Smooth approximations for MAX(X,0) and MIN(X,0)

[aileen]

The use of min and max in a model make some derivatives discontinuous and the model type DNLP needs to be used and solvers get stuck at the point with discontinuous derivatives. How can I find a smooth approximation for max(x,0), and min(x,0)?

[aileen]

Here is the answer from Prof. Ignacio Grossmann (Carnegie Mellon University):

Use the approximation:

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f(x) := ( sqrt( sqr(x) + sqr(epsilon) )  + x ) / 2

for max(x,0), where sqrt is the square root and sqr is the square.

The error err(x) = abs(f(x)-max(x,0)) in the above approximation is maximized at 0 (the point of non differentiability), where err(0) = epsilon/2. As x goes to +/- infinity, err(x) goes to 0. One can shift the function so the error at 0 becomes 0 but takes on epsilon/2 as x goes to +/- infinity:

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g(x) := ( sqrt( sqr(x) + sqr(epsilon) )  + x - epsilon ) / 2

Because min(x,0) = -max(-x,0), you can use the above approximations for min(x,0) as well. Epsilon is a small positive constant.