Linear Programming Active-Set Selection for Real-Time Control
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Solution Overview
Problem
Existing linear programming problem solving systems, such as those using interior point methods and simplex methods, face challenges in estimating the time to converge to an optimal solution and may fail to find the optimal solution within a control cycle, compromising optimality or violating constraints.
Innovation Solution
A linear programming problem solving system that includes a dual solution candidate searcher and an optimal solution calculation device, which determines active sets and finds a basic feasible solution by selecting an optimal active set based on the inner product of the constraint vector and dual solution candidates, ensuring the solution meets constraints and is optimal.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If interior point methods or simplex methods are used to solve linear programming problems, then the optimal solution can be found, but the time to converge cannot be reliably estimated and may exceed the control cycle
Solution Approach 1:
The patent pre-calculates and stores all dual solution candidates and their corresponding active sets before the control cycle begins. This preliminary action eliminates the need for iterative convergence during real-time control, as the system simply selects from pre-computed candidates based on the current constraint vector, thereby resolving the contradiction between solution optimality and convergence time.
2Manufacturing precision
If iterative calculation methods are used, then the solution can be refined to meet constraints, but the calculation may not complete within the control cycle
Solution Approach 1:
All dual solution candidates satisfying the constraints are pre-calculated and stored with their corresponding active sets before runtime. During the control cycle, the system performs a direct selection from these pre-validated candidates based on the objective function, eliminating iterative refinement and ensuring both constraint satisfaction and real-time performance.
3Productivity
If pre-calculated data is stored for high-speed solution, then the calculation speed increases, but the solution may compromise optimality or violate constraints
Solution Approach 1:
The system pre-calculates all possible dual solution candidates that satisfy the constraints and stores them with their corresponding active sets. During runtime, the optimal solution is determined by evaluating the objective function against these pre-validated candidates, ensuring both high-speed calculation and solution optimality without compromise.
Solution Approach 2:
The system uses the inner product of the constraint vector and dual solution candidates as a selection criterion to identify the optimal active set from pre-calculated candidates. This feedback mechanism ensures that the selected solution remains optimal while leveraging pre-computed data for high-speed calculation.
Data Source
AI summary
A dual solution candidate searcher receives an input of information about a constraint coefficient matrix and a cost vector, determines a dual problem of a linear programming problem being a primal problem and all active sets representing combinations of active formulas in constraints of the dual problem, finds, for each of the active sets, a feasible dual solution candidate meeting constraints, and stores the dual solution candidate into a storage in a manner associated with a corresponding one of the active sets. An optimal solution calculation device receives an input of a constraint vector as, selects an optimal one of the active sets as an optimal active set based on an inner product of the constraint vector and the dual solution candidate stored in the storage, and finds and outputs a basic feasible solution corresponding to the selected active set as an optimal solution.