On Well-Posedness of Some Constrained Variational Problems
Abstract
:1. Introduction
2. Preliminaries and Problem Formulation
3. Well-Posedness Associated with (CVP)
- (i)
- It has a unique solution ;
- (ii)
- Each approximating sequence of (CVP) will converge to this unique solution .
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Tykhonov, A.N. On the stability of the functional optimization problems. USSR Comput. Math. Math. Phys. 1966, 6, 631–634. [Google Scholar] [CrossRef]
- Hu, R.; Fang, Y.P. Levitin-Polyak well-posedness by perturbations of inverse variational inequalities. Optim. Lett. 2013, 7, 343–359. [Google Scholar] [CrossRef]
- Jiang, B.; Zhang, J.; Huang, X.X. Levitin-Polyak well-posedness of generalized quasivariational inequalities with functional constraints. Nonlinear Anal. Theory Methods Appl. 2009, 70, 1492–1503. [Google Scholar] [CrossRef]
- Lalitha, C.S.; Bhatia, G. Levitin-Polyak well-posedness for parametric quasivariational inequality problem of the Minty type. Positivity 2012, 16, 527–541. [Google Scholar] [CrossRef]
- Levitin, E.S.; Polyak, B.T. Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 1996, 7, 764–767. [Google Scholar]
- Lignola, M.B.; Morgan, J. Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria, Decision and Control in Management Science; Zaccour, G., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002; pp. 367–378. [Google Scholar]
- Virmani, G.; Srivastava, M. On Levitin-Polyak α-well-posedness of perturbed variational-hemivariational inequality. Optimization 2015, 64, 1153–1172. [Google Scholar] [CrossRef]
- Čoban, M.M.; Kenderov, P.S.; Revalski, J.P. Generic well-posedness of optimization problems in topological spaces. Mathematika 1989, 36, 301–324. [Google Scholar] [CrossRef]
- Dontchev, A.L.; Zolezzi, T. Well-Posed Optimization Problems; Springer: Berlin, Germany, 1993. [Google Scholar]
- Furi, M.; Vignoli, A. A characterization of well-posed minimum problems in a complete metric space. J. Optim. Theory Appl. 1970, 5, 452–461. [Google Scholar] [CrossRef]
- Huang, X.X. Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 2001, 53, 101–116. [Google Scholar] [CrossRef]
- Huang, X.X.; Yang, X.Q. Generalized Levitin-Polyak well-posedness in constrained optimization. SIAM J. Optim. 2006, 17, 243–258. [Google Scholar] [CrossRef] [Green Version]
- Lignola, M.B.; Morgan, J. Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim. 2000, 16, 57–67. [Google Scholar] [CrossRef]
- Lin, L.J.; Chuang, C.S. Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint. Nonlinear Anal. 2009, 70, 3609–3617. [Google Scholar] [CrossRef]
- Lucchetti, R. Convexity and Well-Posed Problems; Springer: New York, NY, USA, 2006. [Google Scholar]
- Zolezzi, T. Extended well-posedness of optimization problems. J. Optim. Theory Appl. 1996, 91, 257–266. [Google Scholar] [CrossRef]
- Lignola, M.B. Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory. Appl. 2006, 128, 119–138. [Google Scholar] [CrossRef] [Green Version]
- Ceng, L.C.; Hadjisavvas, N.; Schaible, S.; Yao, J.C. Well-posedness for mixed quasivariational-like inequalities. J. Optim. Theory Appl. 2008, 139, 109–125. [Google Scholar] [CrossRef]
- Fang, Y.P.; Hu, R. Estimates of approximate solutions and well-posedness for variational inequalities. Math. Meth. Oper. Res. 2007, 65, 281–291. [Google Scholar] [CrossRef]
- Lalitha, C.S.; Bhatia, G. Well-posedness for parametric quasivariational inequality problems and for optimization problems with quasivariational inequality constraints. Optimization 2010, 59, 997–1011. [Google Scholar] [CrossRef]
- Heemels, P.M.H.; Camlibel, M.K.C.; Schaft, A.J.V.; Schumacher, J.M. Well-posedness of the complementarity class of hybrid systems. In Proceedings of the IFAC 15th Triennial World Congress, Barcelona, Spain, 21–26 July 2002. [Google Scholar]
- Chen, J.W.; Wang, Z.; Cho, Y.J. Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems. Math. Meth. Oper. Res. 2013, 77, 33–64. [Google Scholar] [CrossRef]
- Fang, Y.P.; Hu, R.; Huang, N.J. Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 2008, 55, 89–100. [Google Scholar] [CrossRef] [Green Version]
- Ceng, L.C.; Yao, J.C. Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems. Nonlinear Anal. 2008, 69, 4585–4603. [Google Scholar] [CrossRef]
- Xiao, Y.B.; Yang, X.M.; Huang, N.J. Some equivalence results for well-posedness of hemivariational inequalities. Glob. Optim. 2015, 61, 789–802. [Google Scholar] [CrossRef]
- Lignola, M.B.; Morgan, J. α-Well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. Glob. Optim. 2006, 36, 439–459. [Google Scholar] [CrossRef] [Green Version]
- Jayswal, A.; Jha, S. Well-posedness for generalized mixed vector variational-like inequality problems in Banach space. Math. Commun. 2017, 22, 287–302. [Google Scholar]
- Treanţă, S. A necessary and sufficient condition of optimality for a class of multidimensional control problems. Optim. Control Appl. Meth. 2020, 41, 2137–2148. [Google Scholar] [CrossRef]
- Treanţă, S. Well-posedness of new optimization problems with variational inequality constraints. Fractal Fract. 2021, 5, 123. [Google Scholar] [CrossRef]
- Treanţă, S. On well-posed isoperimetric-type constrained variational control problems. J. Diff. Eq. 2021, 298, 480–499. [Google Scholar] [CrossRef]
- Treanţă, S. Second-order PDE constrained controlled optimization problems with application in mechanics. Mathematics 2021, 9, 1472. [Google Scholar] [CrossRef]
- Treanţă, S. On a class of second-order PDE&PDI constrained robust modified optimization problems. Mathematics 2021, 9, 1473. [Google Scholar]
- Treanţă, S. On a class of isoperimetric constrained controlled optimization problems. Axioms 2021, 10, 112. [Google Scholar] [CrossRef]
- Dridi, H.; Djebabla, A. Timoshenko system with fractional operator in the memory and spatial fractional thermal effect. Rend. Circ. Mat. Palermo II. Ser. 2021, 70, 593–621. [Google Scholar] [CrossRef]
- Jana, S. Equilibrium problems under relaxed α-monotonicity on Hadamard manifolds. Rend. Circ. Mat. Palermo II. Ser. 2021. [Google Scholar] [CrossRef]
- Saunders, D.J. The Geometry of Jet Bundles; London Math. Soc. Lecture Notes Series, 142; Cambridge Univ. Press: Cambridge, UK, 1989. [Google Scholar]
- Usman, F.; Khan, S.A. A generalized mixed vector variational-like inequality problem. Nonlinear Anal. 2009, 71, 5354–5362. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Treanţă, S. On Well-Posedness of Some Constrained Variational Problems. Mathematics 2021, 9, 2478. https://doi.org/10.3390/math9192478
Treanţă S. On Well-Posedness of Some Constrained Variational Problems. Mathematics. 2021; 9(19):2478. https://doi.org/10.3390/math9192478
Chicago/Turabian StyleTreanţă, Savin. 2021. "On Well-Posedness of Some Constrained Variational Problems" Mathematics 9, no. 19: 2478. https://doi.org/10.3390/math9192478
APA StyleTreanţă, S. (2021). On Well-Posedness of Some Constrained Variational Problems. Mathematics, 9(19), 2478. https://doi.org/10.3390/math9192478