Parabolic Target-Space Interior-Point Algorithm for Weighted Monotone Linear Complementarity Problem

In this paper, we revisit the main principles for constructing polynomial-time primal-dual interior-point algorithms (IPAs). Starting from the break-through paper by Gonzaga (1989), their development was related to the barrier methods, where the objective function was added to the barrier for the feasible set. With this construction, using the theory of self-concordant functions proposed by Nesterov and Nemirovski (1994), it was possible to develop different variants of IPAs for a large variety of convex problems. However, in order to solve the initial problem, the most efficient primal-dual methods need to follow several central paths (up to three), which correspond to different stages of the solution process. This multistage structure of the methods significantly reduces their efficiency. In this paper, we come back to the initial idea by Renegar (1988) of using the methods of centers. We implement it for the weighted Linear Complementarity Problem (WLCP), by extending the framework of Parabolic Target Space (PTS), proposed by Nesterov (2008) for primal-dual Linear Programming Problems. This approach has several advantages. It starts from an arbitrary strictly feasible primal-dual pair and travels directly to the solution of the problem in one stage. It has the best known worst-case complexity bound. Finally, it works in a large neighborhood of the deviated central path, allowing very large steps. The latter ability results in a significant acceleration in the end of the process, confirmed by our preliminary computational experiments.