Effectiveness of grid and random approaches for a model parameter vector optimization

Selection of effective initial parameter vectors is important for mathematical models having parameter vectors and differential equations in many science and engineering problems. In this paper, we propose a new mathematical method for an inverse problem of parameter vector optimization. We analyze and compare the effectiveness of grid and random approaches in hyperbox in terms of nonlinear least squares error, maximum improvement factor and number of iterations for an inverse problem of parameter vector optimization in a mathematical model coming from asset flow theory. This analysis is valuable to understand the population dynamics of investors and machine learning applications. For this purpose, we use quasi-Newton (QN) method having the Broyden–Fletcher–Goldfarb–Shanno (BFGS) formula with backtracking line search algorithm to optimize the function F[K̃] for each selected event and initial parameter vector, where F[K̃] represents the sum of exponentially weighted squared differences between the proxy for actual market price values via simulation and the computed market price values. Moreover, we employ Monte Carlo simulations and obtain convergence diagrams. We find that the success of the grid approach is relatively better than that of the random approach based on our simulation data set in the unconstrained optimization problem.