Abstract
In this paper, a global efficiency criterion is established for a class of multidimensional variational control problems governed by first order PDE&PDI constraints and path-independent curvilinear integral cost functionals. More precisely, a minimal efficiency criterion for a local efficient solution to be its global efficient solution in the considered optimization problem is formulated and proved. Also, the theoretical developments derived in the paper are accompanied by an example of a nonconvex optimization problem.
Similar content being viewed by others
References
Arana-Jiménez, M., & Antczak, T. (2017). The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem. Mathematical Methods in the Applied Sciences, 40, 6556–6564.
Arana-Jiménez, M., Cambini, R., & Carosi, L. (2018). A reduced formulation for pseudoinvex vector functions. Annals of Operations Research, 269, 21–27.
Archetti, F., & Schoen, F. (1984). A survey on the global optimization problem: General theory and computational approaches. Annals of Operations Research, 1, 87–110.
Clarke, F. H. (2013). Functional analysis, calculus of variations and optimal control, graduate texts in mathematics (Vol. 264). London: Springer.
Giannessi, F. (2005). Constrained optimization and image space analysis. Volume I: Separation of sets and optimality conditions (pp. 1–395). New York: Springer.
Gupta, P., Cambini, R., & Appadoo, S. S. (2018). Recent advances in optimization theory and applications. Annals of Operations Research, 269, 1–2.
Hiriart-Urruty, J.-B., & Lemaréchal, C. (2001). Fundamentals of convex analysis. Berlin: Springer.
Horst, R. (1982). A note on functions whose local minima are global. Journal of Optimization Theory and Applications, 36, 457–463.
Jayswal, A., & Preeti. (2019). Saddle point criteria for multi-dimensional control optimisation problem involving first-order PDE constraints. International Journal of Control. https://doi.org/10.1080/00207179.2019.1661523.
Jayswal, A., & Preeti. (2020). An exact \(l_{1}\) penalty function method for multi-dimensional first-order PDE constrained control optimisation problem. European Journal of Control, 52, 34–41.
Mititelu, Ş., & Treanţă, S. (2018). Efficiency conditions in vector control problems governed by multiple integrals. Journal of Applied Mathematics and Computing, 57, 647–665.
Polyak, B. T. (1987). Introduction to optimization, optimization software. New York: Publications Division.
Treanţă, S. (2018). On a new class of vector variational control problems. Numerical Functional Analysis and Optimization, 39, 1594–1603.
Treanţă, S. (2019a). Variational analysis with applications in optimisation and control. Cambridge: Cambridge Scholars Publishing. ISBN: 978-1-5275-3728-6.
Treanţă, S. (2019b). KT-geodesic pseudoinvex control problems governed by multiple integrals. Journal of Nonlinear and Convex Analysis, 20, 73–84.
Treanţă, S. (2020). On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion. European Journal of Control, 51, 1–9.
Treanţă, S., & Arana-Jiménez, M. (2018a). KT-pseudoinvex multidimensional control problem. Optimal Control Applications and Methods, 39, 1291–1300.
Treanţă, S., & Arana-Jiménez, M. (2018b). On generalized KT-pseudoinvex control problems involving multiple integral functionals. European Journal of Control, 43, 39–45.
Treanţă, S., & Mititelu, Ş. (2019). Duality with \((\rho, b)\)-quasiinvexity for multidimensional vector fractional control problems. Journal of Information and Optimization Sciences, 40, 1429–1445.
Zang, I., & Avriel, M. (1975). On functions whose local minima are global. Journal of Optimization Theory and Applications, 16, 183–190.
Zang, I., Choo, E. U., & Avriel, M. (1976). A note on functions whose local minima are global. Journal of Optimization Theory and Applications, 18, 555–559.
Zang, I., Choo, E. U., & Avriel, M. (1977). On functions whose stationary points are global minima. Journal of Optimization Theory and Applications, 22, 195–208.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Treanţă, S. On a global efficiency criterion in multiobjective variational control problems with path-independent curvilinear integral cost functionals. Ann Oper Res 311, 1249–1257 (2022). https://doi.org/10.1007/s10479-020-03579-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03579-8
Keywords
- Global efficiency criterion
- Control
- Variational control problem
- Path-independent curvilinear integral cost functional