# A new infeasible-interior-point algorithm for linear programming over symmetric cones

A new infeasible-interior-point algorithm for linear programming over symmetric cones In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ 1, τ 2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r 1.5logε −1) for the Nesterov-Todd (NT) direction, and O(r 2logε −1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x 0, y 0, s 0) in N(τ 1, τ 2, η), the complexity bound is $$O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)$$ O ( r log ε − 1 ) for the NT direction, and O(rlogε −1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# A new infeasible-interior-point algorithm for linear programming over symmetric cones

, Volume 33 (3) – Aug 7, 2017
18 pages

/lp/springer_journal/a-new-infeasible-interior-point-algorithm-for-linear-programming-over-S00sccci4I
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-017-0697-7
Publisher site
See Article on Publisher Site

### Abstract

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ 1, τ 2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r 1.5logε −1) for the Nesterov-Todd (NT) direction, and O(r 2logε −1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x 0, y 0, s 0) in N(τ 1, τ 2, η), the complexity bound is $$O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)$$ O ( r log ε − 1 ) for the NT direction, and O(rlogε −1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Aug 7, 2017

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