Computing shadow prices with multiple Lagrange multipliers

Tao Jie; Gao Yan

doi:10.3934/jimo.2020070

Previous Article
Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming
JIMO Home
This Issue
Next Article
Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers

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 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)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Relationship of Lagrange Multiplier and Shadow Price

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Numerical Example

Figure Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Example of Multiple Lagrange Multipliers

Figure Options

Download full-size image

Download as PowerPoint slide

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

Table Options

Download as excel

$ 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

Table Options

Download as excel

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

[1]	Karla L. Cortez, Javier F. Rosenblueth. Normality and uniqueness of Lagrange multipliers. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3169-3188. doi: 10.3934/dcds.2018138
[2]	Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078
[3]	Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[4]	Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
[5]	Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418
[6]	Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[7]	Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247
[8]	Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723
[9]	Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
[10]	Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[11]	Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027
[12]	Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[13]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[14]	Ronan Costaouec, Haoyun Feng, Jesús Izaguirre, Eric Darve. Analysis of the accelerated weighted ensemble methodology. Conference Publications, 2013, 2013 (special) : 171-181. doi: 10.3934/proc.2013.2013.171
[15]	Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377
[16]	Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343
[17]	Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[18]	Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
[19]	Yan Gu, Nobuo Yamashita. Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 487-510. doi: 10.3934/naco.2020047
[20]	Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399

2019 Impact Factor: 1.366

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.366

Tools

Figures and Tables

2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

Computing shadow prices with multiple Lagrange multipliers

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Computing shadow prices with multiple Lagrange multipliers

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors