Incorporating a Max in optimization objective

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I'm trying to use intlinprog to solve a mixed integer linear program where the objective function has a max function embedded in it. For example, the objective may be of the form:

5*max(10-x, 0)+x

The objective is clearly convex and actually piecewise linear. How can I incorporate this in the form

f'x

so as to be able to use intlinprog?

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cchen 2016-12-5

编辑：cchen 2016-12-5

I see what you are saying. Currently, I'm using cvx to model a larger convex mixed-integer problem, which then calls Mosek. It is slow however, and I was hoping to improve upon this by using a Matlab solver. I thought I could use intlinprog since the cvx input to Mosek is an integer linear program, but reading the model, I cannot fully understand the reformulation.

Oskar Adolfson 2017-10-12

I think it is false to say that a convex function is not linear. A convex function can actually be linear, but it doesn't have to be. Try to verify yourself by testing if y = ax is convex.

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Matt J 2017-10-12

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Splitting this piecewise linear function into regions where it is a linear function is akin to knowing the minimum solution in the first place, so I don't see what use this has.

Not sure why that approach isn't worth pursuing. For this example, you simply add the constraint x<=10 for the first region and x>=10 for the second.