Advanced Search
Article Contents
Article Contents

Local smooth representation of solution sets in parametric linear fractional programming problems

  • * Corresponding author: Y. P. Fang

    * Corresponding author: Y. P. Fang
This work was partially supported by the National Science Foundation of China (No. 11471230)and the Scientific Research Foundation of the Education Department of Sichuan Province (No.16ZA0213)
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • The purpose of this paper is to investigate the structure of the solution sets in parametric linear fractional programming problems. It is shown that the solution set of a parametric linear fractional programming problem with smooth data has a local smooth representation. As a consequence, the corresponding marginal function is differentiable and the solution map admits a differentiable selection. We also give an example to illustrate the result.

    Mathematics Subject Classification: Primary: 90C31, 90C32.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] E. B. Bajalinov, Linear Fractional Programming: Theory, Methods, Applications and Software, Kluwer Acad. Publ., Boston, 2003.
    [2] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.
    [3] A. CambiniS. Schaible and C. Sodini, Parametric linear fractional programming for an unbounded feasible region, J. Global Optim., 3 (1993), 157-169.  doi: 10.1007/BF01096736.
    [4] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Res. Log., 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.
    [5] Y. P. FangN. J. Huang and X. Q. Yang, Local smooth representations of parametric semiclosed polyhedra with applications to sensitivity in piecewise linear programs, J. Optim. Theory Appl., 155 (2012), 810-839.  doi: 10.1007/s10957-012-0089-3.
    [6] Y. P. FangK. W. Meng and X. Q. Yang, Piecewise linear multi-criteria programs: the continuous case and its discontinuous generalization, Oper. Res., 60 (2012), 398-404.  doi: 10.1287/opre.1110.1014.
    [7] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering, 165. Academic Press, Inc., Orlando, FL, 1983.
    [8] D. T. Luc, Smooth representation of a parametric polyhedral convex set with application to sensitivity in optimization, Proc. Amer. Math. Soc., 125 (1997), 555-567.  doi: 10.1090/S0002-9939-97-03507-7.
    [9] D. T. Luc and P. H. Dien, Differentiable selection of optimal solutions in parametric linear programming, Proc. Amer. Math. Soc., 125 (1997), 883-892.  doi: 10.1090/S0002-9939-97-03090-6.
    [10] B. Andrew Martos and V. Whinston, Hyperbolic programming, Naval Res. Logist. Quart., 11 (1960), 135-155.  doi: 10.1002/nav.3800110204.
    [11] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, 1970.
    [12] K. Swarup, Linear fractional functionals programming, Oper. Res., 13 (1965), 1029-1036. 
    [13] L. V. Thuan and D. T. Luc, On sensitivity in linear multiobjective programming, J. Optim. Theory Appl., 107 (2000), 615-626.  doi: 10.1023/A:1026455401079.
    [14] H. Wolf, A parametric method for solving the linear fracional programming problem, Oper. Res., 33 (1985), 835-841.  doi: 10.1287/opre.33.4.835.
    [15] H. Wolf, Parametric analysis in linear fractional programming, Oper. Res., 34 (1986), 930-937.  doi: 10.1287/opre.34.6.930.
    [16] S. J. Xue, Determining the optimal solution set for linear fractional programming, J. Syst. Eng. Electron., 13 (2002), 40-45. 
    [17] S. J. Xue, A way to find the set of optimal solutions in linear fractional programming, Comm. Appl. Math Comput., 16 (2002), 90-96. 
    [18] X. Q. Yang and N. D. Yen, Structure and weak sharp minimum of the Pareto solution set for piecewise linear multiobjective optimization, J. Optim. Theory Appl., 147 (2010), 113-124.  doi: 10.1007/s10957-010-9710-5.
    [19] X. Y. Zheng and X. Q. Yang, The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces, Sci. China Ser. A, 51 (2008), 1243-1256.  doi: 10.1007/s11425-008-0021-3.
  • 加载中

Article Metrics

HTML views(1787) PDF downloads(242) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint