Unrestricted Variable Transformation¶

Definition¶

An unrestricted variable $x_j$ can be replaced everywhere by the difference of two nonnegative variables:

If $x_j\le0$ is required, use $x_j=-x_j^-$ with $x_j^-\ge0$ instead.

Interpretation¶

The simplex method works with nonnegative variables only, but some decisions can be negative, for example a change in stock level. The transformation writes such a free variable as the difference of two nonnegative parts, so the model again has only nonnegative variables while the original variable keeps its full range.

On the number line, $x=2.5$ is written with $x^+=2.5$ and $x^-=0$$0$, and $x=-3$ with $x^+=0$ and $x^-=3$$3$. One part carries the size of the number and the other is zero, so $x=x^+-x^-$ holds with both parts nonnegative.

The substitution has to be made in the objective and in every constraint; afterwards $x_j$ no longer appears, and its value is recovered as $x_j^+-x_j^-$$x_j^{-}$. The representation is not unique: $x=2$ can also be written as $5-3$$3$. This does no harm: the objective and constraints depend on the two parts only through their difference, so all representations of the same $x_j$ are equally feasible and give the same objective value. Both parts stay linear terms, so the model remains a linear program.

When the sign is known, splitting is unnecessary: for $x_j\le0$ the single substitution $x_j=-x_j^-$ is enough, and a variable that already satisfies a nonnegativity constraint needs no change. With the negative RHS transformation and the minimization-to-maximization transformation, it is one of the steps that bring a model into the form the simplex method starts from.

Key Results¶

• Apply the substitution to both the objective and every constraint.
• The difference representation is not unique; it is not necessary to impose a nonlinear restriction preventing both parts from being positive.
• The optimal original variable is the difference of the computed parts.

Examples¶

The two replacement columns are opposites: the column of $x_j^-$ is $-1$ times the column of $x_j^+$. They cannot both be basic in the same basic solution, although a feasible solution that is not basic may have both parts positive.

• $x_2^+=0$ and $x_2^-=10$ give $x_2=-10$$-10$.
• The other-model-forms exercise transforms its nonpositive $x_1$ as $-x_1^-$.

Prerequisites¶

• Nonnegativity Constraint
• Linear Programming

Related Concepts¶

• Augmented Form
• Minimization-to-Maximization Transformation

Sources¶

• OR Modelling · Week 2

Aliases: variable splitting

Last updated: 20 October 2026 03:02