An Adaptive Infeasible-Interior-Point Method with the One-Norm Wide Neighborhood for Semi-definite Programming

Abstract

In this paper, we present a Mehrotra-type predictor–corrector infeasible-interior-point method, based on the one-norm wide neighborhood, for semi-definite programming. The proposed algorithm uses Mehrotra’s adaptive updating scheme for the centering parameter, which incorporates a safeguard strategy that keeps the iterates in a prescribed neighborhood and allows to get a reasonably large step size. Moreover, by using an important inequality that is the relationship between the one-norm and the Frobenius-norm, the convergence is shown for a commutative class of search directions. In particular, the complexity bound is $$\mathcal {O}(n\log \varepsilon ^{-1})$$ for Nesterov–Todd direction, and $$\mathcal {O}(n^{3/2}\log \varepsilon ^{-1})$$ for Helmberg–Kojima–Monteiro directions, where $$\varepsilon $$ is the required precision. The derived complexity bounds coincide with the currently best known theoretical complexity bounds obtained so far for the infeasible semi-definite programming. Some preliminary numerical results are provided as well.