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I have this constraint in an optimization problem:

$$\frac{a}{x_1} + \frac{b}{x_2} + \frac{c}{x_3} = d$$

where $x_i$ are the variables. Other than this my problem is linear. Is it possible to transform this constraint into a linear form?

1ENİGMA1
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Jakob Loegstrup
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  • Hi. Actually my constraint is: a/x1 + b/x2 + c/x3 <= d How do I transform this into a second-order cone problem? – Jakob Loegstrup Dec 21 '17 at 14:33
  • This is a separable equation, so you may be able to use a lambda formulation to generate a piecewise linear approximation, depending on whether you're minimizing or maximizing. – Richard Apr 28 '19 at 23:57

1 Answers1

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No.

Non-linear equality constraints are always non-convex, and as you have other linear constraints, you will probably not be able to make any clever variable change, as that will turn your other constraints non-linear and most likely non-convex instead.

However, depending on the sign of your coefficients, and the objective function, it could be that your problem can be written as a convex problem at least. If your objective is deceasing in $x$ (i.e. smaller $x$ better) and $a$, $b$ and $c$ are positive, you can replace the inequality with $\leq$, as it will be tight at optimality. With that, you have a convex program (and if you want, you can write the fractional constraint using a second-order cone reformulation if you want, and thus solve it using second-order cone programming which is a slight generalization of linear programming, otherwise just use a standard nonlinear solver)

Johan Löfberg
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