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July  2020, 16(4): 2029-2044. doi: 10.3934/jimo.2019041

Robust sensitivity analysis for linear programming with ellipsoidal perturbation

Ruotian Gao and  Wenxun Xing ,

Department of Mathematical Sciences, Tsinghua University, Beijing 100086, China

* Corresponding author

Received  September 2018 Revised  January 2019 Published  May 2019

Fund Project: Funding: This research has been supported by the National Natural Science Foundation of China under Grant numbers 11771243 and 11571029

Sensitivity analysis is applied to the robust linear programming problem in this paper. The coefficients of the linear program are assumed to be perturbed in three perturbation manners within ellipsoidal sets. Our robust sensitivity analysis is to calculate the maximal radii of the perturbation sets to keep some properties of the robust feasible set. Mathematical models are formulated for the robust sensitivity analysis problems and all models are either reformulated into linear programs or convex quadratic programs except for the bi-convex programs where more than one row of the constraint matrix is perturbed. For the bi-convex programs, we develop a binary search algorithm.

Keywords: Linear programming, sensitivity analysis, robust optimization, convex quadratic programming, fractional programming.
Mathematics Subject Classification: Primary: 90C31, 90C05; Secondary: 90C20.
Citation: Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2029-2044. doi: 10.3934/jimo.2019041
References:
[1]

A. Ben-Tal, D. D. Hertog, A. D. Waegenaere, B. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), Pages iv–527. doi: 10.1287/mnsc.1120.1641.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898718829.  Google Scholar

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D. Bertsimas and D. B. Brown, Constructing uncertainty sets for robust linear optimization, Operations Research, 57 (2009), 1483-1495.  doi: 10.1287/opre.1080.0646.  Google Scholar

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D. Bertsimas, V. Gupta and N. Kallus, Data-driven robust optimization, Mathematical Programming, 167 (2018), 235-292.  doi: 10.1007/s10107-017-1125-8.  Google Scholar

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D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

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F. Chandra, D. F. Gayme, L. Chen and J. C. Doyle, Robustness, optimization, and architectures, European Journal of Control, 17 (2011), 472-482.  doi: 10.3166/ejc.17.472-482.  Google Scholar

[7]

G. E. Dullerud and F. G. Paganini, A Course in Robust Control Theory–-A Convex Approach, Springer-Verlag, New York, 2000. Google Scholar

[8]

M. Frank and P. Wolfe, An algorithm for quadratic programming, Naval Research Logistics Quarterly, 3 (1956), 95-110.  doi: 10.1002/nav.3800030109.  Google Scholar

[9]

S. I. Gass, Linear Programming: Methods and Applications, Corrected reprint of the 1995 fifth edition. Dover Publications, Inc., Mineola, NY, 2003.  Google Scholar

[10]

J. Gorski, F. Pfeuffer and K. Klamroth, Biconvex sets and optimization with biconvex functions: A survey and extensions, Mathematical Methods of Operations Research, 66 (2007), 373-407.  doi: 10.1007/s00186-007-0161-1.  Google Scholar

[11]

C.-L. Hwang and A. S. Masud, Multiple Objective Decision Making–-Methods and Applications: A State-of-the-Art Survey, Springer-Verlag, Berlin, 1979.  Google Scholar

[12]

A. Marandi, A. Ben-Tal, D. D. Hertog and B. Melenberg, Extending the scope of robust quadratic optimization, Available on Optimization Online, 2017. Google Scholar

[13]

S. Schaible, Parameter-free convex equivalent and dual programs of fractional programming problems, Zeitschrift für Operations Research, 18 (1974), 187-196.  doi: 10.1007/BF02026600.  Google Scholar

[14]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 22 (1974), 892-898.  doi: 10.1287/opre.22.4.892.  Google Scholar

[15]

G. Xu and S. Burer, Robust sensitivity analysis of the optimal value of linear programming, Optimization Methods and Software, 32 (2017), 1187-1205.  doi: 10.1080/10556788.2016.1256400.  Google Scholar

[16]

K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, NJ, 1996. Google Scholar

show all references

References:
[1]

A. Ben-Tal, D. D. Hertog, A. D. Waegenaere, B. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), Pages iv–527. doi: 10.1287/mnsc.1120.1641.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898718829.  Google Scholar

[3]

D. Bertsimas and D. B. Brown, Constructing uncertainty sets for robust linear optimization, Operations Research, 57 (2009), 1483-1495.  doi: 10.1287/opre.1080.0646.  Google Scholar

[4]

D. Bertsimas, V. Gupta and N. Kallus, Data-driven robust optimization, Mathematical Programming, 167 (2018), 235-292.  doi: 10.1007/s10107-017-1125-8.  Google Scholar

[5]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[6]

F. Chandra, D. F. Gayme, L. Chen and J. C. Doyle, Robustness, optimization, and architectures, European Journal of Control, 17 (2011), 472-482.  doi: 10.3166/ejc.17.472-482.  Google Scholar

[7]

G. E. Dullerud and F. G. Paganini, A Course in Robust Control Theory–-A Convex Approach, Springer-Verlag, New York, 2000. Google Scholar

[8]

M. Frank and P. Wolfe, An algorithm for quadratic programming, Naval Research Logistics Quarterly, 3 (1956), 95-110.  doi: 10.1002/nav.3800030109.  Google Scholar

[9]

S. I. Gass, Linear Programming: Methods and Applications, Corrected reprint of the 1995 fifth edition. Dover Publications, Inc., Mineola, NY, 2003.  Google Scholar

[10]

J. Gorski, F. Pfeuffer and K. Klamroth, Biconvex sets and optimization with biconvex functions: A survey and extensions, Mathematical Methods of Operations Research, 66 (2007), 373-407.  doi: 10.1007/s00186-007-0161-1.  Google Scholar

[11]

C.-L. Hwang and A. S. Masud, Multiple Objective Decision Making–-Methods and Applications: A State-of-the-Art Survey, Springer-Verlag, Berlin, 1979.  Google Scholar

[12]

A. Marandi, A. Ben-Tal, D. D. Hertog and B. Melenberg, Extending the scope of robust quadratic optimization, Available on Optimization Online, 2017. Google Scholar

[13]

S. Schaible, Parameter-free convex equivalent and dual programs of fractional programming problems, Zeitschrift für Operations Research, 18 (1974), 187-196.  doi: 10.1007/BF02026600.  Google Scholar

[14]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 22 (1974), 892-898.  doi: 10.1287/opre.22.4.892.  Google Scholar

[15]

G. Xu and S. Burer, Robust sensitivity analysis of the optimal value of linear programming, Optimization Methods and Software, 32 (2017), 1187-1205.  doi: 10.1080/10556788.2016.1256400.  Google Scholar

[16]

K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, NJ, 1996. Google Scholar

Table 1.  Numerical Results
Instance (m1, n2, pert3, scale4) Lower Bound Maximum Radius Improvement Ratio5 CPU Time
Ave Std Ave Std Ave Std Ave Std
(20, 20, 0.25, 0.5) 0.4151 0.1551 1.1650 0.8647 2.3001 3.0673 15.8480 1.1491
(20, 20, 0.25, 1) td> 0.2283 0.0969 0.7672 0.3066 2.6969 1.3452 15.6160 0.9467
(20, 20, 0.5, 0.5) 0.3056 0.1443 1.0726 0.7539 2.8557 3.0440 17.0120 1.2904
(20, 20, 0.5, 1) 0.1385 0.0818 0.4476 0.2327 2.6981 2.4415 18.3810 1.8375
(20, 100, 0.25, 0.5) 0.1108 0.0106 0.5120 0.1750 3.5828 1.5280 27.2940 1.0984
(20, 100, 0.25, 1) 0.0521 0.0047 0.3152 0.2321 5.0011 4.4017 27.9500 1.7346
(20, 100, 0.5, 0.5) 0.0800 0.0066 0.4237 0.2145 4.3204 2.7268 35.2770 1.7099
(20, 100, 0.5, 1) 0.0435 0.0064 0.1897 0.2614 3.7842 7.3197 33.4340 1.9935
(80, 100, 0.25, 0.5) 0.1381 0.0534 0.5341 0.2676 2.9914 1.9738 118.4410 5.8678
(80, 100, 0.25, 1) 0.0676 0.0148 0.4115 0.1058 5.3278 1.8796 120.0080 7.0440
(80, 100, 0.5, 0.5) 0.1033 0.0238 0.5436 0.2278 4.3561 2.4909 184.1960 9.4721
(80, 100, 0.5, 1) 0.0450 0.0075 0.2548 0.0952 4.7366 2.3818 201.1790 20.2812
1 number of constraints
2 dimension of x
3 ratio of the number of perturbation vectors for one row to m + n
4 scale of perturbation vectors
5 improvement ratio=(maximum radius-lower bound)/lower bound
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Instance (m1, n2, pert3, scale4) Lower Bound Maximum Radius Improvement Ratio5 CPU Time
Ave Std Ave Std Ave Std Ave Std
(20, 20, 0.25, 0.5) 0.4151 0.1551 1.1650 0.8647 2.3001 3.0673 15.8480 1.1491
(20, 20, 0.25, 1) td> 0.2283 0.0969 0.7672 0.3066 2.6969 1.3452 15.6160 0.9467
(20, 20, 0.5, 0.5) 0.3056 0.1443 1.0726 0.7539 2.8557 3.0440 17.0120 1.2904
(20, 20, 0.5, 1) 0.1385 0.0818 0.4476 0.2327 2.6981 2.4415 18.3810 1.8375
(20, 100, 0.25, 0.5) 0.1108 0.0106 0.5120 0.1750 3.5828 1.5280 27.2940 1.0984
(20, 100, 0.25, 1) 0.0521 0.0047 0.3152 0.2321 5.0011 4.4017 27.9500 1.7346
(20, 100, 0.5, 0.5) 0.0800 0.0066 0.4237 0.2145 4.3204 2.7268 35.2770 1.7099
(20, 100, 0.5, 1) 0.0435 0.0064 0.1897 0.2614 3.7842 7.3197 33.4340 1.9935
(80, 100, 0.25, 0.5) 0.1381 0.0534 0.5341 0.2676 2.9914 1.9738 118.4410 5.8678
(80, 100, 0.25, 1) 0.0676 0.0148 0.4115 0.1058 5.3278 1.8796 120.0080 7.0440
(80, 100, 0.5, 0.5) 0.1033 0.0238 0.5436 0.2278 4.3561 2.4909 184.1960 9.4721
(80, 100, 0.5, 1) 0.0450 0.0075 0.2548 0.0952 4.7366 2.3818 201.1790 20.2812
1 number of constraints
2 dimension of x
3 ratio of the number of perturbation vectors for one row to m + n
4 scale of perturbation vectors
5 improvement ratio=(maximum radius-lower bound)/lower bound
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