Complexity analysis of interior point methods for linear programming based on a parameterized kernel function

Abstract

Kernel function plays an important role in defining new search directions for primaldual interior point algorithm for solving linear optimization problems. This problem has attracted the attention of many researchers for some years. The goal of their works is to find kernel functions that improve algorithmic complexity of this problem. In this paper, we introduce a real parameter p > 0 to generalize the kernel function (5) given by Bai et al. in [Y.Q. Bai, M El Ghami and C. Roos, SIAM J. Optim. 15 (2004) 101-128.], and give the corresponding primal-dual interior point methods for linear optimization. This parameterized kernel function yields the similar complexity bound given in [Y.Q. Bai, M El Ghami and C. Roos, SIAM J. Optim. 15 (2004) 101-128.] for both large-update and small-update methods.

A primal-dual interior-point method for the semidefinite programming problem based on a new kernel function

“…We analyzed the primal-dual interior point algorithm based on this kernel function, and established that the worst-case iteration bounds are O( √ n(ln n) pq+1 pq ln n ε ) and O( √ n ln n ε ) for large and small-update methods, respectively. Our results improve the complexity results obtained in [7,12,21] for large-update methods. We also remark that the extensions to second-order cone programming (SOCP), and symmetric cone programming (SCP) deserves to be further investigated.…”

“…Finally, we obtain that the worst-case iteration bound for large-update methods is O( √ n(ln n) pq+1 pq ln n ε ). This parametric kernel function yields the similar complexity bound given in [7,12,18,21]. For small-update methods, we obtain the best known iteration bound, namely, O( √ n ln n ε ).…”

“…From three technical Lemmas in [12], we have the following Lemmas 5.1-5.3, since ψ(t) is a kernel function and ψ (t) is monotonically decreasing (see [6,11,21]).…”

“…Since then, many various kernel functions have been proposed and analyzed. For these, we refer the readers to [11,15,16,17,18,19,20,21] for both linear and semidefinite cases. The key of all these results is to find kernel functions that improve the complexity analysis of LP and SDP problems.…”

Complexity analysis of interior point methods for convex quadratic programming based on a parameterized Kernel function

“…Bai et al [1] presented a large class of eligible kernel functions, which is fairly general and includes the classical logarithmic functions and the self-regular functions, as well as many non-self-regular functions as special cases. For some other related kernel function, we refer to [3,4,5,6,7,8,9,12].…”