Absolute value as a linear programming constraint?

Suppose I want to use absolute values in a constraint equation for linear or mixed integer programming - i.e. suppose I need one of the form abs(x1) + abs(x2) <= 1.

How would I incorporate this in the constraint matrix in MATLAB?

[Matt J]

There are no absolute value expressions in the example you've given.

[Mike Vukovich]

Sorry, it was a typo... corrected.

Answer 1

[Matt J]

abs(x1+x2)<=1

is equivalent to the constraints

   x1+x2<=1

 -(x1+x2)<=1

[Mike Vukovich]

How would I use this in the constraint matrix in linprog, though?

[Matt J]

I've showed you that you can convert abs(x1+x2)<=1 to linear inequalities. You would express the inequalities in matrix form the way you always do.

[Brigham Frandsen]

You didn't answer the question. The original question was how to code abs(x1)+abs(x2)<=1 as a constraint. You changed it to abs(x1+x2)<=1 in your answer, which is not equivalent to what the question was asking.

[James Tursa]

Correct. This constraint abs(x1) + abs(x2) <= 1 is actually inside a diamond with vertices at (1,0), (0,1), (-1,0), and (0,-1). So there would be four inequality constraints involved for the four line segments, not just two.

[Matt J]

The original question was how to code abs(x1)+abs(x2)<=1 as a constraint.

I suspect the original question was abs(x1+x2) and was later edited....

However, abs(x1) + abs(x2) <= 1 can be represented by 4 inequality constraints, as James says:

    [1     1    * [x1;x2] <=[1;1;1;1]
     1    -1
    -1     1
    -1    -1]