Direct Optimization Using Gaussian Quadrature and Continuous Runge-Kutta Methods: Application to an Innovation Diffusion Model Fasma Diele 1 , Carmela Marangi 1 , and Stefania Ragni 2 1 Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, 70126 Bari, Italy 2 Facolt`a di Economia, Universit`a di Bari, Via Camillo Rosalba 56, 70100 Bari, Italy Abstract. In the present paper the discretization of a particular model arising in the economic field of innovation diffusion is developed. It con- sists of an optimal control problem governed by an ordinary differential equation. We propose a direct optimization approach characterized by an explicit, fixed step-size continuous Runge-Kutta integration for the state variable approximation. Moreover, high-order Gaussian quadrature ru- les are used to discretize the objective function. In this way, the optimal control problem is converted into a nonlinear programming one which is solved by means of classical algorithms. 1 Introduction Dynamic optimization represents a challenging problem in several fields of ap- plied science. From biology to engineering or economics, a wide variety of phe- nomena can be described in terms of optimal control problems, where a cost functional has to be optimized with respect to variables which control the sy- stem dynamics. Numerical methods for dynamical optimization fall essentially into two clas- ses: classical, indirect methods, relying on the maximum Pontryagin principle, with the main drawback of a lack of robustness, and so-called direct methods which attempt to find a solution through a direct optimization of the cost fun- ctional. In this paper we focus on the last type of methods and introduce a numerical scheme based on the continuous Runge-Kutta integration ([9], [11]) to solve a specific model in economics. The rationale for this choice is that an efficient and accurate approximation of the system dynamics can be obtained with a reduced number of variable evaluations. A further improvement in accu- racy is achieved by adopting high-order Gauss-Legendre quadrature rules ([1]) for the functional discretization. The approach has been tested on a specific innovation diffusion model re- cently proposed in [5]. The early mathematical models in this field date back to 1950 (see [7]) and represent an interesting research subject in economics, socio- logy, marketing as well as in applied mathematics. The problem is easily stated as the one of maximizing the profit of a monopolistic firm, due to the sale of a M. Bubak et al. (Eds.): ICCS 2004, LNCS 3039, pp. 426–433, 2004. c  Springer-Verlag Berlin Heidelberg 2004