Simulated Bifurcation Algorithm for Constrained 0-1 Optimization
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Solution Overview
Problem
Current methods for solving the Ising problem under linear inequality constraints, such as those in portfolio optimization, require a significant increase in the number of Ising spins, leading to increased computation time and cost due to the need for slack variables, especially in large-scale problems like the Tokyo Stock Exchange's 2000 stocks classification.
Innovation Solution
A solution finding device and method that uses an enhanced simulated bifurcation algorithm to solve 0-1 combinatorial optimization problems under constraint conditions expressed as inequality expressions, updating variables alternately and correcting them to satisfy constraints without requiring slack variables, thereby maintaining the same number of variables as unconstrained problems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If slack variables are used to solve the Ising problem under linear inequality constraints, then the constraint satisfaction is achieved, but the number of Ising spins increases significantly leading to increased computation time and cost
Solution Approach 1:
The patent extracts and removes the slack variables from the Ising problem formulation. Instead of adding W slack variables for each constraint, the invention directly solves the constrained optimization problem using the simulated bifurcation algorithm, eliminating the need for additional variables and reducing the system size from N+W spins to just N spins.
Solution Approach 2:
The patent changes the formulation approach by modifying the objective function to include penalty terms for constraint violations directly, rather than using slack variables. The penalty function P(x) = max(0, Σωᵢxᵢ - W) is added to the original objective function, transforming the constrained problem into an unconstrained one that can be solved with the same number of spins.
2Reliability
If slack variables are used to solve the Ising problem under linear inequality constraints, then the constraint satisfaction is achieved, but the number of Ising spins increases leading to increased computational complexity
Solution Approach 1:
The patent extracts and removes the slack variables from the Ising problem formulation. Instead of adding W slack variables for each constraint, the invention directly solves the constrained optimization problem using the simulated bifurcation algorithm, eliminating the need for additional variables and reducing the system size from N+W spins to just N spins.
Solution Approach 2:
The patent changes the formulation approach by modifying the objective function to include penalty terms for constraint violations directly, rather than using slack variables. The penalty function P(x) = max(0, Σωᵢxᵢ - W) is added to the original objective function, transforming the constrained problem into an unconstrained one that can be solved with the same number of spins.
3Reliability
If the portfolio optimization problem is solved for 2000 stocks with constraint conditions for all 99 groups, then the investment risk is controlled, but the number of Ising spins increases by 50% leading to increased computation cost
Solution Approach 1:
The patent extracts and removes the slack variables from the Ising problem formulation. Instead of adding W slack variables for each constraint, the invention directly solves the constrained optimization problem using the simulated bifurcation algorithm, eliminating the need for additional variables and reducing the system size from N+W spins to just N spins.
Solution Approach 2:
The patent changes the formulation approach by modifying the objective function to include penalty terms for constraint violations directly, rather than using slack variables. The penalty function P(x) = max(0, Σωᵢxᵢ - W) is added to the original objective function, transforming the constrained problem into an unconstrained one that can be solved with the same number of spins.
Data Source
AI summary
A 0-1 combinatorial-optimization-problem is solved under constraint-conditions expressed using inequality expressions. A solution-finding device includes an updating-unit and an output-unit. For each of plural elements associated with first- and second-variables, the updating-unit sequentially updates, for each unit-time between initial-timing and end-timing, the first-variable and the second-variable alternatively. The output-unit outputs the solution of the 0-1 combinatorial-optimization-problem based on the first-variable of each of the plural elements at the end-timing. During the updating operation for each unit-time, for each of one or more constraint-conditions, when the inequality expression in which the first-variable corresponding to each of the plural discrete variables is substituted is unsatisfied; the updating-unit subtracts, from the second-variable of each of the plural elements, a correction value corresponding to the component of the element corresponding to the distance from the boundary of the inequality expression to positions identified by the plural elements.