Introduction to the Simplex Method

The famous simplex method, invented by George B. Dantzig in 1947, moves from vertex to vertex in the underlying polyhedron until achieving an optimal vertex. As one of the most widely used mathematical tools, it has dominated the field of Linear Programming for nearly eighty years. However, it has exponential time complexity, and its performance turned out somehow unsatisfactory when solving some difficult LP problems since the solution process can sink into a degenerate vertex for too long.

The Interior-Point Algorithm

In 1984, Karmarkar published his work on the interior-point algorithm, which goes across the interior of the polyhedron, and which was not only of polynomial time complexity but also appeared fast. As such, it immediately drew the attention of researchers worldwide, giving rise to an upsurge in the interior-point method. Some scholars even considered it the winner against the simplex method for solving large-scale and sparse LP problems.

However, the technique can only approach an optimal solution on the boundary, and it cannot be warmly started; hence, it is not applicable for solving integer LP problems, which form the primary domain of LP applications. The interior-point method failed to shake the domination of the simplex method.

The Face Method

After years of research and exploration, the author proposes to break out of the simplex and interior-point methods. Over the recent years, the author has developed the so-called face method, which moves face by face to achieve an optimal face and solution. As the first book on the topic of face method, the monograph summarizes valuable findings and puts forward the theme to the academic world.