Author: haroonkhan

Consider a function f on interval [a, b]. If the function assumes nonnegative values on that interval, it will define a region below its graph; this is illustrated as the shaded region in Fig. 2.31. Now imagine that we are interested in the area of that region. If the function were constant or linear, we would get the…

The last section of this chapter deals with definite integrals. The concept of integral plays a fundamental role in calculus and applied mathematics and, as we shall see, it is in a sense the opposite operation with respect to taking derivatives. In the book, we use definite integrals essentially to deal with continuous random variables…

Series are another important topic in classical calculus. They have limited use in the remainder, so we will offer a very limited treatment, covering what is strictly necessary. To motivate the study of series, let us consider once again the price of a fixed-coupon bond, with coupon C and face value F, maturing at time T. If we discount…

One of the most fruitful application fields of quantitative methods is revenue management. Revenue management is actually a group of techniques that can be applied in quite diverse settings, such as pricing of aircraft seats or perishable products. In this section we consider an idealized case in which a manufacturer has to find an optimal price…

Convexity and concavity play a major role in optimization. Consider a one-dimensional optimization problem, ; this problem is unconstrained, since x can be any point on the real line. Furthermore, assume that f is convex on the whole real line  and that x* is a stationarity point. Property 2.18 applies to x*: for any , but this implies that x* is a global minimizer. We have proved the following theorem.…

Convexity can be easily generalized to functions by applying the idea of convexity for sets to the epigraph of the function. For functions of a single variable, which can be plotted on a plane, the epigraph of the function is just the set of points lying above the function graph. The idea generalizes to an arbitrary number…

Convexity can be introduced as a fairly intuitive concept that applies to n-dimensional subsets of . Spaces with multiple dimensions will be the subject of next chapter, but we can visualize things on a plane, which is just the set  of points with two coordinates. We use boldface characters when referring to a point , with coordinates (x1, x2). Subscripts…

Earlier we plotted the polynomial function whose graph is reported again for convenience in Fig. 2.23. The stationarity points can be found by setting its derivative to zero: Using numerical methods, we find the following roots of f′(x): which are indeed the points at which f is stationary. Observing the graph, we see that x1 is the global minimum, x2 is a local…

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…

Reading on, you will notice that a large part of deals with uncertainty. Uncertainty comes in many forms: One way to deal with uncertainty is to rely on the tools of probability theory and statistics. The main limitation of these tools is that they may require a lot of past data to characterize uncertainty, assuming…