# Looking for a code in Matlab optimization

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Benson Gou on 30 Oct 2019
Commented: Benson Gou on 18 Nov 2019
Dear All,
I do not know if a code exists in Matlab optimization for the following integer linear programming.
Min ||A*x - b||_1
S.T. x is an integer variable, x>= 0 and x <= 1. A is a 1 x n row vector, and b is a scalar.
Thanks a lot.
Benson

John D'Errico on 30 Oct 2019
Edited: John D'Errico on 30 Oct 2019
No. There is no direct code to do so, at least not that I know of. Should I write it? sigh.
But if you spend some time reading, you will find this is just a binary integer linear programming problem. That is...
A 1-norm minimization of that form can be converted into a standard LP problem using slack variables. (Its been many years since I wrote code for that, but I recall doing so.) At that point, it becomes a simple call to intlinprog.
A quick search shows the way...

Benson Gou on 1 Nov 2019
Because my goal is to solve for the integer variable x, that is why I added x in the objective function. If I do not add it, what solution of x could be? I am very sorry, I am not quite clear.
Thanks a lot.
Benson
John D'Errico on 4 Nov 2019
Adding x to the objective function means you will get the wrong answer. We told you how to solve it. The objective function includes ONLY the slack variables. You cannot just make the objective function anything you want and have it be correct.
Or, far betteryet, just use Matt's code, as writing code to do somethign for which well written code already exists is just a bad idea. Unless of course, you already completely understand the problem, and how to solve it, AND you have considerable expertise in coding. You have none of that, so it is far better to use Matt's code.
Benson Gou on 18 Nov 2019
Thanks, John and Matt. I define the objective function as J = w_x*sum(x) + w_s*sum(s). In order to leave only s in the objective and at the same time add the integer contraint on x, I set w_x=0 when using intlinprog.m. It now works. Thanks a lot for your great hekp.