Abstract
In this paper, we examine duality for fractional programming problems in the face of data uncertainty within the framework of robust optimization. We establish strong duality between the robust counterpart of an uncertain convex–concave fractional program and the optimistic counterpart of its conventional Wolfe dual program with uncertain parameters. For linear fractional programming problems with constraint-wise interval uncertainty, we show that the dual of the robust counterpart is the optimistic counterpart in the sense that they are equivalent. Our results show that a worst-case solution of an uncertain fractional program (i.e., a solution of its robust counterpart) can be obtained by solving a single deterministic dual program. In the case of a linear fractional programming problem with interval uncertainty, such solutions can be found by solving a simple linear program.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barros, A., Frenk, J.B.G., Schaible, S., Zhang, S.: Using duality to solve generalized fractional programming problems. J. Glob. Optim. 8, 139–170 (1996)
Frenk, J.B.G., Schaible, S.: Fractional Programming. Handbook of Generalized Convexity and Monotonicity, pp. 333–384. Springer, Berlin (2004)
Liang, Z.A., Huang, H.X., Pardalos, P.M.: Optimality conditions and duality for a class of nonlinear fractional programming problems. J. Optim. Theory Appl. 110, 611–619 (2001)
Stancu-Minasian, I.M.: A sixth bibliography of fractional programming. Optimization 55, 405–428 (2006)
Chinchuluun, A., Yuan, D., Pardalos, P.M.: Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154, 133–147 (2007)
Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)
Ben-Tal, A., Nemirovski, A.: Robust optimization—methodology and applications. Math. Program., Ser. B 92, 453–480 (2002)
Ben-Tal, A., Nemirovski, A.: A selected topics in robust convex optimization. Math. Program., Ser. B 112, 125–158 (2008)
Ben-Tal, A., Boyd, S., Nemirovski, A.: Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107, 63–89 (2006)
Bertsimas, D., Pachamanova, P., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32, 510–516 (2004)
Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)
Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Craven, B.D.: Fractional Programming. Heldermann, Berlin (1988)
Schaible, S.: Fractional programming: a recent survey, Generalized convexity, generalized monotonicity, optimality conditions and duality in scaler and vector optimization. J. Stat. Manag. Syst. 5, 63–86 (2002)
Schaible, S.: Parameter-free convex equivalent and dual programs of fractional programming problems. ZOR, Z. Oper.-Res. 18, 187–196 (1974)
Jeyakumar, V., Li, G.: Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Oper. Res. Lett. 38, 188–194 (2010)
Jeyakumar, V., Li, G., Lee, G.M.: A robust von-Neumann minimax theorem for zero-sum games under bounded payoff uncertainty. Oper. Res. Lett. 39, 109–114 (2011)
Li, G., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74, 2327–2341 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jeyakumar, V., Li, G.Y. Robust Duality for Fractional Programming Problems with Constraint-Wise Data Uncertainty. J Optim Theory Appl 151, 292–303 (2011). https://doi.org/10.1007/s10957-011-9896-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9896-1