Recent developments in constrained optimization

Abstract

Constrained optimization problems occur in many applications of science, engineering, social science and medicine, and hence it is useful for practitioners in these areas to be informed about numerical methods for solving such problems. However, improvements in optimization typically do not spread quickly to the non-specialist literature. This paper contains a brief summary of two recent developments in the solution of nonlinearly constrained problems: direct use of the Lagrangian function, and linearization of nonlinear constraints. In order to highlight the new approaches, we also discuss techniques used in older methods. To illustrate the capabilities of modern optimization methods, a brief description is given of the development of methods to solve large nonlinearly constrained problems that arise in the electrical power industry. The general nonlinear programming problem may be stated in the following form: