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Because of the following relation,

\begin{equation*} \inf(S) = -\sup(-S), \end{equation*} minimization and maximization is essentially the same thing. However, take any optimization routine in R for example (optim, nlm, etc.), and the default setting is to minimize.

Even for something named maximum likelihood estimation, it is normal to minimize the (negative) log-likelihood i.e. implementation is done as

\begin{equation*} \widehat \theta_n = \operatorname*{arg min}_{\theta \in \Theta} - l(\theta;x_1,\ldots, x_n) \end{equation*}

and I have been taught to do this even when I was searching analytically.

Historically, what is the reasoning behind this more frequent use of minimization compared to maximization? I have a slight suspicion that it has something to do with how convex and concave functions historically have been analyzed but that may just come from thin air.

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  • $\begingroup$ You can tell pretty accurately how close to zero you are. Determining how close to +infinity you are is much harder (such as, which infinity?). $\endgroup$ – Jon Custer Jun 15 '16 at 23:09
  • $\begingroup$ Another track: many quantities that are to be minimized can be interpreted in terms of energy, which, in general, is looked for in its minimal states. $\endgroup$ – Jean Marie Becker Jul 24 '16 at 20:13

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