Augmented Lagrange Solver for Non-Convex Quadratic Programming
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Solution Overview
Problem
Existing methods for solving linear constraint/0-1 integer non-convex quadratic programming problems, such as the branch and bound method and approximate solution methods, face inefficiencies due to exponential calculation increases and require significant computational resources, often resulting in poor objective function values and complex hyperparameter tuning.
Innovation Solution
A solver apparatus that uses an augmented Lagrange function, including objective function terms, Lagrange terms, and a penalty term, to efficiently calculate solutions by iteratively updating candidate values and coefficients, thereby reducing the need for large penalty coefficients and improving computational performance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the branch and bound method is used to solve the linear constraint/0-1 integer non-convex quadratic programming problem, then a global optimal solution can be obtained, but the calculation amount increases exponentially with the number of decision variables
Solution Approach 1:
The patent transforms the original non-convex quadratic programming problem into a different parameter space by introducing auxiliary variables and reformulating the objective function. This parameter transformation allows the problem to be solved more efficiently while maintaining solution accuracy, avoiding the exponential complexity of the branch and bound method.
Solution Approach 2:
The patent introduces an intermediary quadratic function with auxiliary variables that acts as a bridge between the original complex problem and the Ising machine solver. This intermediary formulation enables the problem to be expressed in a form suitable for quantum annealing without requiring exhaustive search methods.
2Reliability
If the approximate solution method uses a large penalty parameter to handle constraints, then constraint satisfaction is improved, but the coefficient of the input matrix becomes larger requiring much calculation performance
Solution Approach 1:
The patent reformulates the penalty parameter issue by changing the parameter space through auxiliary variables. The penalty parameters remain in their original scale while the auxiliary variables absorb the constraint enforcement, preventing coefficient explosion and maintaining calculation efficiency.
Solution Approach 2:
The patent segments the constraint handling from the objective function by introducing separate auxiliary variables for each constraint. This segmentation allows constraints to be enforced with moderate penalty parameters while the auxiliary variables manage the complexity, avoiding the need for excessively large coefficients.
3Measurement precision
If the approximate solution method requires hyperparameter tuning on the penalty parameter to reach a good solution, then solution quality can be improved, but processing becomes complicated
Solution Approach 1:
The patent introduces intermediary quadratic functions with auxiliary variables that serve as mediators between the objective function and constraints. This intermediary layer eliminates the need for penalty parameter tuning by providing a direct mathematical relationship that naturally enforces constraints without requiring iterative hyperparameter adjustment.
Data Source
AI summary
A good solution is efficiently calculated. The solver apparatus calculates a solution to a problem minimizing an objective function as a non-convex quadratic function under a condition satisfying J simultaneous-linear-equations and satisfying K simultaneous-linear-inequalities. The solver apparatus includes an acquisition unit, an update unit, and an output unit. The update unit repeats, in predetermined order, first-processing of acquiring candidate values of solutions of the I-decision-variables minimizing a first function including the I-decision-variables generated using an augmented-Lagrange function, second-processing of acquiring candidate values of solutions of the K dependent-variables minimizing a second function including K dependent-variables generated using the augmented-Lagrange function, and the coefficient-processing of updating a coefficient included in the augmented-Lagrange function. The augmented-Lagrange function includes an objective function term including an objective function, J first Lagrange terms corresponding to the J simultaneous-linear-equations, K second Lagrange terms corresponding to the K simultaneous-linear-inequalities, and a penalty term.