Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 31, 07 March 2024
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An in-depth analysis and derivation of extremum conditions based on gradient information
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Abstract
This paper explores the necessary conditions for extremum in both constrained and unconstrained problems by extracting fundamental principles of constraint conditions. We provide a precise geometric understanding of the Lagrange Multiplier, prioritizing analytical insight. Beginning with a geometric interpretation of the gradient, we leverage the expansion of functions and their images to comprehend extremum and detail the Lagrangian derivation process.We expand the base vectors of the constraint surface into those of the full space and use a transition matrix to assess the function's extremum. This demonstrates how the second derivative matrix is transformed into its full-space representation to discern extrema in optimization problems. Additionally, we introduce incremental variables to optimize the second-order derivative matrix in full space, providing a novel perspective to solve extremal necessary conditions.
Keywords
gradient, Hessian matrix, multivariate function expansion, Lagrange Multiplier, sufficient and necessary conditions for extremal
References
1. Li AD 2006 Journal of West Anhui University 22(2), 4.
2. Yan C and Chen T 2017 The Geometric Interpretation of the Lagrange Multiplier Method Studies in College Mathematics 20(2), 3.
3. Liu SM and Li XY 2004 Journal of Henan University of Science and Technology(Natural Science) 25(6), 3.
4. Hu HC 1985 Chinese Journal of Theoretical and Applied Mechanics.(05), 40-48.
5. Widom H 1969 Extremal polynomials associated with a system of curves in the complex plane. Advances in Mathematics 3(2), 127-232.
6. Xu CS (user name Shangguan Zhengshen) 2022 July 20th On Zhihu website Answer Title [How Lagrange came up with the Lagrange multiplier method] retrieved from URL https://www.zhihu.com/question/37137053/answer/2584665110
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-317-3
- ISBN (Online)
- 978-1-83558-318-0
- Published Date
- 07 March 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/31/20241121
- Copyright
- 07 March 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited