Optimization problem involving the sum of reciprocals of variables

Question

I'm trying to solve an optimization problem involving reciprocals.

The problem has the following simple form. Can this form be transformed into a problem form that commercial optimization solvers can solve (eg LP, QP ..)?

$$\begin{split} &\text{min } x+y+z\\ &\text{s.t.}\\ &\frac1x + \frac1y +\frac1z \geq 1\\ &x,y,z \geq 1 \end{split}$$

The answer would be 3, however, what I want to ask is how to re-formulate it when there is reciprocal constraints in a problem where the objective function and other constraints are linear combinations.

Thank you.

Answer 1

You can make piecewise linearization of each of the fraction (reciprocals) using Taylor series. This technique will not be accurate and the bounds may not be tight. In solver problems usually its like

Min $ x+y+z$
s.t.
(1) $ x_1+y_1+z_1 \ge 1$
(2) $ x_1 \cdot x - 1 = 0$
(3) $ y_1 \cdot y - 1 = 0$
(4) $ z_1 \cdot z - 1 = 0$
$ 0 \le x_1,y_1, z_1 \le 1$

Most solvers wont have a problem with constraints (2)-(4)

Answer 2

With the substitutions $a = 1/x, b = 1/y, c = 1/z$, it is equivalent to the following convex problem: \begin{align*} &\min_{a, b, c} \quad \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\\ &\mathrm{s.t.}\quad a + b + c \ge 1; \\ &\qquad\,\, a, b, c > 0; \quad a, b, c \le 1. \end{align*}

CVX can solve it.