Conditional constraints in R

Question

I am solving in R using GLPK a classical optimization problem:

min c*x 

s.t. Ax<=b 

Where x is my vector of variables, c is a vector of constants, A is the constraints matrix and b is the constraints vector.

Now my professor wants me to add the constraint that the positive terms in x must sum to 1 and the negative terms to -1.

To simplify things I though to implement it using the constraints

sum((x)_+) = 1

and

sum(x) = 0

I do not have much experience with R and I do not know which solvers allow me to write sum((x)_+) = 1 as a conditional constraint. I cannot use GLPK of course can someone help me to figure this out?

Answer 1

This is called a variable splitting technique. The formulation is

  xplus[i] - xmin[i] = x[i]
  xplus[i]*xmin[i] = 0   (make sure one of them is zero) 
  xplus[i], xmin[i] >= 0

Some solvers can handle this as is. GLPK is not one of them. Luckily the model can be linearized as:

  xplus[i] - xmin[i] = x[i]
  xplus[i] <= M*b 
  xmin[i] <= M*(1-b)  
  xplus[i], xmin[i] >= 0
  b ∈ {0,1}

Here M is bound on x: -M <= x <= M. This is a constant. b[i] is a binary variable.

Then your additional constraints are:

  sum(i, xplus[i]) = 1
  sum(i, xmin[i]) = 1

And, of course, you should talk to your professor if you don't understand the assignment.