A path following interior-point method for linear complementarity problems over circular cones

Circular cones are a new class of regular cones that include the well-known second-order cones as a special case. In this paper, we study the algebraic structure of the circular cone and show that based on the standard inner product this cone is nonsymmetric while using the new-defined circular inner product this cone not only is symmetric but also the algebra associated with it, is a Euclidean Jordan algebra. Then, using the machinery of Euclidean Jordan algebras and the Nestrov–Todd search directions, we propose a primal-dual path-following interior-point algorithm for linear complementarity problems over the Cartesian product of the circular cones. The convergence analysis of the algorithm is shown and it is proved that this class of mathematical problems is polynomial-time solvable.