gams:smooth_approximations_for_max_x_0_and_min_x_0

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 one find a a smooth approximation for max(x,0), and min(x,0)?

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

Use the approximation

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:

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.

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gams/smooth_approximations_for_max_x_0_and_min_x_0.txt · Last modified: 2020/05/20 18:17 by Michael Bussieck