I am implementing an optimization program on 2 variables with a constraint of the form:
2*|x1| + 3*|x2| <= 2.25 * (|x1| + |x2|)
Given that the effective coefficients on the two abs terms are + and -, I'm unsure if and how this constraint can be linearized. Any suggestions on how to proceed?
Sorry, this forms a non-convex feasible region, so there is no way to do this in a pure continuous LP (even though some other answers -- now deleted- - seem to indicate otherwise) unless there are other special properties of the model we can exploit. Note that any LP has a convex feasible region.
Your particular constraint can be rewritten as $|x_2|\le \frac{1}{3}|x_1|$, and the feasible region looks like:
We do not have to give up however. From the picture it is clear we can solve this in two parts: first with $x_1\ge 0$ and then with $x_1\le 0$ and pick the best solution of the two sub-models.
Another approach is to use just one model, but with binary variables. We can model this constraint with the aid of two binary variable $\delta_1,\delta_2 \in \{0,1\}$ as follows: $$ \begin{align} &x_1 = x_1^{plus} - x_1^{min}\\ &x_1^{plus} \le M \delta_1 \\ &x_1^{min} \le M (1-\delta_1) \\ &x_2 = x_2^{plus} - x_2^{min}\\ &x_2^{plus} \le \frac{1}{3}M \delta_2 \\ &x_2^{min} \le \frac{1}{3}M (1-\delta_2) \\ &x_2^{plus}+x_2^{min} \le \frac{1}{3}(x_1^{plus}+x_1^{min})\\ & x_1 \in [-M,M] \\ & x_2 \in [-\frac{1}{3}M,\frac{1}{3}M] \\ & x_1^{plus}, x_1^{min},x_2^{plus}, x_2^{min} \ge 0\\ & \delta_1,\delta_2 \in \{0,1\} \end{align} $$ Note that $M$,$\frac{1}{3}M$ form bounds on the $x_1$ and $x_2$ variables. Because of the binary variables you need a MIP solver to handle this.
A slightly more compact formulation uses just one binary variable: $$ \begin{align} &-\frac{1}{3}x_1 - \frac{2}{3}M(1-\delta) \le x_2 \le \frac{1}{3} x_1 + \frac{2}{3}M(1-\delta)\\ & \frac{1}{3}x_1 - \frac{2}{3}M\delta \le x_2 \le -\frac{1}{3} x_1 + \frac{2}{3}M\delta\\ & x_1 \in [-M,M] \\ & x_2 \in [-\frac{1}{3}M,\frac{1}{3}M] \\ & \delta \in \{0,1\} \end{align} $$