Re: Objective function defined with max, min.

On Thu, 5 Jan 2017, Michael Hennebry wrote:

> The objective function includes  crop(s) = median(0, s, 1)
> where the range of s includes both negative values and values > 1 .
>
> The set defined is not convex and so cannot be defined purely with
> linear constraints.
> One can get around the need for a binary by using optimality.
>
> Add nonnegative auxillary variables p0, n0, p1 and n1.
> s = p0-n0
> s = p1-n1+1
> Adjust the objective to ensure that p0 or n0 is zero
> at optimality and that p1 or n1 is zero at optimality.

Oops.  That does not work.
There are situations in which the optimality condition is useful,
but your function is neither convex nor concave.

You will need at least one binary.

The convex hull of (s, crop(s) has vertices
(smin, 0) (0, 0) (smax, 1) (1, 1)
in that order.

> 0<=crop(s)<=1
> crop(s)<=p0
> crop(s)>=1-n1

-- 
Michael   [email protected]
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