What we have learned so far about function derivatives suggests that in order to optimize a function, assuming that it is differentiable, a good starting point is to set its first-order derivative to zero. However, we know that this first-order, stationarity condition may not be enough, as it does not even discriminate between a maximum and a minimum. In practice, the matter is further complicated by two features of optimization models:
- They involve a possibly large number of decision variables, whereas we are dealing here with functions of a single variable.
- They involve constraints, as we have seen in the optimal mix example of Section 1.1.2. Generally speaking, our decisions are constrained to stay within a set S called feasible region or feasible set.
We will learn a lot more about these issues. However, it would be nice to find a feature that makes an optimization problem relatively easy. This feature is termed convexity. We start by defining convexity for sets; then, we generalize the concept to functions;23 finally, we illustrate what role convexity plays in economic modeling and in optimization. Before doing so, though, it may be useful to clarify what we mean by local and global optimality formally.
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