A new algorithm for minimizing convex functions over convex sets

An algorithm for minimizing a convex function over a convex set is given. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope is central to the algorithm. The algorithm has a much better rate of global convergence than the ellipsoid algorithm. A by-product of the algorithm is an algorithm for solving linear programming problems that performs a total of O(mn/sup 2/L+M(n)nL) arithmetic operations in the worst case, where m is the number of constraints, n the number of variables, and L a certain parameter. This gives an improvement in the time complexity of linear programming for m>n/sup 2/. >