Application of global search methods for parameter estimation in models with poorly defined objective function surfaces
Nikhil S Pillai (1), Eric A. Sherer (2), Mark Sale (3). Rob Bies (1)
(1) University at Buffalo The State University of New York, (2) Louisiana Tech University, (3) Nuventra Pharma Sciences
Objectives: Assess performance of global search methods for parameter estimation in nonlinear mixed effect modeling.
Methods: Re-analysis of existing clinical data.
Most nonlinear mixed effect modeling uses quasi-Newton, gradient based methods for parameter estimation. These methods are efficient, finding a solution with few evaluations of the objective function. They are, however, not robust with at least two conditions that result in failure to find the global minimum: an initial estimate whose gradients lead to local (rather than global) minima and the failure to estimate a useful gradient in complex highly nonlinear models.
Alternatives to gradient based methods are global search methods. These methods, including genetic algorithm, simulated annealing and particle swarm optimization, do not require a gradient and can be robust to local minima. We investigated the use of Genetic Algorithm/Nelder-Mead minimization for parameter estimation in a model were problems with local minima occurred using quasi-Newton methods.
We reanalyzed 17-dimethylaminoethylamino-17-demethoxygeldanamycin (DMAG) data [1], from 67 patients. These data and model were known to have irregular objective function surface, and location of global minima by quasi-Newton methods require evaluation at multiple initial parameter estimates. We compare the results from NONMEM with multiple initial parameter estimates with the results from Genetic Algorithm/Nelder Mead [2] using multiple initial seed values.
Results: A model for DMAG was previously established (Aregbe, 2012). Objective function value (OBJ) from different initial estimates using NONMEM are given below.
OBJ
9794.9
10367.9
9923.6
9799.1
9799.1
The same model and data set were used to estimate parameters using the GA package in R. Options for the minimization included population size of 1000, 1000 generations and 32 bits of precision. The search space for GA included the range of the initial estimates for NONMEM +/- 0.5 logs, except for exponential covariate relationships for which the search space was -4 to +4. The resulting objective function from 4 different seeds are given below:
OBJ
9795.4
9798.5
9795.6
9795.8
Conclusions: Genetic Algorithm-Nelder/Mead is an alternative parameter estimation method. As expected, this method is much less efficient than quasi-Newton, but appears to be robust to at least some problems associated with gradient based methods. Also as expected, Genetic algorithm has difficulty finding the optimal value in a search space, but is robust in finding “near optimal” solutions.
References:
[1] Aregbe AO. Sherer EA, Egorin MJ, Scher HI, Solit DB, Ramanathan RK, Ramalingam S, Belani CP, Ivy PS, Bies RR. (2012) Population pharmacokinetic analysis of 17-dimethylaminoethylamino-17-demethoxygeldanamycin (17-DMAG) in adult patients with solid tumors. Cancer Chemother Pharmacol
[2] 70:201–205 Scrucca, L. (2016). On some extensions to GA package: hybrid optimisation, parallelisation and islands evolution. Submitted to R Journal. http://arxiv.org/abs/1605.01931.