Constrained optimization: Big picture
If we have a function of (x, y) and bounded region R, then there are two possibilities: Either the maximum occurs at a critical point, meaning that the x derivative at this point equals 0 and the y derivative equals 0. Or another way of is saying it is that the gradient at this point equals 0. One possibility is that the maximum occurs at a critical point. This is true when the gradient vanishes, which means that both the derivative of x and the derivative of y vanish. Another possibility is that the maximum occurs at a boundary point. If F(x,y) bounded region R either 1) max. occur at a crit. point 2) max. occur at a boundary point [Graph] Perhaps it’s helpful to observe that the function may be visualized as a graph in the x and plane, where it looks like an upside-down bowl. A point (x0, yO) is called critical point of a function f(x, y) if f(x0, yo) = 0. The maximum of the function is at the top of the bowl. Example 2: Let R be the region defined by y = f(x). Suppose the maximum value of f occurs along an edge of R, as shown above. If has a slope at some point on this edge other than where it is it equal to zero (a critical point), the slope will be negative at that point. What if in this first example of the bowl, instead of being rotated downwards, it was rotated upwards? What would be the maximum then? If we have an upward bowl, then the minimum will be somewhere in the middle. This is a critical point. The maximum will be along the edge. If the bowl is perfectly symmetric, then there could be a tie. So how can we choose x and y to make fbig enough? There could be tie There could be several different, maybe even infinitely different. many different x’s and y’s that give the same value, which is the biggest value. That big value is the maximum. It occurs at all of those points so it could occur at all of the points all the way around the edge. To find the maximum, you take your function and its interval. find the critical points inside that interval, and check the boundary of those points. If there are two critical points on a boundary of an interval, then you have found your maximum. Well, let’s do the same thing. We can find all of the critical points. There are typically a few critical points in the domain. Then we have to check the boundary. The biggest difference is that checking the boundary is much more complicated because it’s a curve and there are infinitely many points on it. So we have to really think about how to handle it.
