Holomorphic Embedding for Economic Strategy Equilibrium
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Solution Overview
Problem
Conventional methods for analyzing economic strategy equilibria in economic models require significant computing resources, making it difficult for enterprises to adjust strategies in a timely manner and maximize profits.
Innovation Solution
A fast and flexible holomorphic embedding method using arc-length parametrization and polynomial homotopy functions to efficiently solve polynomial equations, allowing for the analysis and selection of equilibria solutions to optimize economic strategies.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional methods are used to analyze equilibrium polynomial equations, then all cases of equilibriums can be obtained, but a huge amount of computing resources are consumed
Solution Approach 1:
The patent segments the solution process by dividing the polynomial system into multiple subsets and solving them separately using homotopy continuation. Each subset is solved independently, reducing the computational complexity compared to solving the entire system at once, while still obtaining all equilibrium solutions.
Solution Approach 2:
The patent performs preliminary actions by constructing a homotopy function and computing its gradient before solving the main polynomial system. This preliminary setup includes defining the homotopy path and preparing the numerical framework, which enables efficient subsequent solution of the equilibrium equations.
2Adaptability or versatility
If conventional equilibrium analysis methods are used, then comprehensive strategy assessment is possible, but strategy adjustment timing is delayed
Solution Approach 1:
The patent changes parameters by introducing a homotopy parameter that transforms the original polynomial system into a more tractable form. This parameter transformation enables faster numerical solution while preserving all equilibrium solutions, allowing timely strategy adjustment without compromising comprehensive assessment.
3Measurement precision
If polynomial systems are solved using traditional numerical methods, then equilibrium solutions can be found, but computing resource consumption is excessive
Solution Approach 1:
The patent substitutes traditional mechanical numerical methods with a homotopy continuation approach that uses gradient information and path-following techniques. This substitution replaces brute-force numerical solving with a more efficient mathematical framework that maintains precision while reducing computational resource consumption.
Data Source
AI summary
Disclosed in the present invention is a fast and flexible holomorphic embedding method and apparatus for economic strategy adjustment. The method includes: obtaining product data and establishing equilibrium polynomial equations based on a Bertrand model; describing the equilibrium polynomial equations as a polynomial system and constructing a polynomial homotopy function; solving the equilibrium polynomial equations to obtain solutions of the equations: and analyzing equilibriums based on the solutions of the equations and performing economic strategy selection. The apparatus includes a memory and a processor configured to perform the fast and flexible holomorphic embedding method for economic strategy adjustment. The present invention can efficiently solve equilibrium polynomial equations based on which an economic strategy can be adjusted, to improve performance of enterprises. The fast and flexible holomorphic embedding method and apparatus for economic strategy adjustment provided by the present invention can be used extensively in the field of strategy adjustment.