A Family of Polynomial Affine Scaling Algorithms for Positive SemiDefinite Linear Complementarity Problems

In this paper the new polynomial affine scaling algorithm of Jansen, Roos, and Terlaky for linear programming (LP) is extended to positive semidefinite (PSD) linear complementarity problems. The algorithm is immediately further generalized to allow higher order scaling. These algorithms are also new for the LP case. The analysis is based on Ling's proof for the LP case; hence, it allows an arbitrary interior feasible pair to start with. With the scaling of Jansen, Roos, and Terlaky, the complexity of the algorithm is $$ {\cal O}\left(\frac{n}{\rho^2(1-\rho^2)}\ln \frac{(x^{(0)})^Ts^{(0)}}{\e}\right), $$ where $\rho^2$ is a uniform bound for the ratio of the smallest and largest coordinate of the iterates in the primal-dual space. We also show that Monteiro, Adler, and Resende's polynomial complexity result for the classical primal-dual affine scaling algorithm can easily be derived from our analysis. In addition, our result is valid for arbitrary, not necessarily centered, initial points. Finally, some computational results are presented, which indicate the influence of the order of the scaling on the numerical performance.