Genetic Optimization Using Derivatives

Abstract

We describe a new computer program that combines evolutionary algorithm methods with a derivative-based, quasi-Newton method to solve difficult unconstrained optimization problems. The program, called GENOUD (GENetic Optimization Using Derivatives), effectively solves problems that are nonlinear or perhaps even discontinuous in the parameters of the function to be optimized. When a statistical model's estimating function (for example, a log-likelihood) is nonlinear in the model's parameters, the function to be optimized will usually not be globally concave and may contain irregularities such as saddlepoints or discontinuous jumps. Optimization methods that rely on derivatives of the objective function may be unable to find any optimum at all. Or multiple local optima may exist, so that there is no guarantee that a derivative-based method will converge to the global optimum. We discuss the theoretical basis for expecting GENOUD to have a high probability of finding global optima. We conduct Monte Carlo experiments using scalar Normal mixture densities to illustrate this capability. We also use a system of four simultaneous nonlinear equations that has many parameters and multiple local optima to compare the performance of GENOUD to that of the Gauss-Newton algorithm in SAS's PROC MODEL.