Computational complexity of solving real algebraic formulae

where e e IR", b e IR" and A is an m x n matrix. In the last four years there has been a vast amount of work on "interior point" algorithms, motivated by Karmarkar's algorithm [16]. Unlike the traditional simplex method which moves from vertex to vertex around the feasible region {x;>4x > b}, interior point methods proceed through the interior {x;Ax > b] of the feasible region. Karmarkar's algorithm is a "projective" interior point algorithm, the basic computation for each iteration being a projective transformation. In the last four years another breed of interior point algorithms has received a lot of attention, "path-following" algorithms. These are closer to traditional numerical analysis than are projective algorithms, having Newton's method at their heart. The best upper bounds known for the complexity of linear programming are based on the analysis of particular path-following algorithms. Following are the simple ideas behind the first path-following algorithm proven to have a polynomial-time bound. Let af denote the /-th row of the constraint matrix A. The center of the system of linear inequalities Ax ^ b is the point z which maximizes Yl^fx — b/), viewed as a function restricted to the feasible region. The center exists and is unique if the feasible region is bounded and has nonempty interior, as we assume in what follows. Equivalently, then, the center is