Abstract
The aim of this paper is to study certain aspects of stability and scalarization of a nonconvex vector optimization problem through improvement sets. This paper attempts to investigate an open problem on stability posed by Chicco et al. The notion of stability is studied through Painlevé–Kuratowski set-convergence, where we establish sufficiency conditions for the lower and upper set-convergences of optimal solution sets of a family of perturbed vector problems, both in the given space and its image space. The perturbations are performed both on the objective function and the feasible set. Further, by using a nonlinear scalarization function defined in terms of an improvement set, we establish lower and upper Painlevé–Kuratowski set-convergences of sequences of approximate solution sets of certain scalarized problems. We then link these set-convergences with the set-convergences of optimal solution sets of the perturbed problems. Finally, we investigate the stability and scalarization of a linear vector optimization problem in finite dimensional spaces.
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References
Lucchetti, R.E., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53, 517–528 (2004)
Lalitha, C.S., Chatterjee, P.: Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities. J. Optim. Theory Appl. 155, 941–961 (2012)
Luc, D.T.: Scalarization of vector optimization problems. J. Optim. Theory Appl. 55, 85–102 (1987)
Miglierina, E.: Characterization of solutions of multiobjective optimization problem. Rend. Circ. Mat. Palermo 50, 153–164 (2001)
Qiu, J.H., Hao, Y.: Scalarization of Henig properly efficient points in locally convex spaces. J. Optim. Theory Appl. 147, 71–92 (2010)
Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)
Chatterjee, P., Lalitha, C.S.: Scalarization of Levitin–Polyak well-posedness in vector optimization using weak efficiency. Optim. Lett. (2014). doi:10.1007/s11590-014-0745-7
Gerstewitz, C.: Nichtkonvexe dualität in der vectoroptimierung. Wiss. Z. Tech. Hochsch. Leuna-Merseburg 25, 357–364 (1983)
Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004)
Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)
Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions via scalarization for a new \(\varepsilon \text{-efficiency }\) concept in vector optimization problems. Eur. J. Oper. Res. 201, 11–22 (2010)
Gutiérrez, C., Jiménez, B., Novo, V.: A generic approach to approximate efficiency and applications to vector optimization with set-valued maps. J. Glob. Optim. 49, 313–342 (2011)
Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150, 516–529 (2011)
Debreu, G.: Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Wiley, New York (1959)
Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)
Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)
Quang, P.H., Yen, N.D.: New proof of a theorem of F. Giannessi. J. Optim. Theory Appl. 68, 385–387 (1991)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Flores-Bazán, F., Hernández, E.: Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. J. Glob. Optim. 56, 299–315 (2013)
Zhao, K.Q., Yang, X.M.: A unified stability result with perturbations in vector optimization. Optim. Lett. 7, 1913–1919 (2013)
Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304–311 (2012)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer-Verlag, Berlin (1998)
Lalitha, C.S., Chatterjee, P.: Stability for properly quasiconvex vector optimization problem. J. Optim. Theory Appl. 155, 492–506 (2012)
Caruso, A.O., Khan, A.A., Raciti, F.: Continuity results for a class of variational inequalities with applications to time-dependent network problems. Numer. Funct. Anal. Optim. 30, 1272–1288 (2009)
Salinetti, G., Wets, R.J.-B.: On the convergence of sequences of convex sets in finite dimensions. SIAM Rev. 21, 18–33 (1979)
Acknowledgments
The authors are grateful to Prof. F. Giannessi for his valuable comments and suggestions which helped in improving the paper and led to Sect. 5 of the paper. Research of C.S. Lalitha is supported by R&D Doctoral Research Programme for University Faculty.
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Lalitha, C.S., Chatterjee, P. Stability and Scalarization in Vector Optimization Using Improvement Sets. J Optim Theory Appl 166, 825–843 (2015). https://doi.org/10.1007/s10957-014-0686-4
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DOI: https://doi.org/10.1007/s10957-014-0686-4