Abstract
In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.
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Aubin J.P., Ekeland I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Auslender A., Cominetti R., Haddou M.: Asymptotical analysis for penalty and barrier methods in convex and linear programming. Math. Oper. Res. 22, 43–62 (1997)
Benoist J., Popovici N.: Characterizations of convex and quasiconvex set-valued maps. Math. Methods Oper. Res. 57, 427–435 (2003)
Borwein J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13, 183–199 (1977)
Chen G.Y., Huang X.X., Yang X.Q.: Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)
Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)
Cominetti, R.: Nonlinear averages and convergence of penalty trajectories in convex programming. In: Ill-posed Variational Problems and Regularization Techniques (Trier, 1998), 65–78. Lecture Notes in Econom. and Math. Systems, 477. Springer, Berlin (1999)
Corley H.W.: An existence result for maximizations with respect to cones. J. Optim. Theory Appl. 31, 277–281 (1980)
Corley H.W.: Existence and Lagrange duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54, 489–501 (1987)
Corley H.W.: Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58, 1–10 (1988)
Deng S.: Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl. 96, 123–131 (1998)
Dontchev A.L., Zolezzi T.: Well-Posed Optimization Problems, Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
Flores-Bazan F.: Ideal, weakly efficient solutions for vector optimization problems. Math. Program. Ser. A. 93, 453–475 (2002)
Flores-Bazan F., Vera C.: Characterization of the nonemptiness and compactness of solution sets in convex and nonconvex vector optimization. J. Optim. Theory Appl. 130, 185–207 (2006)
Gotz A., Jahn J.: The Lagrange multiplier rule in set-valued optimization. SIAM J. Optim. 10, 331–344 (1999)
Hiriart-Urruty J.B., Lemarechal C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)
Huang X.X., Yang X.Q.: Characterizations of the nonemptiness and compactness of the set of weakly efficient solutions for convex vector optimization and applications. J. Math. Anal. Appl. 264, 270–287 (2001)
Huang, X.X., Yang, X.Q.: Further study on the Levitin-Polyak well-posedness of constrained convex optimization problems. Nonliear Anal. Theory Methods Appl. 75, 1341–1347 (2012)
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. Glob. Optim. 37, 287–304 (2007)
Huang X.X., Teo K.L., Yang X.Q.: Calmness and exact penalization in vector optimization with cone constraints. Comput. Optim. Appl. 35, 47–67 (2006)
Huang X.X., Yang X.Q., Teo K.L.: Characterizing the nonemptiness and compactness of solution set of a convex vector optimization problem with cone constraints and applications. J. Optim. Theory Appl. 123, 391–407 (2004)
Huang X.X., Yang X.Q., Teo K.L.: Convergence analysis of a class of penalty methods for vector optimization problems with cone constraints. J. Glob. Optim. 36, 637–652 (2006)
Kort B.W., Bertsekas D.P.: Combined primal-dual and penalty methods for convex programming. SIAM J. Control Optim. 14, 268–294 (1976)
Konsulova A.S., Revalski J.P.: Constrained convex optimization problems-well-posedness and stability. Numer. Funct. Anal. Optim. 15, 889–907 (1994)
Levitin E.S., Polyak B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 7, 764–767 (1966)
Loridan P.: Well-Posed Vector Optimization, Recent Developments in Well-Posed Variational Problems, Mathematics and its Applications, vol. 331. Kluwer, Dordrecht (1995)
Luc D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991)
Luc D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Pardalos, P.M., Siskos, Y., Zopounidis, C. (eds): Advances in Multicriteria Analysis. Kluwer, New York (1995)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970)
Rockafellar R.T., Wets R.J.B.: Variational Analysis. Springer, Berlin (1998)
Sawaragi Y., Nakayama H., Tanino T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)
Tykhonov A.N.: On the stability of the functional optimization problem. USSR Compt. Math. Math. Phys. 6, 28–33 (1966)
Yang X.Q., Huang X.X.: A nonlinear Lagrangian approach to constrained optimization problems. SIAM J. Optim. 11, 1119–1144 (2001)
Zopounidis, C., Pardalos, P.M. (eds): Handbook of Multicriteria Analysis. Springer, New York, NJ (2010)
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Huang, X.X., Yao, J.C. Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems. J Glob Optim 55, 611–626 (2013). https://doi.org/10.1007/s10898-012-9846-y
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DOI: https://doi.org/10.1007/s10898-012-9846-y