Polynomial Dynamics Control via Degree-Reduced Optimization
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Solution Overview
Problem
Existing optimization techniques fail to solve polynomial optimization problems under strict timing constraints due to large linear transformation sizes and weak relaxation bounds, leading to increased computational times and inefficiencies in controlling systems with polynomial dynamics.
Innovation Solution
A device that transforms polynomial optimization problems into a higher dimension space by introducing a minimum number of additional variables, reducing the degree of the polynomial function to a target degree through a reformulation, and solving a mixed-integer optimization problem to find an optimal solution among all possible encodings, constrained by the number of introduced variables.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If polynomial optimization problems are transformed into linear programs in higher dimension space, then the problems can be solved, but the resulting large transformation size increases computational times
Solution Approach 1:
The patent transforms polynomial optimization problems into a higher dimension space by introducing additional variables. Specifically, it converts degree-d polynomial problems into degree-2 polynomial problems in higher dimensions, enabling the use of efficient quadratic programming techniques while maintaining solution accuracy.
Solution Approach 2:
The patent changes the parameters of the optimization problem by transforming the degree-d polynomial into a degree-2 polynomial through variable substitution. This parameter transformation reduces computational complexity while preserving the essential optimization structure.
2Device complexity
If weak relaxation is obtained from linear transformation, then the transformation is simpler, but many nodes must be explored in iterative optimization increasing computational times
Solution Approach 1:
The patent improves the relaxation bound by changing the transformation parameters from simple linear transformation to a structured quadratic transformation. This parameter change yields tighter relaxation bounds that reduce the search space in branch-and-bound algorithms, decreasing computational time despite increased transformation complexity.
Solution Approach 2:
The patent introduces additional variables as intermediaries to represent products of original variables. These intermediary variables enable tighter relaxation bounds by capturing second-order interactions, which improves optimization efficiency without excessively increasing complexity.
3Productivity
If a minimum number of additional variables are introduced to reduce polynomial degree, then computational efficiency improves, but constraints on the structure of additional variables increase problem complexity
Solution Approach 1:
The patent segments the polynomial optimization problem by introducing additional variables that represent specific products of original variables. This segmentation allows the high-degree polynomial to be broken down into quadratic components, improving efficiency while managing complexity through structured variable definitions.
Data Source
AI summary
A device for controlling an operation of a system performing a task determines a current control input based on the feedback signal by solving a polynomial optimization of a polynomial function with a reformulation derived by introducing additional variables reducing the degree of the polynomial function till a target degree subject to constraints on a structure of the additional variables. The device solves a mixed-integer optimization problem to find a subset of encodings among all possible encodings of factorizations of the polynomial function that reduce the degree of the polynomial function to the target degree with a predetermined minimum number of additional variables and selects an optimal encoding from the subset of encodings with an optimal relaxation bound.