Primal-dual interior point methods for Semidefinite programming based on a new type of kernel functions

Abstract

In this paper, we propose the first hyperbolic-logarithmic kernel function for Semidefinite programming problems. By simple analysis tools, several properties of this kernel function are used to compute the total number of iterations. We show that the worst-case iteration complexity of our algorithm for large-update methods improves the obtained iteration bounds based on hyperbolic [24] as well as classic kernel functions. For small-update methods, we derive the best known iteration bound.

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

“…Then, we parametrized the new function. The obtained class of KFs contains the hyperbolic-logarithmic KF proposed in [51] as a special case up to a multiplicative constant and improves its theoretical complexity bound to the currently best iteration bound for large-update methods namely, O( √ n log n log n ). We structure our paper as follows.…”

“…Remark 4.1. This function can also be considered as a generalization, up to the multiplicative constant 1/a, of the KF ψ 8 introduced in [51] (see Table 1).…”

“…Recently, Touil and Chikouche [51] introduced the first IPM based on a hyperboliclogarithmic KF for SDP. They proved that the corresponding interior-point algorithm meets O(n 2 3 log n ) iterations as the worst case complexity bound for the large-update method.…”

A Primal-Dual Interior-Point Algorithm Based on a Kernel Function with a New Barrier Term

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In this paper, we propose a path-following interior-point method (IPM) for solving linear optimization (LO) problems based on a new kernel function (KF). The latter differs from other KFs in having an exponential-hyperbolic barrier term that belongs to the hyperbolic type, recently developed by I. Touil and W. Chikouche [22,23]. The complexity analysis for large-update primal-dual IPMs based on this KF yields an O √ n log 2 n log n ϵ iteration bound which improves the classical iteration bound.

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“…See [9,24,6,10,12,16,11,26] for more informations on interior point algorithms based on KFs. It's worth noting that the latest type is the hyperbolic one which was recently introduced by Touil and Chikouche [22,23].…”