Ellipsoid method
iterative method for minimizing convex functions
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.
When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size.
History
The ellipsoid method has a long history. As an iterative method, a preliminary version was introduced by Naum Z. Shor. In 1972, an approximation algorithm for real convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin).
As an algorithm for solving linear programming problems with rational data, the ellipsoid algorithm was studied by Leonid Khachiyan; Khachiyan's achievement was to prove the polynomial-time solvability of linear programs. This was a notable step from a theoretical perspective: The standard algorithm for solving linear problems at the time was the simplex algorithm, which has a run time that typically is linear in the size of the problem, but for which examples exist for which it is exponential in the size of the problem. As such, having an algorithm that is guaranteed to be polynomial for all cases was a theoretical breakthrough.
Khachiyan's work showed, for the first time, that there can be algorithms for solving linear programs whose runtime can be proven to be polynomial.
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