Ellipsoid method
Introduction
The Ellipsoid method is a technique used in mathematical optimization to solve convex optimization problems. It was developed by Naum Z. Shor and is based on the theory of ellipsoids. The method is an iterative process that, at each step, constructs an ellipsoid that contains the optimal solution. The method is notable for its theoretical importance, as it was the first method proven to be capable of solving all convex optimization problems in polynomial time.
History
The Ellipsoid method was introduced in the 1970s by the Soviet mathematician Naum Z. Shor, who was working on methods for non-smooth optimization. The method was further developed and popularized by other Soviet mathematicians, including Leonid Khachiyan, who proved that the method could solve all convex optimization problems in polynomial time. This was a significant breakthrough in the field of optimization, as it was the first time that a polynomial-time algorithm had been found for a broad class of optimization problems.
Theory
The Ellipsoid method is based on the theory of ellipsoids. An ellipsoid is a generalization of a sphere that can be elongated along one or more axes. In the context of the Ellipsoid method, an ellipsoid is used to represent a region of space that contains the optimal solution to a convex optimization problem.
At each step of the Ellipsoid method, an ellipsoid is constructed that contains the optimal solution. This ellipsoid is then cut in half, and the half that contains the optimal solution is selected. This process is repeated until the optimal solution is found, or until the size of the ellipsoid is reduced to a sufficiently small value.
The key to the Ellipsoid method is the way in which the ellipsoid is cut. This is done by constructing a hyperplane that separates the optimal solution from the rest of the ellipsoid. The location of this hyperplane is determined by the gradient of the objective function at the current solution.
Algorithm
The Ellipsoid method is an iterative algorithm that operates in a series of steps. At each step, the algorithm constructs an ellipsoid that contains the optimal solution, and then cuts this ellipsoid in half.
The algorithm begins by constructing an initial ellipsoid that contains the feasible region of the optimization problem. This initial ellipsoid can be constructed in various ways, depending on the specific problem.
At each step of the algorithm, the current solution is evaluated, and the gradient of the objective function at this solution is computed. This gradient is used to construct a hyperplane that separates the optimal solution from the rest of the ellipsoid.
The half of the ellipsoid that contains the optimal solution is then selected, and a new ellipsoid is constructed that contains this half. The new ellipsoid is smaller than the original ellipsoid, and it is centered at a point that is closer to the optimal solution.
This process is repeated until the optimal solution is found, or until the size of the ellipsoid is reduced to a sufficiently small value.
Applications
The Ellipsoid method has been applied to a wide range of optimization problems, including linear programming, quadratic programming, and convex programming. It has also been used in the field of machine learning, where it has been used to solve problems such as support vector machines and logistic regression.
Despite its theoretical importance, the Ellipsoid method is not often used in practice. This is because the method is relatively slow compared to other optimization methods, such as the simplex method or interior-point methods. However, the Ellipsoid method remains an important tool in the field of optimization, due to its theoretical properties and its ability to solve a wide range of problems.
See Also
