A New Infeasible-Interior-Point Algorithm for Linear Programming over Symmetric Cones

• Published in: Acta Mathematicae Applicatae Sinica Vol. 33; no. 3; pp. 771 - 788
• Main Authors: Liu, Chang-he, Shang, You-lin, Han, Ping
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2017; Springer Nature B.V
• Edition: English series
• Subjects: Algorithms; Applications of Mathematics; Complexity; Cones; Linear programming; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; 不可行内点算法; 复杂性; 对称; 搜索方向; 收敛性; 日志; 精度要求; 线性规划; polynomial complexity; 90C51; interior-point methods; Euclidean Jordan algebra; 90C25; 90C22; symmetric cone; linear programming; 90C05
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-017-0697-7

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N（ t1, t2, 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（r1.5 log ε-1） for the Nesterov-Todd （NT） direction, and O（r2 log ε-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 （x0, y0, s0） in N（t1, t2, η）, the complexity bound is O（ √ r log ε-1） for the NT direction, and O（r log ε-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.