# Optimization

The parameters of plasma input TACs and compartmental models are estimated (fitted) using non-linear least-squares optimization algorithms (NLLSs), such as Particle swarm optimization (PSO), artificial immune network (AIN), and gravitational search algorithm (GSA).

In times of very limited computing resources the traditional methods such as Newton-Gaussian or Levenberg-Marquardt algorithms were also used; being very fast, those methods are very dependent on the initial parameter guesses. These methods converge to the local optimum, but could be used as a part of global optimization routines. For example, the commonly used Nelder-Mead algorithm (Downhill Simplex) (Nelder & Mead, 1965; Price et al., 2002) forms the basis of globalized bounded Nelder-Mead (GBNM) algorithm (Luersen et al., 2004a and 2004b). Local optimization method can be used as part of simulated annealing (SA) algorithm (Wong et al., 2002; Yaqub et al., 2006). Topographical global optimization (TGO) is one of the methods that can be used to get good initial parameter estimates for local optimization routines (Sederholm, 2003; Henderson et al., 2017).

Population-based heuristic optimization methods are affected by the selection of the initial population of parameters. Initial population is usually set randomly using pseudo-random number generators such as Mersenne Twister. However, it is more important that the initial population is evenly distributed than random, and therefore quasi-random low-discrepancy sequences are preferred over pseudo-random numbers (Maaranen et al., 2004).

Several stopping rules for terminating global optimization methods have been proposed (Betrò and Schoen, 1986; Boender and Kan, 1991; Hart, 1998; Lagaris and Tsoulos, 2008).

While simple models with few parameters can be fitted with any algorithm, more effort is needed to select and refine the algorithm as the complexity of the model increases; no algorithm works well for all optimization problems (Wolpert & Macready, 1997).

## Linearisation

Alternatively, some non-linear models can be linearised to estimate the macroparameter of interest using linear regression, for example the multiple-time graphical analyses, or individual model parameters using for example NNLS or GLLS (Feng et al., 1995, 1996, 1999). Based on simulations, Muzic and Christian (2006) have shown that iteratively re-weighted least squares (IRLS) and variations of extended least squares (ELS0, ELS1, ELS3) perform better than methods (WLS, GLLS, PWLS) that determine weights based directly on the measured data.

## Literature

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Updated at: 2022-03-26
Created at: 2017-08-07
Written by: Vesa Oikonen