nonlinear Control 32:5837–5847
Jayswal A, Baranwal A (2023) Robust approach for uncertain multi-dimensional fractional control optimization problems. Bull Malays Math Sci Soc 46:1–17
Minh VT, Afzulpurkar N (2005) Robust model predictive control for input saturated and softened state constraints. Asian J Control 7:319–325
Kim MH, Kim GS (2015) Optimality conditions and duality in fractional robust optimization problems. East Asian Math J 31:345–349
Antczak T (2021) Parametric approach for approximate efficiency of robust multiobjective fractional programming problems. Math Methods Appl Sci 44:11211–11230
Jayswal A, Baranwal A, Jiménez MA (2022) \(G\)-penalty approach for multi-dimensional control optimization problem with non-linear dynamical system. Int J Control 96:1165–1176
Antczak T, Treanţă S (2023) 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
Antczak T (2009) Exact penalty functions method for mathematical programming problems involving invex functions. Eur J Oper Res 198:29–36
Antczak T (2012) The vector exact \(l_1\) penalty method for nondifferentiable convex multiobjective programming problems. Appl Math Comput 218:9095–9106
Pitea A, Postolache M (2012) Minimization of vectors of curvilinear functionals on the second order jet bundle: necessary conditions. Optim Lett 6:459–470
Pitea A (2022) Multiobjective optimization problems on jet bundles. Front Phys 10:875847
Dinkelbach W (1967) On nonlinear fractional programming. Manage Sci 13:492–498
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The research of the first author is financially supported by the MATRICS, SERB-DST, New Delhi, India (No. MTR/ 2021/ 000002).
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Jayswal, A., Baranwal, A. & Antczak, T. Solving convex uncertain PDE-constrained multi-dimensional fractional control problems via a new approach. J Eng Math 145, 8 (2024). https://doi.org/10.1007/s10665-024-10338-2
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DOI: https://doi.org/10.1007/s10665-024-10338-2
Keywords
- Exact absolute penalty function method
- Parametric method
- PDE-constrained multi-dimensional fractional control problem with uncertainty
- Robust optimization
- Uncertainty modeling