Abstract
This paper is devoted to the study of continuity properties of Pareto solution maps for parametric semi-infinite vector optimization problems (PSVO). We establish new necessary conditions for lower and upper semicontinuity of Pareto solution maps under functional perturbations of both objective functions and constraint sets. We also show that the necessary condition becomes sufficient for the lower and upper semicontinuous properties in the special case where the constraint set mapping is lower semicontinuous at the reference point. Examples are given to illustrate the obtained results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Brosowski B.: Parametric semi-infinite linear programming. I. Continuity of the feasible set and of the optimal value. Sensitivity, stability and parametric analysis. Math. Program. Stud. 21, 18–42 (1984)
Cánovas M.J., López M.A., Parra J., Todorov M.I.: Stability and well-posedness in linear semi-infinite programming. SIAM J. Optim. 10(1), 82–98 (1999)
Cánovas M.J., López M.A., Parra J., Toledo F.J.: Lipschitz continuity of the optimal value via bounds on the optimal set in linear semi-infinite optimization. Math. Oper. Res. 31(3), 478–489 (2006)
Cánovas M.J., Klatte D., López M.A., Parra J.: Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM. J Optim. 18(3), 717–732 (2007)
Chen G.Y., Craven B.D.: Existence and continuity of solutions for vector optimization. J. Optim. Theory Appl. 81(3), 459–468 (1994)
Colgen R., Schnatz K.: Continuity properties in semi-infinite parametric linear optimization. Numer. Funct. Anal. Optim. 3(4), 451–460 (1981)
Gayá V.E., López M.A., Vera de Serio V.N.: Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems. Optimization 52(6), 693–713 (2003)
Goberna M.A.: Linear semi-infinite optimization: recent advances. Continuous optimization, 3–22, Appl. Optim., 99. Springer, New York (2005)
Goberna M.A., López M.A., Todorov M.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17(4), 730–743 (1996)
Goberna M.A., López M.A., Todorov M.: Stability theory for linear inequality systems. II. Upper semicontinuity of the solution set mapping. SIAM J. Optim. 7(4), 1138–1151 (1997)
Goberna M.A., Gómez S., Guerra F., Todorov M.I.: Sensitivity analysis in linear semi-infinite programming: perturbing cost and right-hand-side coefficients. Eur. J. Oper. Res. 181(3), 1069–1085 (2007)
López M.A., Vera de Serio V.N.: Stability of the feasible set mapping in convex semi-infinite programming semi-infinite programming (Alicante, 1999), 101–120. Nonconvex Optim. Appl., 57. Kluwer Acad. Publ., Dordrecht (2001)
Luc D.T.: Theory of Vector Optimization Lecture Notes in Economics and Mathematical Systems, 319. Springer, Berlin (1989)
Mordukhovich B.S.: Variational Analysis and Generalized Differentiation. I. Basic theory, Grundlehren der Mathematischen Wissenschaften, 330. Springer, Berlin (2006)
Mordukhovich B.S.: Variational Anlysis and Generalized Differentiation. II. Applications, Grundlehren der Mathematischen Wissenschaften, 331. Springer, Berlin (2006)
Reemtsen, R., Rückmann J.-J. (eds): Semi-Infinite Programming, Nonconvex Optimization and its Applications, 25. Kluwer Academic Publishers Boston, MA (1998)
Weber G.-W.: Generalized Semi-Infinite Optimization and Related Topics, Research and Exposition in Mathematics, 29. Heldermann, Lemgo (2003)
Xiang S.W., Zhou Y.H.: Continuity properties of solutions of vector optimization. Nonlinear Anal. 64, 2496–2506 (2006)
Xiang S.W., Yin W.S.: Stability results for efficient solutions of vector optimization problems. J. Optim. Theory Appl. 134, 385–398 (2007)
Yu J.: Essential weak efficient solution in multiobjective optimization problems. J. Math. Anal. Appl. 166, 230–235 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chuong, T.D., Huy, N.Q. & Yao, J.C. Stability of semi-infinite vector optimization problems under functional perturbations. J Glob Optim 45, 583–595 (2009). https://doi.org/10.1007/s10898-008-9391-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9391-x