Linear Programming Feasible Region Analysis via Iterative Constraint Segmentation
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Solution Overview
Problem
Current methods for solving linear programming problems, particularly those involving large numbers of constraints, are complex and often require non-negative variables or starting feasible points, limiting their applicability and efficiency in analyzing constraint sets without an objective function.
Innovation Solution
A method that selects a subset of constraints to calculate an initial feasible region, iteratively updates this region by incorporating additional constraints, and declares the final feasible region, allowing for the identification of active and non-active constraints, thereby simplifying the analysis of large constraint sets without requiring non-negative variables or initial feasible points.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If current methods (simplex algorithm) are used to solve linear programming problems, then the solution can be found for standard LP problems, but the computational complexity increases significantly when the number of constraints is large
Solution Approach 1:
The patent segments the constraint set into two parts: a subset of P constraints (where P > N) used to define the initial feasible region, and the remaining constraints processed iteratively. This segmentation reduces the initial computational burden by not processing all M constraints simultaneously, thereby lowering computational complexity while maintaining solution accuracy.
Solution Approach 2:
The patent performs preliminary action by first selecting P constraints out of M total constraints to calculate an initial feasible region before processing the remaining constraints. This preliminary region calculation establishes a foundation that simplifies subsequent iterative updates, reducing the overall computational complexity compared to processing all constraints from the beginning.
2Reliability
If the simplex algorithm is used, then LP problems can be solved, but additional assumptions are required (non-negative variables, non-negative right hand side constants, known starting feasible point)
Solution Approach 1:
The patent creates a universal method that can handle general constraint sets without requiring specific assumptions about variable signs or the presence of an objective function. The algorithm works for both optimization problems and pure constraint analysis, making it adaptable to various problem types including those without non-negative variable requirements or known starting feasible points.
Solution Approach 2:
Instead of starting with assumptions about non-negative variables and objective functions and working forward to find a solution, the patent inverts the approach by first analyzing the constraint set structure and identifying the feasible region defined by the constraints themselves. This inversion allows the method to handle cases where traditional LP assumptions do not apply.
3Measurement precision
If all M constraints are processed to find the feasible region, then the complete constraint analysis is obtained, but the computational time and complexity increase
Solution Approach 1:
The patent segments the constraint processing into two phases: an initial phase where P constraints are processed to define the feasible region, and a subsequent phase where remaining constraints are processed iteratively. This segmentation allows the method to achieve complete constraint analysis while reducing computational time by establishing the feasible region structure early, before processing all constraints.
Solution Approach 2:
The patent applies partial action by initially processing only P constraints (where P > N but P < M) to calculate the initial feasible region. This partial processing provides sufficient information to define the region structure, and only the remaining constraints need to be processed iteratively to refine the solution, thereby reducing total computational time while maintaining analysis completeness.
Data Source
AI summary
Method and system for analyzing a linear programming problem or any other problem involving inequalities constraints set relating to multiple variables. An initial feasible region is calculated based on a sub-set of the constraints set. The feasible region is updated based on the additional constraints added one at a time. The method checks for feasibly, identifies active constraints, and provides end-points of the feasible region. The method may be applied to a control system or to a crossbar switch handling routing between multiple input and multiple outputs, such as digital data networking switch used to route TDM digital data streams being packet, frame or cell based, in a LAN, WAN, MAN or Internet application.