## A Strongly Polynomial Rounding Procedure Yielding A Maximally Complementary Solution for $P_*(\kappa)$ Linear Complementarity Problems

### T. Illés, J. Peng, C. Roos and T. Terlaky

We deal with Linear Complementarity Problems (LCPs) with $P_*(\kappa)$ matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter $\mu$, as usual. Our elementary proof reproduces the known result that the variables on, or close to the central path fall apart in three classes in which these variables are O(1), O(\mu) and O(sqrt(\mu)), respectively. The constants hidden in these bounds are expressed in, or bounded by, the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and close enough to the central path, the rounding procedure produces a maximally complementary solution in at most O(n^3) arithmetic operations. The result implies that Interior Point Methods (IPMs) not only converge to a complementary solution of $P_*(\kappa)$ LCPs but, when furnished with our rounding procedure, they can produce a maximally complementary (exact) solution in polynomial time.

Report 98-15,
Faculty of Information Technology and Systems
Subfaculty of Technical Mathematics and Informatics,
Department of Statistics, Stochastic and Operations Research
Delft University of Technology,
P.O. Box 5031, 2600 GA Delft, The Netherlands.

Contact: terlaky@ssor.twi.tudelft.nl