computer science

Lemke's algorithm

Lemke's algorithm is a computer 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 Lemke's algorithm rather than just read about it. In short: In mathematical optimization, Lemke's algorithm is a procedure for solving linear complementarity problems, and more generally mixed linear complementarity problems. It is named after Carlton E.

Key takeaways

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

Reference excerpt

In mathematical optimization, Lemke's algorithm is a procedure for solving linear complementarity problems, and more generally mixed linear complementarity problems. It is named after Carlton E. Lemke. Lemke's algorithm is of pivoting or basis-exchange type. Similar algorithms can compute Nash equilibria for two-person matrix and bimatrix games.

References Cottle, Richard W.; Pang, Jong-Shi; Stone, Richard E. (1992). The linear complementarity problem. Computer Science and Scientific Computing. Boston, MA: Academic Press, Inc. pp. xxiv+762 pp. ISBN 0-12-192350-9. MR 1150683. Murty, K. G. (1988). Linear complementarity, linear and nonlinear programming. Sigma Series in Applied Mathematics. Vol. 3. Berlin: Heldermann Verlag. pp. xlviii+629 pp. ISBN 3-88538-403-5. Archived from the original on 2010-04-01. (Available for download at the website of Professor Katta G. Murty.) MR 0949214

External links OMatrix manual on Lemke Chris Hecker's GDC presentation on MLCPs and Lemke Linear Complementarity and Mathematical (Non-linear) Programming Siconos/Numerics open-source GPL implementation in C of Lemke's algorithm and other methods to solve LCPs and MLCPs

Worked examples

Example 1 — a first encounter with Lemke's algorithm

Start with the simplest possible case. Write down what Lemke's algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer 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 Lemke's algorithm 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 Lemke's algorithm 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 Lemke's algorithm

In research: Lemke's algorithm appears in computer 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 Lemke's algorithm 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: Lemke's algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algorithms and data structures stubs, Optimization algorithms and methods, so understanding it makes those chapters shorter.
In everyday life: Look for Lemke's algorithm 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 Lemke's algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lemke's algorithm 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 Lemke's algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lemke's algorithm in simple terms?

In mathematical optimization, Lemke's algorithm is a procedure for solving linear complementarity problems, and more generally mixed linear complementarity problems. It is named after Carlton E.

Why does Lemke's algorithm matter?

Because it connects several computer 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 Lemke's algorithm?

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 Lemke's algorithm.