Abstract
BackgroundThe estimation of parameter values for mathematical models of biological systems is an optimization problem that is particularly challenging due to the nonlinearities involved. One major difficulty is the existence of multiple minima in which standard optimization methods may fall during the search. Deterministic global optimization methods overcome this limitation, ensuring convergence to the global optimum within a desired tolerance. Global optimization techniques are usually classified into stochastic and deterministic. The former typically lead to lower CPU times but offer no guarantee of convergence to the global minimum in a finite number of iterations. In contrast, deterministic methods provide solutions of a given quality (i.e., optimality gap), but tend to lead to large computational burdens.ResultsThis work presents a deterministic outer approximation-based algorithm for the global optimization of dynamic problems arising in the parameter estimation of models of biological systems. Our approach, which offers a theoretical guarantee of convergence to global minimum, is based on reformulating the set of ordinary differential equations into an equivalent set of algebraic equations through the use of orthogonal collocation methods, giving rise to a nonconvex nonlinear programming (NLP) problem. This nonconvex NLP is decomposed into two hierarchical levels: a master mixed-integer linear programming problem (MILP) that provides a rigorous lower bound on the optimal solution, and a reduced-space slave NLP that yields an upper bound. The algorithm iterates between these two levels until a termination criterion is satisfied.ConclusionThe capabilities of our approach were tested in two benchmark problems, in which the performance of our algorithm was compared with that of the commercial global optimization package BARON. The proposed strategy produced near optimal solutions (i.e., within a desired tolerance) in a fraction of the CPU time required by BARON.
Highlights
The estimation of parameter values for mathematical models of biological systems is an optimization problem that is challenging due to the nonlinearities involved
Optimization approach The method devised for globally optimizing the nonlinear programming (NLP) that arises from the reformulation of the parameter estimation problem (Eqs. 13–17) is based on an outer approximation algorithm [8] used by the authors in previous works [13,14,15,16,17]
The master problem is a relaxation of the original NLP and provides a rigorous lower bound on its global optimum
Summary
The estimation of parameter values for mathematical models of biological systems is an optimization problem that is challenging due to the nonlinearities involved. Deterministic global optimization methods overcome this limitation, ensuring convergence to the global optimum within a desired tolerance. Global optimization techniques are usually classified into stochastic and deterministic. The former typically lead to lower CPU times but offer no guarantee of convergence to the global minimum in a finite number of iterations. Mathematical modelling of biological systems is nowadays becoming an essential partner of experimental work. One of the most challenging tasks and protein expression data, the latter are still very rare In this context, there is a strong motivation for developing systematic techniques for building dynamic biological models from experimental data. The parameter estimation of these models gives rise to dynamic optimization problems which are hard to solve
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