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Stability preservation in Galerkin-type projection-based model order reduction
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 |
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.
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. |
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