Characterizing Shadow Price via Lagrange Multiplier for Nonsmooth Problem

doi:10.1007/s40305-022-00416-6

Journal of the Operations Research Society of China ›› 2023, Vol. 11 ›› Issue (4): 827-838.doi: 10.1007/s40305-022-00416-6

收稿日期:2021-05-22 修回日期:2022-03-02 出版日期:2023-12-30 发布日期:2023-12-26
通讯作者: Yan Gao E-mail:gaoyan@usst.edu.cn

Characterizing Shadow Price via Lagrange Multiplier for Nonsmooth Problem

Yan Gao

School of Management, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received:2021-05-22 Revised:2022-03-02 Online:2023-12-30 Published:2023-12-26
Contact: Yan Gao E-mail:gaoyan@usst.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China (No. 72071130).

摘要/Abstract

Abstract: In this paper, the relation between the shadow price and the Lagrange multiplier for nonsmooth optimization problem is explored. It is obtained that the Lagrange multipliers are upper bounds of the shadow price for a convex optimization problem and a class of Lipschtzian optimization problems. This result can be used in pricing mechanisms for nonsmooth situation. Several nonsmooth functions involved in this class of Lipschtzian optimizations are listed. Finally, an application to electricity pricing is discussed.

Key words: Shadow price, Lagrange multiplier, Nonsmooth optimization

中图分类号:

49J52
91B24

. [J]. Journal of the Operations Research Society of China, 2023, 11(4): 827-838.

Yan Gao. Characterizing Shadow Price via Lagrange Multiplier for Nonsmooth Problem[J]. Journal of the Operations Research Society of China, 2023, 11(4): 827-838.

参考文献

[1] Nicholson, W., Snyder, C.: Microeconomic Theory: Basic Principles and Extensions, 12th edn. Cengage Learning, Boston (2016)
[2] Kantorovich, L.V.: The Best Use of Economic Resources. Harvard University Press, Cambridge (1965)
[3] Koopmans, T.C.: Concepts of optimality and their uses. Math. Program. 11(1), 212-228 (1976)
[4] Kantorovich, L.V.: Mathematics in economics: achievements, difficulties, perspectives. Math. Program. 11(1), 204-211 (1976)
[5] Bazaraa, M.S., Sherali, H.D., Shett, C.M.: Nonlinear Programming Theory and Algorithms. John Wiley and Sons, New York (1993)
[6] Bertsekas, D.P.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)
[7] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
[8] Lee, C.Y., Zhou, P.: Directional shadow price estimation of CO₂, SO₂ and NOx in the United States coal power industry 1990-2010. Energy Econom. 51, 493-502 (2015)
[9] Col, B., Durnev, A., Molchanov, A.: Foreign risk, domestic problem: capital allocation and firm performance under political instability. Manage. Sci. 64(5), 2102-2125 (2017)
[10] Ke, T.T., Shen, Z., Villas-Boas, J.M.: Search for information on multiple products. Oper. Res. 58(4), 389-392 (2018)
[11] Deng, R., Yang, Z., Chow, M.Y., et al.: A survey on demand response in smart grids: mathematical models and approaches. IEEE Trans. Ind. Inf. 11(3), 570-582 (2015)
[12] Xu, Z., Guo, L., Gao, Y., et al.: Real-time pricing of smart grid based on piece-wise linear functions. J. Syst. Sci. Inform. 7(4), 295-316 (2019)
[13] Zhang, L., Gao, Y., Zhu, H., et al.: Bi-level stochastic real-time pricing model in multi-energy generation system: a reinforcement learning approach. Energy 121, 121926 (2022)
[14] Clarke, F.H., Leda Yu, S., Stern, R.J., et al.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
[15] Gao, Y.: Nonsmooth Optimization, 2nd edn. Science Press, Beijing (2018) (in Chinese)
[16] Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Peterlang, Frankfurt am Main (1995)

Characterizing Shadow Price via Lagrange Multiplier for Nonsmooth Problem

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

编辑推荐

Metrics

本文评价