Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming

We present an analysis of the full-Newton step infeasible interior-point algorithm for semidefinite optimization, which is an extension of the algorithm introduced by Roos [C. Roos, A full-Newton step 𝒪(n) infeasible interior-point algorithm for linear optimization, SIAM J. Optim. 16 (2006), pp. 1110–1136] for the linear optimization case. We use the proximity measure σ(V)?=?‖I???V ²‖ to overcome the difficulty of obtaining an upper bound of updated proximity after one full-Newton step, where I is an identity matrix and V is a symmetric positive definite matrix. It turns out that the complexity analysis of the algorithm is simplified and the iteration bound obtained is improved slightly.     

A large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be $O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)$ . This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.     

线性权互补问题的新全牛顿步可行内点算法

基于一个连续可微函数,通过等价变换中心路径,给出求解线性权互补问题的一个新全牛顿步可行内点算法.该算法每步迭代只需求解一个线性方程组,且不需要进行线搜索.通过适当选取参数,分析了迭代点的严格可行性,并证明算法具有线性优化最好的多项式时间迭代复杂度.数值结果验证了算法的有效性.     

The Complexity of Self-Regular Proximity Based Infeasible IPMs

Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. In this paper a self-regular proximity based Infeasible Interior Point Method (IIPM) is proposed for linear optimization problems. First we mention some interesting properties of a specific self-regular proximity function, studied recently by Peng and Terlaky, and use it to define infeasible neighborhoods. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, large-update IIPMs emerge as the only natural choice. Then, we apply these results to design a specific self-regularity based dynamic large-update IIPM in large neighborhood. The new dynamic IIPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established. Finally, we report the main results of our computational experiments.     

A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING*

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.     

Convergence of the homotopy path for a full-Newton step infeasible interior-point method

Roos [C. Roos, A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16 (4) (2006) 1110-1136 (electronic)] proposed a new primal-dual infeasible interior-point method for linear optimization. This new method can be viewed as a homotopy method. In this work, we show that the homotopy path has precisely one accumulation point in the optimal set. Moreover, this accumulation point is the analytic center of a subset of the optimal set and depends on the starting point of the infeasible interior-point method.     

A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm for solving linear optimization problems. In this paper we present a new kernel function which yields an algorithm with the best known complexity bound for both large- and small-update methods.     

一个新的求解半正定规划问题的原始对偶内点算法

选择合适的核函数对设计求解线性规划与半正定规划的原始对偶内点算法以及复杂性分析都十分重要.Bai等针对线性规划提出三种核函数,并给出求解线性规划的大步迭代复杂界,但未给出数值算例验证算法的实际效果(Bai Y Q,Xie W,Zhang J.New parameterized kernel functions for linear optimization.J Global Optim,2012.DOI 10.1007/s10898-012-9934-z).基于这三种核函数设计了新的求解半正定规划问题的原始对内点算法.进一步分析了算法关于大步方法的计算复杂性界,同时通过数值算例验证了算法的有效性和核函数所带参数对计算复杂性的影响.     

A Full-Newton Step Infeasible Interior-Point Algorithm for Linear Programming Based on a Kernel Function

This paper proposes an infeasible interior-point algorithm with full-Newton step for linear programming, which is an extension of the work of Roos (SIAM J. Optim. 16(4):1110–1136, 2006). The main iteration of the algorithm consists of a feasibility step and several centrality steps. We introduce a kernel function in the algorithm to induce the feasibility step. For parameter p∈[0,1], the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods, that is, O(nlog n/ε). This work was supported in part by the National Natural Science Foundation of China under Grant No. 10871098.     

基于全牛顿求解P*(κ)阵水平线性互补问题的内点算法

提出一种求解P_*(k)阵水平线性互补问题的全牛顿内点算法,全牛顿算法的优势在于每次迭代中不需要线性搜寻.当给定适当的中心路径邻域的阈值和更新势垒参数,证明算法中心邻域的全牛顿是局部二次收敛的,最后给出算法迭代复杂性O(√n)log(n+1+k)/εμ₀.     

基于局部self-concordant障碍函数的全牛顿步多项式时间算法

由Nesterov和Nemirovski[4]创立的self-concordant障碍函数理论为解线性和凸优化问题提供了多项式时间内点算法.根据self-concordant障碍函数的参数,就可以分析内点算法的复杂性.在这篇文章中,我们介绍了基于核函数的局部self-concordant障碍函数,它在线性优化问题的中心路径及其邻域内满足self-concordant性质.通过求解此障碍函数的局部参数值,我们得到了求解线性规划问题的基于此局部Self-concordant障碍函数的纯牛顿步内点算法的理论迭代界.此迭代界与目前已知的最好的理论迭代界是一致的.     

求解凸二次规划问题的一种加权路径跟踪内点算法

基于Darvay提出用加权路径跟踪内点算法解线性规划问题的相关工作,本文致力于将此算法推广于解凸二次规划问题,并证明此算法具有局部二次收敛速度和目前所知的最好的多项式时间算法复杂性.