An Oblivious Ellipsoid Algorithm for Solving a System of (In)Feasible Linear Inequalities

Abstract

The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables [Formula: see text] when its set of solutions has positive volume. However, when [Formula: see text] is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two fundamental algorithms for tackling [Formula: see text], namely, the simplex and interior-point methods, each of which can be easily implemented in a way that either produces a solution of [Formula: see text] or proves that [Formula: see text] is infeasible by producing a solution to the alternative system [Formula: see text]. This paper develops an oblivious ellipsoid algorithm (OEA) that either produces a solution of [Formula: see text] or produces a solution of [Formula: see text]. Depending on the dimensions and other condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modified versions of OEA, whose computational complexity is superior to that of OEA when [Formula: see text]. This is achieved in the first modified version by proving infeasibility without producing a solution of [Formula: see text], and in the second version by using more memory. Funding: J. Lamperski and R. M. Freund were supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0240].