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It is a rather theoretical question to better understand some definitions.

Can descision variables in descrete optimization problems be only integers? If I have a set of real numbers which are not only natural numbers is it a continious optimization problem?

An example of a problem:

If there is a set of variables:

animal value
frog 0.54
cat -9.12
duck 0.001
dog 4
snake -300.09

is that a continious or descrete problem?

From the Wiki definition: A continuous variable is a variable whose value is obtained by measuring, i.e., one which can take on an uncountable set of values. For example, a variable over a non-empty range of the real numbers is continuous, if it can take on any value in that range. The reason is that any range of real numbers between a and b with a , b ∈ R ; a ≠ b is uncountable. A variable is a discrete variable if and only if there exists a one-to-one correspondence between this variable and N, the set of natural numbers.

Theoretically I could say that the example is a non-empty range of real numbers between -300.09 and 4 and see it is a continuous problem. From the Wikipedia definition I understand that descrete variables must belong to a set of natural numbers, so it is not my case. Or do I undestand it wrong?

Alex
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  • [computerscience.se] is a more appropriate site for this question, as it is not about programming (code) or use of a programmer's tool. Also, you'll find your experiences here will be much better if you spend some time taking the [tour] and reading the [help] pages to learn how the site works before you begin posting. – Ken White Dec 06 '22 at 02:48
  • Mixed-integer programming is the canonical example of an discrete-optimization problems which has continuous as well as discrete variables (*mixed*). Without discrete-variables, one cannot really capture decisions or discontinuous parts on the optimization-surface (e.g. "if you set x=3; set y=5"). On the other hand, one will lose gradients which make cont. opt so attractive. Imho the general rule will say: treat as much as possible in an continuous way. – sascha Dec 06 '22 at 12:51

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