A POLYNOMIAL-TIME PRIMAL-DUAL AFFINE SCALING ALGORITHM FOR LINEAR AND CONVEX QUADRATIC PROGRAMMING AND ITS POWER SERIES EXTENSION*t

We describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate. If r > 1 is the order of approximation used, we show that our algorithm has time complexity O(n t(+l/r)L(l+l/r)) iterations and O(n3 + n2r) arithmetic operations per iteration, where n is the dimension of the problem and L is the size of the input data. When r = 1, we show that the algorithm can be interpreted as an affine scaling algorithm in the primal-dual setup. 1. Introduction. After the presentation of the new polynomial-time algorithm for linear programming by Karmarkar in his landmark paper [15], several so-called interior point algorithms for linear and convex quadratic programming have been proposed. These algorithms can be classified into three main groups: (a) Projective algorithms, e. The algorithms of class (a) are known to have polynomial-time complexity requiring O(nL) iterations. However, these methods appear not to perform well in practice [30]. In contrast, the algorithms of group (b), while not known to have polynomial-time complexity, have exhibited good behavior on real world linear programs [1], [20], [23], [31]. Most path following algorithms of group (c) have been shown to require O(/n L) iterations. These algorithms use Newton's method to trace the path of minimizers for the logarithmic barrier family of problems, the so-called central path. The logarithmic barrier function approach is usually attributed to Frisch [12] and is formally studied in Fiacco and McCormick [11] in the context of nonlinear optimization. Continuous trajectories for interior point methods were proposed by Karmarkar [16] and are extensively studied in Bayer and Lagarias [6] [7], Megiddo [21] and Megiddo and Shub [22]. Megiddo [21] related the central path to the classical barrier path in the framework of the primal-dual complementarity relationship. Kojima, Mizuno and Yoshise [19] used this framework to describe a primal-dual interior point algorithm that traces the central trajectory and has a worst time complexity of O(nL) iterations.