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Variable splitting

Mathematical function used in optimisation

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Source updated September 12, 2026

In applied mathematics and computer science, variable splitting is a decomposition method that relaxes a set of constraints.

01Details

When the variable $x$ appears in two sets of constraints, it is possible to substitute the new variables $x_{1}$ in the first constraints and $x_{2}$ in the second, and then join the two variables with a new "linking" constraint, which requires that

$x_{1}=x_{2}$

This new linking constraint can be relaxed with a Lagrange multiplier; in many applications, a Lagrange multiplier can be interpreted as the price of equality between $x_{1}$ and $x_{2}$ in the new constraint.

For many problems, relaxing the equality of split variables allows the system to be broken down, enabling each subsystem to be solved separately. This significantly reduces computation time and memory usage. Solving the relaxed problem with variable splitting can give an approximate solution to the initial problem. Using an approximate solution as a "warm start" facilitates the iterative solving of the original problem with only the variable $x$ .

This was first introduced by Jörnsten, Näsberg, and Smeds in 1985. At the same time, M. Guignard and S. Kim introduced the same idea under the name "Lagrangean Decomposition" (their papers appeared in 1987).

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