Complex-Variable Optimization Model for Faster Combinatorial Solving
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Solution Overview
Problem
Current methods for solving combinatorial optimization problems, such as the traveling salesman problem, are inefficient and require excessive computational resources and time due to exponential complexity, especially with large numbers of variables, making them impractical for real-world applications.
Innovation Solution
A method utilizing a multi-direction bifurcation quantum heuristic algorithm with phase encoding, mapping the problem to a spin model like the Potts model, and using a dynamic equation with a control variable to drive complex variables into multiple stable phases, reducing the number of spins and computation required.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If exhaustive or enumeration methods are used to solve combinatorial optimization problems, then complete solution accuracy can be achieved, but computational time and resource consumption increase exponentially with problem scale
Solution Approach 1:
The patent transforms the combinatorial optimization problem from discrete variable space to continuous complex variable space by mapping decision variables to complex numbers with specific phases. This parameter transformation allows the use of continuous dynamic systems instead of discrete enumeration, achieving polynomial-time complexity while maintaining solution accuracy through the relationship between phase values and discrete decisions.
Solution Approach 2:
The patent replaces traditional computational algorithms (exhaustive search, enumeration) with a physics-based dynamic system described by differential equations. The system evolves complex variables according to dynamic equations with damping terms, naturally converging to stable states that correspond to optimal solutions, thereby substituting mechanical computation with physical evolution processes.
2Productivity
If parallel computing with clusters of supercomputers is used to reduce solution time, then computational speed improves, but energy and resource consumption become enormous
Solution Approach 1:
The patent changes the fundamental parameters of the problem representation by using complex variables with phase encoding instead of traditional discrete variables. This allows the optimization problem to be solved through continuous dynamic evolution rather than parallel discrete computation, achieving high solution speed with significantly reduced energy consumption by avoiding the need for large supercomputer clusters.
Solution Approach 2:
The patent creates a simplified continuous model (copy) of the original discrete optimization problem that can be solved efficiently. By working with the continuous complex variable representation rather than the original discrete structure, the system achieves fast solutions without requiring enormous computational resources, effectively solving a copied version of the problem that preserves the essential optimization characteristics.
3Adaptability or versatility
If the number of decision variables increases to handle larger problems, then problem coverage improves, but the complexity and difficulty of finding solutions increase exponentially
Solution Approach 1:
The patent applies parameter transformation by mapping discrete decision variables to continuous complex variables with phase encoding. This transformation allows the system to handle larger problem scales polynomially rather than exponentially, as the continuous dynamic system can evolve solutions efficiently regardless of the number of variables, thereby improving adaptability without proportionally increasing computational complexity.
Solution Approach 2:
The patent introduces a new dimension by using complex numbers instead of real numbers, adding the imaginary dimension to the variable representation. This dimensional extension allows phase encoding that captures discrete decision information in continuous form, enabling the system to handle larger-scale problems by utilizing the additional dimensional space for more efficient information representation and processing.
Data Source
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AI summary
A method for solving a combinatorial optimization problem, an apparatus, and a system are provided. The method includes: receiving a combinatorial optimization problem input by a user; mapping the combinatorial optimization problem to a model, where the model includes a to-be-solved complex variable and an objective function, the to-be-solved complex variable is used to describe a solution in which permutation and combination of things meet a requirement as a discrete variable in a complex number field, and the objective function is used to describe the permutation and combination of a plurality of things under a given constraint condition; determining a dynamic equation for the model in the complex number field, where the dynamic equation is used to describe a time-based evolution process of the to-be-solved complex variable, and the dynamic equation includes at least a control variable, a multi-bifurcation parameter, and an action parameter; solving the dynamic equation, and obtaining q stable phases of the to-be-solved complex variable by changing a value of the control variable, where a value of q is determined by the multi-bifurcation parameter; and obtaining a solution to the combinatorial optimization problem based on the q stable phases. In this application, for solving the combinatorial optimization problem, a spin number can be reduced and time for finding a solution can be shortened.