Month: February 2023

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… Continue reading Convex functions

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… Continue reading Convex sets

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… Continue reading Local and global optimality

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… Continue reading CONVEXITY AND OPTIMIZATION

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… Continue reading Sensitivity Analysis

The derivative tells us something about the rate at which a function f increases or decreases at some point x. This rate is the slope of the tangent line to the graph of f at x. So, the derivative tells us something about the “linear” behavior of a function. However, this does not tell us anything about its curvature. To visualize… Continue reading HIGHER-ORDER DERIVATIVES AND TAYLOR EXPANSIONS

The derivative is the slope of the tangent line to the graph of a function. Hence, the sign of the derivative at a point tells us whether the function is increasing or decreasing there and how rapidly. We can use this to figure out essential features of a function and to sketch its graph. Example… Continue reading USING DERIVATIVES FOR GRAPHING FUNCTIONS

The rules of previous section do not help us in finding derivatives of functions like the square root or, given the derivative of logarithm, in finding the derivative of the exponential. We need a rule to deal with the derivative of an inverse function. THEOREM 2.9 (Derivative of an inverse function) Let x = g(y) be the… Continue reading Derivative of inverse functions

Given two functions g and h, we may build a new function by composition, namely, g o h. It would be nice to have a way of finding the derivative of the composite function by decomposing the task and exploiting knowledge about the derivatives of g and h. THEOREM 2.8 (Chain rule) Given functions g and h, we obtain the derivative of their composition as… Continue reading Derivative of composite functions

Given two functions f and g, there are a few easy ways to build other functions by ordinary arithmetic operations such as sum, multiplication, and division. If we are able to find the derivative of f and g, the following theorem shows how to find the derivative of functions defined by the mechanisms above. THEOREM 2.7 Let f and g be functions… Continue reading Derivative of functions obtained by sum, multiplication, and division