Polynomial convergence of Mehrotra-type predictor-corrector algorithm for the Cartesian P∗(κ)-LCP over symmetric cones

• Published in: Optimization Vol. 64; no. 4; pp. 815 - 837
• Main Authors: Liu, Xinze, Liu, Hongwei, Wang, Weiwei
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.04.2015; Taylor & Francis LLC
• Subjects: (κ) property; Algorithms; Cartesian P; Euclidean Jordan algebra; Euclidean space; interior-point methods; linear complementarity problem; Linear programming; Mathematical problems; polynomial complexity; Polynomials; Studies; Symmetry
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2013.820300

We propose a new infeasible variant of Mehrotra-type predictor-corrector interior-point algorithms which can be regarded as an extension of Salahi et al. (SIAM J. Optim. 18: 1377-1397, 2007) for linear programming to the Cartesian linear complementarity problems over symmetric cone. That algorithm incorporates a safeguard in Mehrotra's original algorithm, which allows us to prove polynomial iteration complexity. In our algorithm, the safeguard strategy is implemented by bounding the central parameter, which is different from the algorithm of Salahi et al. We modify the maximum step size in the affine scaling step and the Newton system in the corrector step, and extend the algorithm to symmetric cones using the machinery of Euclidean Jordan algebras. We show that the iteration-complexity bound of the proposed algorithm is , where is the rank of the associated Euclidean Jordan algebras, is the handicap of the problem, is the condition number of matrix and is the required precision. Some numerical tests are reported to illustrate our theoretical results.