Let us consider how we may represent a semicontinuous variable within an LP framework. As we have mentioned, a common requirement on the level of an activity is that, if it is undertaken, its level should be in the interval [m,M]. Note that this is not equivalent to requiring that. Rather, we want something like 

which is a non-convex set (recall that the union of convex sets need not be convex). Using the same big-M trick as above, we may introduce a binary variable  and write 

Semicontinuous variables may be used when the amount of an asset in a portfolio must be above a minimum threshold, if the asset is included in the portfolio.

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mxM