Abstract
In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.
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References
Bednarczuk, E.: Well-posedness of vector optimization problems. In: Jahn, J., Krabs, W. (eds.), Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Econom. and Math. Systems, vol. 294, pp. 51–61. Springer, Berlin (1987)
Bednarczuk, E.: An approach to well-posedness in vector optimization consequences to stability. Parametric optimization. Control Cybern. 23, 107–122 (1994)
Crespi, G.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213–226 (2007)
Deng, S.: Coercivity properties and well-posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003)
Dentcheva, D., Helbig, S.: On variational principles, level sets, well-posedness, and \(\varepsilon \text{-solutions }\) in vector optimization. J. Optim. Theory Appl. 89, 325–349 (1996)
Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems, vol. 1543. Springer, Berlin (1993)
Huang, X.X., Yang, X.Q.: Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J. Optim. 17, 243–258 (2006)
Huang, X.X., Yang, X.Q.: Levitin–Polyak well-posedness of constrained vector optimization problems. J. Global Optim. 37, 287–304 (2007)
Huang, X.X., Yang, X.Q.: Further study on the Levitin–Polyak well-posedness of constrained convex vector optimization problems. Nonlinear Anal. 75, 1341–1347 (2012)
Ioffe, A.D., Lucchetti, R.E., Revalski, J.P.: Almost every convex or quadratic programming problem is well posed. Math. Oper. Res. 29, 369–382 (2004)
Kettner, L.J., Deng, S.: On well-posedness and Hausdorff convergence of solution sets of vector optimization problems. J. Optim. Theory Appl. 153, 619–632 (2012)
Konsulova, A.S., Revalski, J.P.: Constrained convex optimization problems-well-posedness and stability. Numer. Funct. Anal. Optim. 15, 889–907 (1994)
Lalitha, C.S., Chatterjee, P.: Well-posedness and stability in vector optimization using Henig proper efficiency. Optimization. 62, 155–165 (2013)
Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 7, 764–767 (1966)
Loridan, P. (1995) Well-posedness in vector optimization. In: Lucchetti, R., Revalski, J. (eds.) Recent Developments in Well-Posed Variational Problems. Math. Appl. vol. 331, pp. 171–192. Kluwer, Dordrecht
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Econom, and Math, Systems, vol. 319. Springer, Berlin (1989)
Lucchetti, R.: Convexity and Well-Posed Problems. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. vol. 22. Springer, New York (2006)
Lucchetti, R.: Well-posedness, towards vector optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Econom, and Math. Systems, vol. 294, pp. 194–207. Springer, Berlin (1987)
Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003)
Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)
Tanaka, T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math Optim. 36, 313–322 (1997)
Todorov, M. et al.: Well-posedness in the linear vector semi-infinite optimization. Multiple criteria decision making expand and enrich the domains of thinking and applications. In: Yu, P.L. (ed.) Proceedings of the Tenth International Conference, pp. 141–150. Springer, New York (1994)
Todorov, M.: Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization. Eur. J. Oper. Res. 94, 610–617 (1996)
Tykhonov, A.N.: On the stability of the functional optimization problem. U.S.S.R. Comput. Math. Math. Phys. 6, 28–33 (1966)
Zolezzi, T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. 25, 437–453 (1995)
Zolezzi, T.: Well-posedness and optimization under perturbations. Ann. Oper. Res. 101, 351–361 (2001)
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Research of C. S. Lalitha was supported by R&D Doctoral Research Programme funds for university faculty.
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Lalitha, C.S., Chatterjee, P. Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. J Glob Optim 59, 191–205 (2014). https://doi.org/10.1007/s10898-013-0103-9
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DOI: https://doi.org/10.1007/s10898-013-0103-9