ellipsoid method

mathematics

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July 21, 2026

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The ellipsoid method is a mathematical algorithm used in linear programming to solve optimization problems. It finds the maximum or minimum value of a linear function subject to constraints. While the simplex method was a significant advancement, it faced computational limitations with complex problems. The ellipsoid method, discovered by Leonid Khachiyan, offered a polynomial-time solution, meaning the computational steps grow as a power of the number of variables, rather than exponentially. However, in practice, the simplex method often proved faster.1

An ellipsoid itself is a closed surface where all plane cross-sections are ellipses or circles. If all three principal semiaxes are equal, it's a sphere. When two axes are equal, it becomes an ellipsoid of revolution, also known as a spheroid, formed by rotating an ellipse around one of its axes. The Earth's shape is approximated as an oblate spheroid.23

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linear programming
However, Khachiyan’s algorithm (called the ellipsoid method) was slower than the simplex method when practically applied. In 1984 Indian mathematician Narendra Karmarkar discovered another polynomial-time algorithm, the interior point method, that proved competitive with the simplex method.
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However, Khachiyan’s algorithm (called the ellipsoid method) was slower than the simplex method when practically applied. In 1984 Indian mathematician Narendra Karmarkar discovered another polynomial-time algorithm, the interior point method, that proved competitive with the simplex method.
Read More