Accuracy Certificates for Computational Problems with Convex Structure
Mathematics of Operations Research
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The Ellipsoid Algorithm (EA) for linear programming attracted recently great attention. EA was proposed in [N76] and developed in [K79, G81] and other works. It is a modification of Method of Centralized Splitting presented in [L65], which differs from EA in two essential respects. Firstly, [L65] uses simplexes instead of ellipsoids; it is admitted, secondly, that, several (q(n))splittings of the n-dimensional simplex may be needed before the remaining polyhedron can be enclosed into a simplex of a smaller volume. Only a very rough upper bound q(n) 0, where A is an m 脳 n matrix of rank n. We normalize solutions by a restriction (e ċ Ax) = 1 where e 0. On every step the algorithm considers a simplex BAx ≥ 0 containing all solutions, where B is a non-negative n 脳 m matrix with det(BA) ≠ 0. Let us denote this simplex by ΔB, its volume by VB and its center by CB. Initially we take an arbitrary B and e = BT(1,..,1).