Linear optimization problem with additional constant cost for non-zero variables

Question

I have a linear optimization problem with integer variables of the form

minimize $a_1 x_1 + ... + a_n x_n$

under a set of constraints

Which R package lpSolve solves nicely even though its quite large.

But now I realized I need to add a constant cost term to the cost-function for each $x_i$ which is non zero.

Question: Is there a trick to minimize a cost-function with terms of the form $a_1 x_1 + f(a_1) + ... + a_n x_n + f(a_n)$ where

$f(x)={0.1 \text{ if } x>0\text{, and }0\text{ otherwise}} $

as if it were truely linear?

@JohanLöfberg lpSolve supports binary variables, but I don't see how to use them for this problem, since I cannot multiply an additional binary variable $b_1$ with $a_1$ to get the desired effect. — Beginner, Mar 13 '15 at 14:07
Introduce binaries $z_i$, add the constraints $a_i \leq constant_i z_i$, and add $0.1\sum z_i$ to the objective. — Johan Löfberg, Mar 13 '15 at 14:10
@JohanLöfberg of course! thank you :) please post as answer, so I can accept. — Beginner, Mar 13 '15 at 14:12

score 1 · Accepted Answer · answered Mar 13 '15 at 14:13

1

Introduce binaries $z_i$, add the constraints $a_i \leq constant_i z_i$, and add $0.1\sum z_i$ to the objective.

answered Mar 13 '15 at 14:13

Johan Löfberg

1 Answers1