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When mathematicians speak of 'optimising' do they always mean 'minimising'? Or does the term refer to a basket of techniques?

For example Wolfram MathWorld describes Optimization Theory as, "A branch of mathematics which encompasses many diverse areas of minimization and optimization".

Is optimization synonymous with minimization? Are all optimisation problems usually viewed in terms of minimising some cost or energy function? If you read in a paper that some value is to be "optimised", do you assume the author means "minimized"?

For context if a paper refers to an rms (root mean square) value being "optimised", one would assume the idea is to minimise it.

Robinson
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  • I would argue that the difference is that "minimizing" suggests that you know exactly what function you'd like to be as small as possible, whereas "optimizing" suggests that this point is not settled. For example, when we speak of optimizing the fit of a line to some data, you might be speaking of Least Squares, but then you might mean something else (Weighted Least Squares, say, or you might want to use a measure of error other than squared distance). In general curve fitting, for example, you certainly want low errors but you might also want some degree of smoothness. – lulu Sep 09 '15 at 11:28
  • Added a little context to the question - assuming we're using rms (root mean square) across some value in some domain as our measure, there's no possibility we don't mean "minimise" whenever we say "optimise"? Sorry to sound so pedantic but the distinction is important for my current task. – Robinson Sep 09 '15 at 11:45
  • It's not a question of pedantry, it's a question of exactly what the situation calls for. If you are asked "find the parameters which minimize the root mean square error in some model" then that is a straight minimization question, I agree. In real world situations, however, people often look at the result of that and then say that they want some other property in addition. But, yes, the problem you describe sounds like straight minimization. – lulu Sep 09 '15 at 11:49

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No, optimizing is about to find the best while minimizing is about to find the least.

Now when minimizing a function $f$ on a set $D$ you want to find the $m\in D$ such that $f(m) \le f(x)$ for all $x\in D$ (in some cases you might settle for the value $f(m)$, but often you want to have the argument as well).

Now optimizing means that you want to find the "best" $m$ in $D$, but that relies on evaluating which element is better than the other. Often you describe this relation by assigning a "badness" to each element, that is a function $f$ such that $f(a) < f(b)$ means that $a$ is better than $b$. This would reduce the optimization problem to a minimization problem.

skyking
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