Abstract
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.
Highlights
Across the physical, biological and social sciences, mathematical models are formulated and studied to better understand real-world phenomena
We demonstrate our techniques on examples from biology, where polynomial models often arise as the steadystate descriptions of mass-action chemical reaction networks
We demonstrate our methods on problems in biomedicine: cell death activation, synthetic biology, epidemiology and multisite phosphorylation
Summary
Biological and social sciences, mathematical models are formulated and studied to better understand real-world phenomena. It becomes necessary to choose between different models, for example, based on their fit with noisy experimental data. Unless f and g are convex, solving (1.1) is a non-convex problem, which can be challenging as standard local solvers run the risk of getting trapped in local minima (especially in high dimensions). This can be mitigated somewhat with techniques such as simulated annealing [6,7] or convex relaxation that has been successful for model invalidation [8,9,10], but there is generally no guarantee that a global minimum will be found
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