Quadratic Programming Algorithm for Infeasible Initial Guesses
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Solution Overview
Problem
Existing real-time quadratic programming methods face challenges in maintaining consistency and feasibility when dealing with infeasible initial guesses for actuator commands, leading to potential constraint violations and suboptimal solutions in dynamic systems.
Innovation Solution
The algorithm solves the Equality-Constrained Quadratic Program (EQP) without relaxing the problem, using the initial guess for the active set to maintain consistency and feasibility, and introduces an extra variable to relax non-critical constraints, ensuring critical constraints are not violated, thereby retaining the benefits of a consistent active set hot start.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the initial guess for the quadratic program is infeasible and the problem is relaxed using the big-K formulation, then a feasible solution can be obtained, but the solution will violate the relaxed constraint and the objective function may increase from one iteration to another
Solution Approach 1:
The algorithm performs preliminary classification of constraints into critical and non-critical sets before solving. Critical constraints are identified a priori and enforced strictly, while non-critical constraints are relaxed only when necessary. This preliminary action prevents unnecessary relaxation and maintains solution quality.
Solution Approach 2:
Different constraint types are treated differently: critical constraints are enforced with strict feasibility requirements, while non-critical constraints allow controlled violations through relaxation variables. This local differentiation allows the algorithm to maintain high solution quality for important constraints while permitting flexibility elsewhere.
2Reliability
If constraints are relaxed to handle infeasible initial guesses, then feasibility can be achieved, but potentially promising constraints are dropped from the active set
Solution Approach 1:
The algorithm performs preliminary classification of constraints into critical and non-critical sets before solving. Critical constraints are identified a priori and enforced strictly, while non-critical constraints are relaxed only when necessary. This preliminary action prevents unnecessary relaxation and maintains solution quality.
Solution Approach 2:
Different constraint types are treated differently: critical constraints are enforced with strict feasibility requirements, while non-critical constraints allow controlled violations through relaxation variables. This local differentiation allows the algorithm to maintain high solution quality for important constraints while permitting flexibility elsewhere.
3Productivity
If the initial guess for the active set is used without relaxation, then computational efficiency is improved, but the solution may be infeasible
Solution Approach 1:
The algorithm performs preliminary classification of constraints into critical and non-critical sets before solving. Critical constraints are identified a priori and enforced strictly, while non-critical constraints are relaxed only when necessary. This preliminary action prevents unnecessary relaxation and maintains solution quality.
Solution Approach 2:
The algorithm dynamically adjusts the level of relaxation based on the problem state. It starts with minimal relaxation using the initial active set guess, then progressively relaxes only non-critical constraints if needed, rather than applying fixed relaxation from the beginning. This dynamic approach maintains efficiency while ensuring feasibility.
4Measurement precision
If consistency is maintained between the current iterate and active set, then solution quality is improved, but constraints must be dropped when the initial guess is inconsistent
Solution Approach 1:
The algorithm performs preliminary classification of constraints into critical and non-critical sets before solving. Critical constraints are identified a priori and enforced strictly, while non-critical constraints are relaxed only when necessary. This preliminary action prevents unnecessary relaxation and maintains solution quality.
Solution Approach 2:
Different constraint types are treated differently: critical constraints are enforced with strict feasibility requirements, while non-critical constraints allow controlled violations through relaxation variables. This local differentiation allows the algorithm to maintain high solution quality for important constraints while permitting flexibility elsewhere.
Data Source
AI summary
The present invention provides an algorithm that does not relax the problem at the very onset, even if xf is infeasible. Instead, it solves the EQP with the initial guess for the active set without relaxing the problem. If this solution to the first EQP is not optimal, but nevertheless feasible, we can use this as our guess for the feasible point. This has the advantage of being a feasible point that is consistent with the initial active set, whereas the initial guess used in the previous method is not necessarily so.