Abstract
New results are established for multiobjective DC programs with infinite convex constraints (MOPIC) that are defined on Banach spaces (finite or infinite dimensional) with objectives given as the difference of convex functions. This class of problems can also be called multiobjective DC semi-infinite and infinite programs, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Such problems have not been studied as yet. Necessary and sufficient optimality conditions for the weak Pareto efficiency are introduced. Further, we seek a connection between multiobjective linear infinite programs and MOPIC. Both Wolfe and Mond-Weir dual problems are presented, and corresponding weak, strong, and strict converse duality theorems are derived for these two problems respectively. We also extend above results to multiobjective fractional DC programs with infinite convex constraints. The results obtained are new in both semi-infinite and infinite frameworks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dinh, N., Mordukhovich, B., Nghia, T.T.A.: Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs. Math. Program. Ser. B 123, 101–138 (2010)
Dinh, N., Goberna, M.A., López, M.A., Son, T.Q.: New Farkas-type results with applications to convex infinite programs. ESAIM Control Optim. Calc. Var. 13, 580–597 (2007)
Shapiro, A.: Semi-infinite programs, duality, discretization and optimality conditions. Optimization 58, 133–161 (2009)
Li, D.H., Qi, L.Q., Tam, J., Wu, S.Y.: A smoothing Newton method for semi-infinite programs. J. Glob. Optim. 30, 169–194 (2004)
Chen, C.F., Wu, S.Y.: Duality theorems and algorithms for linear programs in measure spaces. J. Glob. Optim. 30, 207–233 (2004)
López, M., Still, G.: Semi-infinite programs. Eur. J. Oper. Res. 180, 491–518 (2007)
Huang, N.J., Li, J., Wu, S.Y.: Optimality conditions for vector optimization problems. J. Optim. Theory Appl. 142, 323–342 (2009)
Glover, B.M., Jeyakumar, V., Rubinov, A.M.: Dual conditions characterizing optimality for convex multi-objective. Math. Program. 84, 201–217 (1999)
van Rooyen, M., Zlobec, X: A saddle point characterization of Pareto optima. Math. Program. 67, 77–88 (1994)
Bot, R.I., Wanka, G.: Duality for multiobjective optimization problems with convex objective functions and DC constraints. J. Math. Anal. Appl. 315, 526–543 (2006)
Ye, J.J.: Nondifferentiable multiplier rules for optimization and bileve optimization problems. SIAM J. Optim. 15, 252–274 (2004)
Caristi, G., Ferrara, M., Stefanescu, A.: Semi-infinite multiobjective programs with generalized invexity, http://www.csm.ro/reviste/Mathematical-Reports/Pdfs/2010/3/Caristi.pdf
Chen, J.W., Wan, Z.P., Zheng, Y: Optimality conditions and duality for nonsmooth multiobjective optimization problems with cone constraints and applications, http://www.optimization-online.org/DBFILE/2010/07/2672.pdf
Li, L.: Mond-Weir duality for multiobjective optimization problems with cone constraints. Southeast Asian Bull. Math. 34, 322–332 (2010)
Jeyakumar, V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)
Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005)
Hanson, M.A.: On the sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Noor, M.A.: On generalized preinvex functions and monotonicities. J. Inequal. Pure Appl. Math. 5, 1–9 (2004)
Chen, X.H.: Optimization duality theorem for the multiobjective fractional programs with the generalized (F, \(\rho \))-convexity. J. Math. Anal. Appl. 273, 190–205 (2002)
Huang, L.: Optimality conditions and duality for a class of nonlinear fractional programs. J. Optim. Theory Appl. 11, 611–619 (2001)
Zalmai, G.J., Zhang, Q.H.: Semiparametric duality models for semiinfinite multiobjective fractional programs problems containing generalized \((\alpha,\eta,\rho )\)-V-invex functions. Southeast Asian Bull. Math. 32, 779–804 (2008)
Yang, Y: \(F_{\varepsilon }\)-convex function and fractional semi-infinite programming. In: International Conference of Information Science and Management Engineering (2010)
Acknowledgments
The authors are very grateful to the editor in chief and the anonymous referee who have made many constructive comments that have helped to improve the presentation and results of the paper. This work was supported by Natural Scientific Foundation of China (Nos. 71201040, 11201099) and by Major Program of the National Natural Science Foundation of China (No. 71031003). This work was also supported by Postdoctoral Science-Research Development Foundation of Heilongjiang Province (No. LBH-Q11118) and by NOLFR-314-000-080-720.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qu, S., Goh, M., Wu, SY. et al. Multiobjective DC programs with infinite convex constraints. J Glob Optim 59, 41–58 (2014). https://doi.org/10.1007/s10898-013-0091-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0091-9