Closed form solution

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This is where one could express a solution (e.g., optimal) by an equation of the

form:

LaTeX: 
x^* = F(p),

where LaTeX: F is some function of the parameters, LaTeX: p. This is typically meant to suggest that no algorithm is needed, but one must be careful since LaTeX: F is not limited in any way, such as being "easy" to evaluate. For example, under mild assumptions, the coordinates of a global maximum of a function, LaTeX: f, on the domain LaTeX: X, is given by the following integral:

LaTeX: 
\displaystyle
x^*_j = \lim_{w \rightarrow \infty}
\frac{\int_{X} x_j e^{wf(x)} dx} {\int_{X} e^{w f(x)} dx}.

(The idea is to weight "mass" at the global maximum, and it is valid if the set of global maxima is convex.) This is not too useful, and the integral would typically have to be evaluated with a numerical method - i.e., an algorithm.


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