Numerical Algorithms for Optimization Problems in Genetical Analysis

The focus of this thesis is on numerical algorithms for efficient solution of QTL analysis problem in genetics. Firstly, we consider QTL mapping problems where a standard least-squares model is used for computing the model fit. We develop optimization methods for the local problems in a hybrid global-local optimization scheme for determining the optimal set of QTL locations. Here, the local problems have constant bound constraints and may be non-convex and/or flat in one or more directions. We propose an enhanced quasi-Newton method and also implement several schemes for constrained optimization. The algorithms are adopted to the QTL optimization problems. We show that it is possible to use the new schemes to solve problems with up to 6 QTLs efficiently and accurately, and that the work is reduced with up to two orders magnitude compared to using only global optimization. Secondly, we study numerical methods for QTL mapping where variance component estimation and a REML model is used. This results in a non-linear optimization problem for computing the model fit in each set of QTL locations. Here, we compare different optimization schemes and adopt them for the specifics of the problem. The results show that our version of the active set method is efficient and robust, which is not the case for methods used earlier. We also study the matrix operations performed inside the optimization loop, and develop more efficient algorithms for the REML computations. We develop a scheme for reducing the number of objective function evaluations, and we accelerate the computations of the derivatives of the log-likelihood by introducing an efficient scheme for computing the inverse of the variance-covariance matrix and other components of the derivatives of the log-likelihood.