Computing shadow prices with multiple Lagrange multipliers

Tao Jie; Gao Yan

doi:10.3934/jimo.2020070

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September 2021, 17(5): 2307-2329. doi: 10.3934/jimo.2020070

Computing shadow prices with multiple Lagrange multipliers

Tao Jie and Gao Yan ^,

516 Jungong Road, Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

^* Corresponding author: Gao Yan

Received August 2019 Revised November 2019 Published September 2021 Early access March 2020

Fund Project: Tao Jie is supported by National Natural Science Foundation of China grant No. 71601117 and Soft Science Foundation of Shanghai grant No. 19692104600

There is a wide consensus that the shadow prices of certain resources in an economic system are equal to Lagrange multipliers. However, this is misleading with respect to multiple Lagrange multipliers. In this paper, we propose a new type of Lagrange multiplier, the weighted minimum norm Lagrange multiplier, which is a type of shadow price. An attractive aspect of this type of Lagrange multiplier is that it conveys the sensitivity information when resources are required to be proportionally input. To compute the weighted minimum norm Lagrange multiplier, we propose two algorithms. One is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of $ O(\frac{1}{k^2}) $. Furthermore, we propose a two-phase procedure to compute a particular subset of shadow prices that belongs to the set of bounded Lagrange multipliers. This subset is particularly attractive since all its elements are computable shadow prices. We report the numerical results for randomly generated problems.

Keywords: Shadow Price, Multiple Lagrange Multipliers, Weighted Norm Lagrange Multiplier, Accelerated Gradient Method, Convex Programming.

Mathematics Subject Classification: Primary: 90C25, 90C30; Secondary: 90C31.

Citation: Tao Jie, Gao Yan. Computing shadow prices with multiple Lagrange multipliers. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2307-2329. doi: 10.3934/jimo.2020070

References:

[1]	M. Akgul, A note on shadow prices in linear programming, Journal of the Operational Research Society, 35 (1984), 425-431. Google Scholar
[2]	D. C. Aucamp and D. I. Steinberg, The computation of shadow prices in linear programming, Journal of the Operational Research Society, 33 (1982), 557-565. doi: 10.1057/jors.1982.118. Google Scholar
[3]	M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons Inc., Hoboken, NJ, 2006. doi: 10.1002/0471787779. Google Scholar
[4]	D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, Belmont, 2015. Google Scholar
[5]	D. P. Bertsekas, A. Nedi, A. E. Ozdaglar, et al., Convex Analysis and Optimization, Athena Scientific, Belmont, 2003. Google Scholar
[6]	D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a lagrange multiplier theory for constrained optimization, Journal of Optimization Theory and Applications, 114 (2002), 287-343. doi: 10.1023/A:1016083601322. Google Scholar
[7]	J. P. Caulkins, D. Grass, G. Feichtinger and G. Tragler, Optimizing counter-terror operations: Should one fight fire with"fir" or"wate"?, Computers & Operations Research, 35 (2008), 1874-1885. doi: 10.1016/j.cor.2006.09.017. Google Scholar
[8]	T.-L. Chen, J. T. Lin and S.-C. Fang, A shadow-price based heuristic for capacity planning of tft-lcd manufacturing, Journal of Industrial & Management Optimization, 6 (2010), 209-239. doi: 10.3934/jimo.2010.6.209. Google Scholar
[9]	B. Col, A. Durnev and A. Molchanov, Foreign risk, domestic problem: Capital allocation and firm performance under political instability, Management Science, 64 (2018), 1975-2471. doi: 10.1287/mnsc.2016.2638. Google Scholar
[10]	M. E. Dyer, The complexity of vertex enumeration methods, Mathematics of Operations Research, 8 (1983), 381-402. doi: 10.1287/moor.8.3.381. Google Scholar
[11]	J. Gauvin, Shadow prices in nonconvex mathematical programming, Mathematical Programming, 19 (1980), 300-312. doi: 10.1007/BF01581650. Google Scholar
[12]	M. Hessel and M. Zeleny, Optimal system design: towards new interpretation of shadow prices in linear programming, Computers & Operations Research, 14 (1987), 265-271. doi: 10.1016/0305-0548(87)90063-3. Google Scholar
[13]	B. Jansen, J. De Jong, C. Roos and T. Terlaky, Sensitivity analysis in linear programming: Just be careful!, European Journal of Operational Research, 101 (1997), 15-28. doi: 10.1016/S0377-2217(96)00172-5. Google Scholar
[14]	T. T. Ke, Z.-J. M. Shen and J. M. Villas-Boas, Search for information on multiple products, Management Science, 62 (2016), 3576-3603. doi: 10.1287/mnsc.2015.2316. Google Scholar
[15]	R. Kutsuzawa, A. Yamashita, N. Takemura, J. Matsumoto, M. Tanaka and N. Yamanaka, Demand response minimizing the impact on the consumers' utility towards renewable energy, in Smart Grid Communications (SmartGridComm), 2016 IEEE International Conference on, IEEE, 2016, 68–73. doi: 10.1109/SmartGridComm.2016.7778740. Google Scholar
[16]	J. Kyparisis, On uniqueness of kuhn-tucker multipliers in nonlinear programming, Mathematical Programming, 32 (1985), 242-246. doi: 10.1007/BF01586095. Google Scholar
[17]	C.-Y. Lee and P. Zhou, Directional shadow price estimation of co2, so2 and nox in the united states coal power industry 1990–2010, Energy Economics, 51 (2015), 493-502. Google Scholar
[18]	O. L. Mangasarian, Uniqueness of solution in linear programming, Linear Algebra and Its Applications, 25 (1979), 151-162. doi: 10.1016/0024-3795(79)90014-4. Google Scholar
[19]	W. Meng and X. Wang, Distributed energy management in smart grid with wind power and temporally coupled constraints, IEEE Transactions on Industrial Electronics, 64 (2017), 6052-6062. doi: 10.1109/TIE.2017.2682001. Google Scholar
[20]	K. Schittkowski, More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, 282. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-61582-5. Google Scholar
[21]	N. Walter, Microeconomic Theory: Basic Principles and Extensions., Nelson Education, Canada, 2005. Google Scholar
[22]	Q. Wei and H. Yan, A method of transferring polyhedron between the intersection-form and the sum-form, Computers & Mathematics with Applications, 41 (2001), 1327-1342. doi: 10.1016/S0898-1221(01)00100-6. Google Scholar
[23]	L. Zhang, D. Feng, J. Lei, C. Xu, Z. Yan, S. Xu, N. Li and L. Jing, Congestion surplus minimization pricing solutions when lagrange multipliers are not unique, IEEE Transactions on Power Systems, 29 (2014), 2023-2032. doi: 10.1109/TPWRS.2014.2301213. Google Scholar

show all references

References:

[1]	M. Akgul, A note on shadow prices in linear programming, Journal of the Operational Research Society, 35 (1984), 425-431. Google Scholar
[2]	D. C. Aucamp and D. I. Steinberg, The computation of shadow prices in linear programming, Journal of the Operational Research Society, 33 (1982), 557-565. doi: 10.1057/jors.1982.118. Google Scholar
[3]	M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons Inc., Hoboken, NJ, 2006. doi: 10.1002/0471787779. Google Scholar
[4]	D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, Belmont, 2015. Google Scholar
[5]	D. P. Bertsekas, A. Nedi, A. E. Ozdaglar, et al., Convex Analysis and Optimization, Athena Scientific, Belmont, 2003. Google Scholar
[6]	D. P. Bertsekas and A. E. Ozdaglar, Pseudonormality and a lagrange multiplier theory for constrained optimization, Journal of Optimization Theory and Applications, 114 (2002), 287-343. doi: 10.1023/A:1016083601322. Google Scholar
[7]	J. P. Caulkins, D. Grass, G. Feichtinger and G. Tragler, Optimizing counter-terror operations: Should one fight fire with"fir" or"wate"?, Computers & Operations Research, 35 (2008), 1874-1885. doi: 10.1016/j.cor.2006.09.017. Google Scholar
[8]	T.-L. Chen, J. T. Lin and S.-C. Fang, A shadow-price based heuristic for capacity planning of tft-lcd manufacturing, Journal of Industrial & Management Optimization, 6 (2010), 209-239. doi: 10.3934/jimo.2010.6.209. Google Scholar
[9]	B. Col, A. Durnev and A. Molchanov, Foreign risk, domestic problem: Capital allocation and firm performance under political instability, Management Science, 64 (2018), 1975-2471. doi: 10.1287/mnsc.2016.2638. Google Scholar
[10]	M. E. Dyer, The complexity of vertex enumeration methods, Mathematics of Operations Research, 8 (1983), 381-402. doi: 10.1287/moor.8.3.381. Google Scholar
[11]	J. Gauvin, Shadow prices in nonconvex mathematical programming, Mathematical Programming, 19 (1980), 300-312. doi: 10.1007/BF01581650. Google Scholar
[12]	M. Hessel and M. Zeleny, Optimal system design: towards new interpretation of shadow prices in linear programming, Computers & Operations Research, 14 (1987), 265-271. doi: 10.1016/0305-0548(87)90063-3. Google Scholar
[13]	B. Jansen, J. De Jong, C. Roos and T. Terlaky, Sensitivity analysis in linear programming: Just be careful!, European Journal of Operational Research, 101 (1997), 15-28. doi: 10.1016/S0377-2217(96)00172-5. Google Scholar
[14]	T. T. Ke, Z.-J. M. Shen and J. M. Villas-Boas, Search for information on multiple products, Management Science, 62 (2016), 3576-3603. doi: 10.1287/mnsc.2015.2316. Google Scholar
[15]	R. Kutsuzawa, A. Yamashita, N. Takemura, J. Matsumoto, M. Tanaka and N. Yamanaka, Demand response minimizing the impact on the consumers' utility towards renewable energy, in Smart Grid Communications (SmartGridComm), 2016 IEEE International Conference on, IEEE, 2016, 68–73. doi: 10.1109/SmartGridComm.2016.7778740. Google Scholar
[16]	J. Kyparisis, On uniqueness of kuhn-tucker multipliers in nonlinear programming, Mathematical Programming, 32 (1985), 242-246. doi: 10.1007/BF01586095. Google Scholar
[17]	C.-Y. Lee and P. Zhou, Directional shadow price estimation of co2, so2 and nox in the united states coal power industry 1990–2010, Energy Economics, 51 (2015), 493-502. Google Scholar
[18]	O. L. Mangasarian, Uniqueness of solution in linear programming, Linear Algebra and Its Applications, 25 (1979), 151-162. doi: 10.1016/0024-3795(79)90014-4. Google Scholar
[19]	W. Meng and X. Wang, Distributed energy management in smart grid with wind power and temporally coupled constraints, IEEE Transactions on Industrial Electronics, 64 (2017), 6052-6062. doi: 10.1109/TIE.2017.2682001. Google Scholar
[20]	K. Schittkowski, More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, 282. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-61582-5. Google Scholar
[21]	N. Walter, Microeconomic Theory: Basic Principles and Extensions., Nelson Education, Canada, 2005. Google Scholar
[22]	Q. Wei and H. Yan, A method of transferring polyhedron between the intersection-form and the sum-form, Computers & Mathematics with Applications, 41 (2001), 1327-1342. doi: 10.1016/S0898-1221(01)00100-6. Google Scholar
[23]	L. Zhang, D. Feng, J. Lei, C. Xu, Z. Yan, S. Xu, N. Li and L. Jing, Congestion surplus minimization pricing solutions when lagrange multipliers are not unique, IEEE Transactions on Power Systems, 29 (2014), 2023-2032. doi: 10.1109/TPWRS.2014.2301213. Google Scholar

Figure 1. Numerical Example in Schttfkowski (1987)

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Figure 2. Relationship of Lagrange Multiplier and Shadow Price

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Figure 3. Numerical Example

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Figure 4. Convergence of the $ \mathcal{PFA} $ algorithm with different penalty parameters

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Figure 5. Example of Multiple Lagrange Multipliers

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Table 1. Computational Times of the $ \mathcal{AGM} $ algorithm on Large - scale Data Sets

$ m $	$ n $	Computational Time
500	5000	10.7504
500	10000	11.4222
500	20000	12.5574
500	50000	15.5700
1000	5000	21.4259
1000	10000	21.8054
1000	20000	22.2733
1000	50000	25.2937
5000	5000	101.1437
5000	10000	102.1762
5000	20000	106.8786
5000	50000	107.3515

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$ m $	$ n $	Computational Time
500	5000	10.7504
500	10000	11.4222
500	20000	12.5574
500	50000	15.5700
1000	5000	21.4259
1000	10000	21.8054
1000	20000	22.2733
1000	50000	25.2937
5000	5000	101.1437
5000	10000	102.1762
5000	20000	106.8786
5000	50000	107.3515

Table 2. Result of the 2-phase Procedure with 23 Vertices of the Bounded Lagrange Multiplier Set

Vertices	Elements
$ v_1 $	0.0000	1.6118	0.0000	0.0000	0.0000	0.0000	0.4939	0.0000	0.0000
$ v_2 $	0.0000	2.1057	0.4939	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$ v_3 $	0.0000	3.0834	0.0000	0.0000	0.1352	0.0000	0.0000	0.6957	0.0000
$ v_4 $	0.0000	3.4207	0.0000	0.0000	0.0026	0.0000	0.0064	0.7603	0.1801
$ v_5 $	0.0000	5.7145	0.0000	0.0000	3.1147	2.5431	3.6277	0.0000	0.0000
$ v_6 $	0.0000	7.1430	0.0000	4.9601	0.9637	0.0000	1.9331	0.0000	0.0000
$ v_7 $	0.0000	7.2983	0.0000	0.0000	0.0000	1.7188	3.3700	0.0000	2.6128
$ v_8 $	0.0000	7.5898	0.0000	4.3192	0.0000	0.0000	2.0493	0.0000	1.0416
$ v_9 $	0.0000	7.7043	2.9184	0.0000	2.4098	1.9675	0.0000	0.0000	0.0000
$ v_{10} $	0.0000	8.1361	1.7390	4.2912	0.8338	0.0000	0.0000	0.0000	0.0000
$ v_{11} $	0.0000	8.5707	1.8287	3.7067	0.0000	0.0000	0.0000	0.0000	0.8939
$ v_{12} $	0.0000	8.8386	2.7554	0.0000	0.0000	1.3515	0.0000	0.0000	2.0545
$ v_{13} $	0.0000	9.0761	0.0000	3.0271	0.9637	0.0000	0.0000	1.9331	0.0000
$ v_{14} $	0.0000	9.1971	0.0000	0.0000	1.9422	1.1568	0.0000	2.7039	0.0000
$ v_{15} $	0.0000	9.6392	0.0000	2.2699	0.0000	0.0000	0.0000	2.0493	1.0416
$ v_{16} $	0.0000	10.0534	0.0000	0.0000	0.0000	0.6918	0.0000	2.5809	1.6740
$ v_{17} $	0.0000	3.4315	0.0050	0.0000	0.0027	0.0000	0.0000	0.7627	0.1807
$ v_{18} $	0.6494	10.2980	0.0000	0.0000	0.0000	0.0000	0.0000	2.4700	1.5827
$ v_{19} $	1.0475	9.6669	0.0000	0.0000	1.7714	0.0000	0.0000	2.5142	0.0000
$ v_{20} $	1.2187	9.3956	2.5332	0.0000	0.0000	0.0000	0.0000	0.0000	1.8526
$ v_{21} $	1.5166	8.1461	0.0000	0.0000	0.0000	0.0000	3.0317	0.0000	2.3055
$ v_{22} $	1.6981	8.6358	2.5864	0.0000	2.0798	0.0000	0.0000	0.0000	0.0000
$ v_{23} $	2.1242	7.1628	0.0000	0.0000	2.6016	0.0000	3.1114	0.0000	0.0000

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Vertices	Elements
$ v_1 $	0.0000	1.6118	0.0000	0.0000	0.0000	0.0000	0.4939	0.0000	0.0000
$ v_2 $	0.0000	2.1057	0.4939	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
$ v_3 $	0.0000	3.0834	0.0000	0.0000	0.1352	0.0000	0.0000	0.6957	0.0000
$ v_4 $	0.0000	3.4207	0.0000	0.0000	0.0026	0.0000	0.0064	0.7603	0.1801
$ v_5 $	0.0000	5.7145	0.0000	0.0000	3.1147	2.5431	3.6277	0.0000	0.0000
$ v_6 $	0.0000	7.1430	0.0000	4.9601	0.9637	0.0000	1.9331	0.0000	0.0000
$ v_7 $	0.0000	7.2983	0.0000	0.0000	0.0000	1.7188	3.3700	0.0000	2.6128
$ v_8 $	0.0000	7.5898	0.0000	4.3192	0.0000	0.0000	2.0493	0.0000	1.0416
$ v_9 $	0.0000	7.7043	2.9184	0.0000	2.4098	1.9675	0.0000	0.0000	0.0000
$ v_{10} $	0.0000	8.1361	1.7390	4.2912	0.8338	0.0000	0.0000	0.0000	0.0000
$ v_{11} $	0.0000	8.5707	1.8287	3.7067	0.0000	0.0000	0.0000	0.0000	0.8939
$ v_{12} $	0.0000	8.8386	2.7554	0.0000	0.0000	1.3515	0.0000	0.0000	2.0545
$ v_{13} $	0.0000	9.0761	0.0000	3.0271	0.9637	0.0000	0.0000	1.9331	0.0000
$ v_{14} $	0.0000	9.1971	0.0000	0.0000	1.9422	1.1568	0.0000	2.7039	0.0000
$ v_{15} $	0.0000	9.6392	0.0000	2.2699	0.0000	0.0000	0.0000	2.0493	1.0416
$ v_{16} $	0.0000	10.0534	0.0000	0.0000	0.0000	0.6918	0.0000	2.5809	1.6740
$ v_{17} $	0.0000	3.4315	0.0050	0.0000	0.0027	0.0000	0.0000	0.7627	0.1807
$ v_{18} $	0.6494	10.2980	0.0000	0.0000	0.0000	0.0000	0.0000	2.4700	1.5827
$ v_{19} $	1.0475	9.6669	0.0000	0.0000	1.7714	0.0000	0.0000	2.5142	0.0000
$ v_{20} $	1.2187	9.3956	2.5332	0.0000	0.0000	0.0000	0.0000	0.0000	1.8526
$ v_{21} $	1.5166	8.1461	0.0000	0.0000	0.0000	0.0000	3.0317	0.0000	2.3055
$ v_{22} $	1.6981	8.6358	2.5864	0.0000	2.0798	0.0000	0.0000	0.0000	0.0000
$ v_{23} $	2.1242	7.1628	0.0000	0.0000	2.6016	0.0000	3.1114	0.0000	0.0000

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