Machine Task Control Using Lifted QP Infeasibility Detection
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Solution Overview
Problem
Quadratic programs often face difficulties due to potential non-convexity and constraints, and may be infeasible when constraints defining different feasible sets do not intersect.
Innovation Solution
The solution involves reformulating the constraints into a higher-dimensional space to ensure intersection of all possible sets defined by different constraints, allowing for detection of infeasibility based on convergence rather than divergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the quadratic program is solved using conventional methods (interior point method, active set method, etc.), then the optimization problem can be addressed, but the method fails when constraints define infeasible sets that do not intersect
Solution Approach 1:
The patent transforms the original quadratic program from an n-dimensional space to an (n+1)-dimensional lifted space by introducing an additional non-negative variable τ. This dimensional extension allows the formulation of a homogeneous quadratic program where infeasibility can be detected through the value of τ. Specifically, the original constraints Ax=b and Cx≤d are lifted to Aξ=bτ and Cξ≤dτ in the higher-dimensional space, creating a system that always has a feasible solution (the origin) while enabling reliable infeasibility detection.
2Difficulty of detecting and measuring
If the quadratic program is reformulated into a different space using methods like augmented Lagrangian, then some difficulties are addressed, but the infeasibility detection remains problematic and computationally expensive
Solution Approach 1:
The homogeneous quadratic program formulation is self-diagnosing through the additional variable τ. When the original QP is infeasible, the optimization naturally drives τ to zero, automatically signaling infeasibility without requiring separate detection mechanisms. The system serves itself by embedding the feasibility check within the optimization objective, eliminating the need for complex external infeasibility detection algorithms and reducing computational overhead.
Solution Approach 2:
The patent performs preliminary transformation of the original QP into a homogeneous form before solving, by introducing the lifting variable τ and reformulating constraints in advance. This preliminary action ensures that the optimization problem is structured to always have a feasible solution (the origin in lifted space), preventing solver failure and enabling straightforward infeasibility detection through the value of τ at optimality.
3Reliability
If the constraints are lifted into a higher-dimensional space, then all constraint sets intersect at least at the origin, but the dimensionality of the problem increases
Solution Approach 1:
The patent extracts the feasibility detection function from the constraint satisfaction problem and embeds it in the objective through the additional variable τ. By separating the feasibility check (indicated by τ=0) from the primary optimization variables, the method handles the increased dimensionality efficiently. The origin (0,0) in lifted space serves as a universal intersection point that guarantees feasibility, while τ acts as an extracted indicator of original problem infeasibility.
Data Source
AI summary
The present disclosure discloses a system and a method for controlling an operation of a machine according to a task. The method comprises formulating an original quadratic program (QP) for optimizing an objective function subject to equality constraints and inequality constraints, lifting the equality constraints and the inequality constraints into a lifted space by a lifting operation introducing an additional non-negative variable, and transforming the objective function of the original QP into a quadratic objective function. The quadratic objective function subject to the lifted equality and inequality constraints forms a homogeneous QP in the lifted space. The method further comprises solving the homogeneous QP to produce a solution in the lifted space and controlling the machine according to an infeasibility protocol when a value of the additional non-negative variable in the solution in the lifted space equals zero.