A Parametric Kernel Function Generating the best Known Iteration Bound for Large-Update Methods for CQSDO

In this paper, we propose a large-update primal-dual interior point algorithm for convex quadratic semidefiniteoptimization (CQSDO) based on a new parametric kernel function. This kernel function is a parameterized version of the kernel function introduced by M.W. Zhang (Acta Mathematica Sinica. 28: 2313-2328, 2012) for CQSDO. The investigation according to it generating the best known iteration bound O for large-update methods. Thus improves the iteration bound obtained by Zhang for large-update methods. Finally, we present few numerical results to show the efficiency of the proposed algorithm.

A new parameterized logarithmic kernel function for linear optimization with a double barrier term yielding the best known iteration bound

In this paper, we propose a large-update primal-dual interior point algorithm for linear optimization. The method is based on a new class of kernel functions which differs from the existing kernel functions in which it has a double barrier term. The investigation according to it yields the best known iteration bound O\sqrt n \log (n)\log \left( {{n \over \in }} \right) for large-update algorithm with the special choice of its parameter m and thus improves the iteration bound obtained in Bai et al. [2] for large-update algorithm.

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

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.

A polynomial-time weighted path-following interior-point algorithm for linear optimization

In this paper, a weighted short-step primal-dual path-following interior-point algorithm for solving linear optimization (LO) is presented. The algorithm uses at each interior-point iteration a full-Newton step, thus no need to use line search, and the strategy of the central-path to obtain an [Formula: see text]-approximated solution of LO. We show that the algorithm yields the iteration bound, namely, [Formula: see text]. This bound is currently the best iteration bound for LO. Finally, some numerical results are reported in order to analyze the efficiency of the proposed algorithm.

An improved and modified infeasible interior-point method for symmetric optimization

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

A wide neighborhood predictor–corrector infeasible-interior-point method for Cartesian P∗(κ)-LCP over symmetric cones

In this paper, we present a predictor–corrector infeasible-interior-point method based on a new wide neighborhood of the central path for linear complementarity problem over symmetric cones (SCLCP) with the Cartesian [Formula: see text]-property. The convergence of the algorithm is proved for commutative class of search directions. Moreover, using the theory of Euclidean Jordan algebras and some elegant tools, the iteration bound improves the earlier complexity of these kind of algorithms for the Cartesian [Formula: see text]-SCLCPs.

A modified infeasible interior-point algorithm for P*(κ)-HLCP over symmetric cones

An improved version of infeasible interior-point algorithm for [Formula: see text] horizontal linear complementarity problem over symmetric cones is presented. In the earlier version (optimization, doi: 10.1080/02331934.2015.1062011) each iteration of the algorithm consisted of one so-called feasibility step and some centering steps. The main advantage of the modified version is that it uses only one feasibility step in each iteration and the centering steps not to be required. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.