We propose a new approach to solving linear programs which combines the advantages of interior point methods and active set methods. Methods of the first type, in particular the Nesterov-Todd interior-point method for self-concordant barriers [8] solve the problem in polynomial time, running \(\mathcal {O}\left( \sqrt{n}\text {log}\left( \frac{1}{\varepsilon }\right) \right) \) iterations, however, it can be shown that it converges to accuracy \(\sqrt{\varepsilon _M}\) when implemented on machines with \(\varepsilon _M\) accuracy, which is unacceptable for some practical problems, such as bank scheduling [14]. On the other hand, the simplex method [4] avoids the argument accuracy problem, since its trajectory traverses only the vertices of the feasible polyhedron. However, the simplex method is known to spend at worst exponential time to find the answer, and in practice the interior point method runs much faster and is preferred. The approach proposed in this paper is a hybrid algorithm which generates iterations as in an interior point method, but at some steps it additionally analytically solves an auxiliary minimization problem on a cone approximating the orthant with marginal additional cost. This allows us to find exactly-zero solution components and lower the dimensionality of the problem.