Simplices by point-sliding and the Yamnitsky–Levin algorithm

Yamnitsky and Levin proposed a variant of Khachiyan's ellipsoid method for testing feasibility of systems of linear inequalities that also runs in polynomial time but uses simplices instead of ellipsoids. Starting with the n-simplex S and the half-space {x|a^Tx ≤ β}, the algorithm finds a simplex SYL of small volume that encloses S ∩ {x|a^Tx ≤ β}. We interpret SYL as a simplex obtainable by point-sliding and show that the smallest such simplex can be determined by minimizing a simple strictly convex function. We furthermore discuss some numerical results. The results suggest that the number of iterations used by our method may be considerably less than that of the standard ellipsoid method.