# 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 & 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.

show all references

##### 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.
 [1] Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343 [2] Behrouz Kheirfam. Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2010, 6 (2) : 347-361. doi: 10.3934/jimo.2010.6.347 [3] Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1 [4] Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611 [5] Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019041 [6] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 [7] Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2018138 [8] Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019 [9] Rong Hu, Ya-Ping Fang. A parametric simplex algorithm for biobjective piecewise linear programming problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 573-586. doi: 10.3934/jimo.2016032 [10] Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1 [11] Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial & Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689 [12] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [13] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 [14] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 [15] Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789 [16] Nguyen Thi Bach Kim, Nguyen Canh Nam, Le Quang Thuy. An outcome space algorithm for minimizing the product of two convex functions over a convex set. Journal of Industrial & Management Optimization, 2013, 9 (1) : 243-253. doi: 10.3934/jimo.2013.9.243 [17] Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225 [18] Wei Gao, Juan Luis García Guirao, Mahmoud Abdel-Aty, Wenfei Xi. An independent set degree condition for fractional critical deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 877-886. doi: 10.3934/dcdss.2019058 [19] Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237 [20] Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233

Impact Factor:

## Metrics

• HTML views (234)
• Cited by (0)

• on AIMS