# American Institute of Mathematical Sciences

March  2019, 9(1): 45-52. doi: 10.3934/naco.2019004

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

 1 Department of Mathematics, Sichuan University, Chengdu 610065, P. R. China 2 Department of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P. R. China 3 Department of Mathematics, Sichuan University, Chengdu 610065, P. R. China

* Corresponding author: Y. P. Fang

Received  November 2017 Revised  May 2018 Published  October 2018

Fund Project: 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).

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.

Citation: Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004
##### References:
 [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. Cambini, S. 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. Fang, N. 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. Fang, K. 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.

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##### References:
 [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. Cambini, S. 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. Fang, N. 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. Fang, K. 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.
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