Ellipsoid method

File:Ellipsoid 2.png
An iteration of the ellipsoid method

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a finite number of steps.

The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.

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.

Following Khachiyan's work, the ellipsoid method was the only algorithm for solving linear programs whose runtime had been proved to be polynomial until Karmarkar's algorithm. However, the interior-point method and variants of the simplex algorithm are much faster than the ellipsoid method in practice. Karmarkar's algorithm is also faster in the worst case.

Notes

1. M. Grötschel, L. Lovász, A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988.
2. L. Lovász: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986.
3. V. Chandru and M.R.Rao, Linear Programming, Chapter 31 in Algorithms and Theory of Computation Handbook, edited by M. J. Atallah, CRC Press 1999, 31-1 to 31-37.
4. V. Chandru and M.R.Rao, Integer Programming, Chapter 32 in Algorithms and Theory of Computation Handbook, edited by M.J.Atallah, CRC Press 1999, 32-1 to 32-45.