An introduction to nonlinear programming—I: Necessary and sufficient conditions

Abstract Parameter optimization problems arise in a wide variety of applications. The importance of the general problem has led to the investigation and development of the field of nonlinear programming and of the special classes of these problems such as linear programming . The study of these problems has reached a sophisticated level and an extensive literature exists, including a number of excellent textbooks. In this and in subsequent articles, the basic results of nonlinear programming are reviewed, developed, and discussed. No attempt is made here to survey the vast literature on the subject but sufficient references are included to provide the reader with a basis for a more detailed study. It is hoped that this series of tutorial articles will make the literature more accessible and will provide a useful perspective on the current status of the field. In this first article, the conditions which a solution must satisfy are discussed. To a lesser extent, conditions are examined which, when satisfied, ensure that a solution has been obtained. The discussion in this article of necessary and sufficient conditions for an optimum provides a basis for the subsequent examination of algorithms which can be used to determine the solution of parameter optimization (i.e. nonlinear programming) problems.