Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods, O(n2/3log⁡(n/ε)), and small-update methods, O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions

Kernel-function Based Primal-Dual Algorithms for

Recently, [Y.Q. Bai, M. El Ghami and C. Roos, SIAM J. Opt. 15 (2004) 101–128] investigated a new class of kernel functions which differs from the class of self-regular kernel functions. The class is defined by some simple conditions on the growth and the barrier behavior of the kernel function. In this paper we generalize the analysis presented in the above paper for P*(κ) Linear Complementarity Problems (LCPs). The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper

Primal-Dual IPMS for Semidefinite Optimization Based on Finite Barrier Functions

In this paper we extend the results obtained for a class of finite kernel functions by Y.Q. Bai M. El Ghami and C.Roos published in SIAM Journal of Optimization, 13(3):766–782, 2003 [3] for linear optimization to semidefinite optimization. We show that the iteration bound for primal dual methods is O ( √ n log n log n ɛ), for large-update methods and O ( √ n log n ɛ), for small-update methods. The iteration complexity obtained for semidefinite programming is the same as the best bound for primal-dual interior point methods in linear optimization

Kernel-function based algorithms for semidefinitie optimization

Recently, Y.Q. Bai, M. El Ghami and C. Roos [3] introduced a new class of so-called eligible kernel functions which are defined by some simple conditions. The authors designed primal-dual interiorpoint methods for linear optimization (LO) based on eligible kernel functions and simplified the analysis of these methods considerably. In this paper we consider the semidefinite optimization (SDO) problem and we generalize the aforementioned results for LO to SDO. The iteration bounds obtained are analogous to the results in [3] for LO.Software TechnologyElectrical Engineering, Mathematics and Computer Scienc