Improvement of the ellipsoid method for nonlinear programming

Abstract

For nolinear programming, the ellipsoid algorithm is extremely robust and competitive with some widely used algorithm with regard to efficiency. But most of the variants suggested so far did not improve the efficiency of the ellipsoid algorithm because the benefit of the larger reductions in ellipsoid volume was offset by the extra effort. We focus on improving the method relative to convergence rate and computational efficiency. We have suggested the two-step search ellipsoid algorithm with a scheme for constructing an elliposid of smaller volume using two successive subgradients. The suggested method improves convergence rate, and our computational results show an improvement in computational efficiency due to the advantage of our algorithm outweighing the extra effort. furthermore we have provided general rank-two update formulas for minimum volume ellipsoid that contains wedge-shaped subset of a given ellipsoid. Using this formulas, deep-cut options and other factors, we have suggested a simple variant of the two-step search ellipsoid algorithm, and it is showed that of all the other ellipsoid algorithms used in the comparison, the variant is most eficient.