science

Variable splitting

Variable splitting is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Variable splitting rather than just read about it. In short: In applied mathematics and computer science, variable splitting is a decomposition method that relaxes a set of constraints. Details When the variable x {\displaystyle x} appears in two sets of constraints, it is possible to substitute the new variables x 1 {\displaystyle x_{1}} in the first constraints and x 2 {\displaystyle x_{2}} in the second, and then join the two variables with a new "linking" constraint, whic…

Key takeaways

Variable splitting belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Variable splitting to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Variable splitting from memory before moving on to harder problems.

Reference excerpt

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

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

x 1 = x 2 {\displaystyle 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 {\displaystyle x_{1}} and x 2 {\displaystyle 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 {\displaystyle 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).

References

Bibliography Adlers, Mikael; Björck, Åke (2000). "Matrix stretching for sparse least squares problems". Numerical Linear Algebra with Applications. 7 (2): 51–65. doi:10.1002/(sici)1099-1506(200003)7:2<51::aid-nla187>3.0.co;2-o. ISSN 1099-1506. Alvarado, Fernando (1997). "Matrix enlarging methods and their application". BIT Numerical Mathematics. 37 (3): 473–505. CiteSeerX 10.1.1.24.5976. doi:10.1007/BF02510237. S2CID 120358431. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Grcar, Joseph (1990). Matrix stretching for linear equations (Technical report). Sandia National Laboratories. arXiv:1203.2377. Bibcode:2012arXiv1203.2377G. SAND90-8723. Vanderbei, Robert J. (July 1991). "Splitting dense columns in sparse linear systems". Linear Algebra and Its Applications. 152: 107–117. doi:10.1016/0024-3795(91)90269-3. ISSN 0024-3795. Jörnsten, Kurt O.; Näsberg, Mikael; Smeds, Per A. (1985). "Variable Splitting: A New Lagrangean Relaxation Approach to Some Mathematical Programming Models". LiTH MAT R. 84–85. University of Linköping, Department of Mathematics: 1–52. Guignard, Monique; Kim, Siwhan (1987). "Lagrangean Decomposition: A Model Yielding Stronger Bounds". Mathematical Programming. 39 (2): 215–228. doi:10.1007/BF02592948. hdl:2027.42/6740.

Worked examples

Example 1 — a first encounter with Variable splitting

Start with the simplest possible case. Write down what Variable splitting claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Variable splitting before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Variable splitting ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Variable splitting

In research: Variable splitting appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Variable splitting in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Variable splitting is common in secondary-school and first-year university syllabi. It links to neighbouring topics Decomposition methods, so understanding it makes those chapters shorter.
In everyday life: Look for Variable splitting outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Variable splitting in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Variable splitting means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Variable splitting out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Variable splitting in simple terms?

In applied mathematics and computer science, variable splitting is a decomposition method that relaxes a set of constraints. Details When the variable x {\displaystyle x} appears in two sets of constraints, it is possible to substitute the new variables x 1 {\displaystyle x_{1}} in the first constr…

Why does Variable splitting matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Variable splitting?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Variable splitting.