An inexact infeasible arc-search interior-point method for linear programming problems (2403.18155v2)

Abstract: Inexact interior-point methods (IPMs) are a type of interior-point methods that inexactly solve the linear equation system for obtaining the search direction. On the other hand, arc-search IPMs approximate the central path with an ellipsoidal arc obtained by solving two linear equation systems in each iteration, while conventional line-search IPMs solve one linear system. Therefore, the improvement due to the inexact solutions of the linear equation systems can be more beneficial in arc-search IPMs than conventional IPMs. In this paper, we propose an inexact infeasible arc-search interior-point method. We establish that the proposed method is a polynomial-time algorithm through its convergence analysis. The numerical experiments for the large benchmark problems show that the proposed method using the conjugate gradient method as the inexact linear system solver can reduce both of the number of iterations and the computation time compared to the existing inexact IPM due to the reduction in computational complexity by the arc-search. Andmore, it can reduce the computation time compared to the existing exact IPMs because the dependence of the computational complexity on the dimension $n$ of the coefficient matrix is smaller for the conjugate gradient method than for the Cholesky factorization.