Polynomial Dynamics Control via Minimal-Variable Reformulation
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Solution Overview
Problem
Current optimization techniques for polynomial optimization problems in systems with polynomial dynamics often fail to solve these problems within strict timing constraints, especially when transforming them into linear programs, leading to inefficiencies in controlling complex systems like production lines and power grids.
Innovation Solution
A device that reformulates polynomial optimization problems by introducing a minimum number of additional variables, reducing the degree of the polynomial function to a target degree through mixed-integer optimization, allowing for efficient control input determination using bilinear or trilinear programming reformulations.
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 computational time exceeds strict timing constraints
Solution Approach 1:
The patent transforms the polynomial optimization problem into a higher-dimensional space by introducing additional variables representing monomial products. This dimensionality change allows the use of linear programming techniques while maintaining polynomial equivalence, resolving the contradiction between solvability and computational efficiency.
Solution Approach 2:
The patent changes the parameters of the optimization problem by reformulating polynomial constraints and objectives into linear constraints and objectives in an extended variable space. This parameter transformation enables the application of efficient linear programming algorithms to polynomial optimization problems within timing constraints.
2Productivity
If additional variables are introduced to reduce polynomial degree, then the degree is reduced to target degree, but the number of additional variables increases computational complexity
Solution Approach 1:
The patent segments the polynomial function into individual monomials and introduces variables only for the unique monomial products that appear in the polynomial. This segmentation approach reduces the number of additional variables compared to introducing variables for all possible monomial combinations, thereby reducing computational complexity while achieving degree reduction.
Solution Approach 2:
The patent creates a universal reformulation framework that handles polynomial optimization problems of any degree by systematically introducing variables for monomial products. This multi-functional approach allows the same reformulation technique to reduce polynomial degree efficiently across different problem instances, balancing computational speed and variable complexity.
Data Source
AI summary
A device for controlling an operation of a system performing a task is disclosed. The device submits a sequence of control inputs to the system thereby changing states of the system according to the task and receives a feedback signal. The device determines a current control input for controlling the system based on the feedback signal including a current measurement of a current state of the system by solving a polynomial optimization of a polynomial function with a reformulation derived by introducing additional variables reducing a 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 an optimal solution among all possible encodings of factorizations of the polynomial function that reduces the degree of the polynomial function till the target degree with a minimum number of additional variables.