Mixed Integer Programming Relaxation for Binary Convergence
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Solution Overview
Problem
Existing combinatorial optimization methods, such as continuous relaxation and penalty methods, face challenges in accurately solving mixed integer programming problems due to difficulties in adjusting penalty coefficients and obtaining local or constraint-violating solutions.
Innovation Solution
A method that applies continuous relaxation annealing to mixed integer programming by continuously relaxing binary variables while keeping real variables intact, using a graph neural network (GNN) to adjust a loss term based on the degree of continuity and discreteness, optimizing the search process for solutions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If continuous relaxation is applied to binary variables in mixed integer programming, then the search space becomes continuous and easier to optimize, but the solution accuracy deteriorates because the relaxed variables may not converge to valid binary values
Solution Approach 1:
The patent changes the parameter γ (gamma) that controls the degree of continuous relaxation. By dynamically adjusting γ from initial values that allow continuous relaxation to final values that enforce binary constraints, the method transitions the optimization problem from an easier continuous form to a more accurate discrete form, resolving the contradiction between optimization ease and solution accuracy
Solution Approach 2:
The patent introduces dynamic adjustment of the relaxation parameter γ during the optimization process. The parameter γ is changed from an initial value γ_initial to a final value γ_final, making the continuous relaxation degree dynamic rather than static. This allows the system to benefit from continuous relaxation initially while ensuring binary convergence at the end
2Reliability
If penalty methods are used to enforce constraints in combinatorial optimization, then constraint satisfaction improves, but the difficulty of adjusting penalty coefficients increases
Solution Approach 1:
The patent makes the system self-adjusting by using the degree of continuous relaxation γ to automatically control the effective penalty strength. Instead of requiring manual tuning of penalty coefficients, the optimization process itself regulates constraint enforcement through the parameter γ, eliminating the need for complex penalty coefficient adjustment while maintaining constraint satisfaction
3Ease of operation
If all variables are continuously relaxed in mixed integer programming, then the optimization problem becomes a standard continuous problem, but the binary variable constraints are violated
Solution Approach 1:
The patent changes the relaxation parameter γ from an initial value allowing continuous relaxation to a final value that enforces binary constraints. This parameter transformation allows the system to temporarily relax binary constraints for easier optimization while ensuring they are satisfied in the final solution
Solution Approach 2:
The patent applies continuous relaxation preliminarily during the optimization process and then removes it in the final stage. By performing relaxation temporarily and then enforcing binary constraints through parameter adjustment, the method achieves both optimization ease and constraint accuracy
Data Source
AI summary
There is provided a non-transitory computer-readable medium storing a calculation program for causing a computer to execute a process. The process includes, in a mixed integer programming problem including real variables and binary variables as variables to be optimized, adding a loss term according to a degree of continuity and discreteness of decision variables obtained by continuously relaxing the binary variables to a cost function obtained by continuously relaxing the binary variables without continuously relaxing the real variables, and changing the loss term according to progress of a search process for searching for a solution using the cost function.