On the mixed monotonicity of polynomial functions
Adam M Tahir
TL;DR
This work proves that every univariate polynomial is mixed monotone on the entire real line with a polynomial decomposition function obtained from the Gram-matrix representation of its derivative, enabling a global and potentially tighter decomposition than local Jacobian-based methods. The decomposition is constructed as g(x,y) = q(x) − r(y), where q and r arise from integrating a pair of PSD matrices obtained from the Gram decomposition of p′. Extending to multivariate polynomials, the authors show that products and sums of mixed-monotone components preserve mixed monotonicity, yielding polynomial decompositions for all polynomials. The paper also discusses tightness via semidefinite programming to choose alpha, U, and V, and presents five illustrative examples—including a reachable-set over-approximation—to demonstrate tightness, monotonicity checks, and applicability to interval observers and set propagation. This approach provides a scalable framework for global mixed-monotone decompositions with practical impact on reachability and interval-based estimation of polynomial systems, while outlining avenues for handling uncertainty and further tightening.
Abstract
In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of polynomial functions. The tightness of polynomial decomposition functions is discussed. Several examples are provided. An example is provided to show that polynomial decomposition functions, in addition to being global decomposition functions, can be much tighter than local decomposition functions constructed using local Jacobian bounds. Furthermore, an example is provided to demonstrate the application to reachable set over-approximation.