Until now we have considered the optimization of a linear function subject to linear constraints. This assumption of linearity is now relaxed and we examine the complex problems of optimizing a function which is not necessarily linear which may possibly be subject to constraints which are also not necessarily linear. This present chapter is concerned with the calculus necessary to identify the optimal points of a continuous function or a functional. This is often called classical optimization, even though many of the results are of relatively recent origin. Occasionally these methods can be used to solve real-world problems. However, it is usual that too many variables are present for the methods to be at all efficient from the point of view of numerical computation. In these cases nonlinear programming algorithms must be developed and some of these are presented in the next chapter. However, most of these algorithms rely on the theoretical development of the present chapter.
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