Incorporating a Max in optimization objective

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?

[Brendan Hamm]

A convex function is not linear, so intlinprog will not solve it. 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.

What is the need for an optimization routine to solve this anyhow? Are you trying to expand the concept to a larger problem?

[cchen]

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]

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.

Answer 1

[Matt J]

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.

Answer 2

[Mario CASTRO GAMA]

You can try to use the method presented in this video.

LP piecewise linear constraints & objectives