Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 37

It is well-known, from basic mathematics, that the optimum of a continuous differentiable function is attached when its derivative is zero. This fact can be formalized in view of the concepts presented in previous sections. Consider a function f : → with a local minimum in x~. A neighborhood is defined as N={x∈R:x=x~±t,|t|<t0} with the following condition:

(2.23)

where t can be positive or negative. If t > 0, then (Equation 2.23) can be divided by t without modifying the direction of the inequality, to then take the limit when t→0+t → 0+ as presented below:

(2.24)

The same calculation can be made if t < 0, just in that case, the direction of the inequality changes as follows:

(2.25)

Notice that this limit is the definition of derivative; hence, f′(x~)≥0 and f′(x~)≤0 These two conditions hold simultaneously when f′(x~)=0. Consequently, the optimum of a differentiable function is the point where the derivative vanishes. This condition is local in the neighborhood N.

This idea can be easily extended to multivariable functions as follows: consider a function f:Rn→R (continuous and differentiable) and a neighborhood given by N={x∈Rn:x=x~+Δx} Now, define a function g(t)=f(x~+tΔx) If x~ is a local minimum of f, then

(2.26)

In terms of the new function g, (Equation 2.26) leads to the following condition:

(2.27)

This condition implies that 0 is a local optimum of g; moreover,

(2.28)

Notice that g is a function of one variable, then optimal conditiong′ = 0 is met, regardless the direction of Δx. Therefore, the optimum of a multivariate function is given when the gradient is zero ∇f(x~)=0). This condition permits to find local optimal points, as presented in the next section. Two questions are still open: in what conditions are the optimum global? And, when is the solution unique? We will answer these relevant questions in the next chapter. For now, let us see how to find the optimum using the gradient.