Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 36

Let us consider a mathematical optimization problem represented as (Equation 2.22).

(2.22)

where f : ⁿ → is the objective function, x are decision variables, Ω is the feasible set, and β are constant parameters of the problem.

A point x~ is a local optimum of the problem, if there exists an open set N(x~), named neighborhood, that contains x~ such that f(x)≥f(x~),∀x∈N(x~). If N=Ω then, the optimum is global. Figure 2.4 shows the concept for two functions in R with their respective neighborhoods N.

Figure 2.4 Example of local and global optima: a) function with two local minima and their respective neighborhoods, b) function with a unique global minimum (the neighborhood is the entire domain of the function).

There are two local minima in the first case, whereas there is a unique global minimum in the second case. This concept is more than a fancy theoretical notion; what good is a local optimum if there are even better solutions in another region of the feasible set? In practice, we require global or close-to-global optimum solutions.

On the other hand, several points may be optimal, as shown in Figure 2.5. In that case, all the points in the interval x₁ ≤ x ≤ x₂ are global optima. Thus, the question is not only if the optimal point is global but also if it is unique. Both globality and uniqueness are geometrical questions with practical implications, especially in competitive markets. Convex optimization allows naturally answering these questions as explained in Chapter 3

Figure 2.5 Example of a function with several optimal points.