关于半定规划的一种宽邻域不可行内点算法的注记

针对半定规划的宽邻域不可行内点算法, 将牛顿法和预估校正法进行结合, 构造出适当的迭代方向, 提出一个修正的半定规划宽邻域不可行内点算法, 并在适当的假设条件下, 证明了该算法具有O(\sqrt{n}L)的迭代复杂界．最后利用Matlab编程, 给出了基于KM方向和NT方向的数值实验结果．     

二阶锥规划的基于自协调指数核函数的原始-对偶内点算法

基于一个自协调指数核函数, 设计求解二阶锥规划的原始-对偶内点算法. 根据自协调指数核函数的二阶导数与三阶导数的特殊关系, 在求解问题的中心路径时, 用牛顿方向代替了负梯度方向来确定搜索方向. 由于自协调指数核函数不具有``Eligible'性质, 在分析算法的迭代界时, 利用牛顿方法求解目标函数满足自协调性质的无约束优化问题的技术, 估计算法内迭代中自协调指数核函数确定的障碍函数的下降量, 得到原始-对偶内点算法大步校正的迭代界O(2N\frac{\log2N}{\varepsilon}), 这里N是二阶锥的个数. 这个迭代界与线性规划情形下的迭代界一致. 最后, 通过数值算例验证了算法的有效性.     

一类凸规划的多项式预估校正内点法

1、引言 1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法．1993年Mizuno,Todd和Y．Ye．给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步．1994年F．A．Potra给出了不可行预估校正内点法,使得可以从一个不可行的初始点开始算法的迭代,并证明了其为二次收敛．     

带有新的迭代格式的内点算法

研究了求解线性规划问题的二阶Mehrotra型预估-矫正内点算法,使用Newton方法求解预估方向和矫正方向,并利用两个方向的一种新的组合方式得到搜索方向.在每次迭代中,要求新的迭代点在中心路径的一个宽邻域内,从而计算出步长参数.通过分析,证明了该算法经过有限次迭代后收敛到问题的一个最优解,并具目前内点算法最好的多项式复杂度O(槡nL).数值实验表明该算法在实践中是有效的.     

基于全牛顿步长求解凸二次规划问题的不可行内点算法

借助于全牛顿步长对凸二次规划问题提出了一种新的不可行内点算法.算法主要迭代由可行迭代步和中心路径邻域迭代步组成.其优点是线性搜寻方向是不需要的.最后证明算法迭代复杂性为O(nlogn/ε),与目前最好的不可行内点算法复杂性一致.     

基于一个有限罚函数的二阶锥优化的原始-对偶内点算法

本文基于一个有限罚函数,设计了关于二阶锥优化问题的原始-对偶路径跟踪内点算法,由于该罚函数在可行域的边界取有限值,因而它不是常规的罚函数,尽管如此,它良好的解析性质使得我们能分析算法并得到基于大步校正和小步校正方法目前较好的多项式时间复杂性分别为O(N~(1/2)log N log N/ε)和O(N~(1/2)log N/ε),其中N为二阶锥的个数．     

一个求解二阶锥规划的光滑牛顿算法

本文研究了二阶锥规划问题.利用新的最小值函数的光滑函数,给出一个求解二阶锥规划的光滑牛顿算法.算法可以从任意点出发,在每一步迭代只需求解一个线性方程组并进行一次线性搜索.在不需要满足严格互补假设条件下,证明了算法是全局收敛和局部二阶收敛的.数值试验表明算法是有效的.     

单调线性互补问题的Mehrotra型预估-校正算法的迭代复杂性

Mehrotra型预估-校正算法是很多内点算法软件包的算法基础,但它的多项式迭代复杂性直到2007年才被Salahi等人证明.通过选择一个固定的预估步长及与Salahi文中不同的校正方向,本文把Salahi等人的算法拓展到单调线性互补问题,使得新算法的迭代复杂性为O(n log((x0)T s0/ε)),同时,初步的数值实验证明了新算法是有效的.     

基于代数等价变换下的凸二次规划不可行内点算法

基于代数等价变换和在KMM算法的框架基础上,在原始-对偶内点方法的牛顿方程里嵌入一种自调节功能.从而对凸二次规划提出了一种新的迭代方向的不可行内点算法,并证明了算法的全局收敛性.     

一种新的可分凸二次规划的不可行内点算法

本文对可分凸二次规划提出了一个新的不可行内点算法 ,证明了该算法是一个多项式时间算法 ,并将迭代复杂性界降至O(nL) .     

Solving semidefinite-quadratic-linear programs using SDPT3

 This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primal-dual path-following algorithms. The software developed by the authors uses Mehrotra-type predictor-corrector variants of interior-point methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported. Received: March 19, 2001 / Accepted: January 18, 2002 Published online: October 9, 2002 Mathematics Subject Classification (2000): 90C05, 90C22     

Simplified infeasible interior-point algorithm for linear optimization based on a simple function

Based on a similar kernel function, we present an infeasible version of the interior-point algorithm for linear optimization introduced by Wang et al. (2016). The property of exponential convexity is still important to simplify the analysis of the algorithm. The iteration bound coincides with the currently best iteration bound for infeasible interior-point algorithms.     

Global convergence in infeasible-interior-point algorithms

This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence theory given by Polak in 1971 for general nonlinear programs. The class of algorithms is characterized as: Move in a Newton direction for approximating a point on the path of centers of the complementarity problem at each iteration. Starting from a strictly positive but infeasible initial point, each algorithm in the class either generates an approximate solution with a given accuracy or provides us with information that the complementarity problem has no solution in a given bounded set. We present three typical examples of our interior-point algorithms, a horn neighborhood model, a constrained potential reduction model with the use of the standard potential function, and a pure potential reduction model with the use of a new potential function.Research supported in part by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.Corresponding author.     

Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier

We propose a new primal-dual infeasible interior-point method for symmetric optimization by using Euclidean Jordan algebras. Different kinds of interior-point methods can be obtained by using search directions based on kernel functions. Some search directions can be also determined by applying an algebraic equivalent transformation on the centering equation of the central path. Using this method we introduce a new search direction, which can not be derived from a usual kernel function. For this reason, we use the new notion of positive-asymptotic kernel function which induces the class of corresponding barriers. In general, the main iterations of the infeasible interior-point methods are composed of one feasibility and several centering steps. We prove that in our algorithm it is enough to take only one centering step in a main iteration in order to obtain a well-defined algorithm. Moreover, we conclude that the algorithm finds solution in polynomial time and has the same complexity as the currently best known infeasible interior-point methods. Finally, we give some numerical results.     

A Mehrotra-type predictor-corrector algorithm with polynomiality andQ-subquadratic convergence

Mehrotra's predictor-corrector algorithm [3] is currently considered to be one of the most practically efficient interior-point methods for linear programming. Recently, Zhang and Zhang [18] studied the global convergence properties of the Mehrotra-type predictor-corrector approach and established polynomial complexity bounds for two interior-point algorithms that use the Mehrotra predictor-corrector approach. In this paper, we study the asymptotic convergence rate for the Mehrotra-type predictor-corrector interior-point algorithms. In particular, we construct an infeasible-interior-point algorithm based on the second algorithm proposed in [18] and show that while retaining a complexity bound ofO(n ^1.5 t)-iterations, under certain conditions the algorithm also possesses aQ-subquadratic convergence, i.e., a convergence ofQ-order 2 with an unboundedQ-factor.Research supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171.     

A full Nesterov–Todd step infeasible-interior-point algorithm for Cartesian P*(κ) horizontal linear complementarity problems over symmetric cones

Euclidean Jordan algebra is a commonly used tool in designing interior-point algorithms for symmetric cone programs. In this paper, we present a full Nesterov–Todd (NT) step infeasible interior-point algorithm for horizontal linear complementarity problems over Cartesian product of symmetric cones. Since the algorithm uses only full-NT feasibility and centring steps, it has the advantage that no line searches are needed. The complexity result obtained here for symmetric cones using NT directions coincides with the best bound obtained for horizontal linear complementarity problems.     

线性互补问题的宽邻域预估校正算法

对线性互补问题提出了一种新的宽邻域预估校正算法,算法是基于经典线性规划路径跟踪算法的思想,将Maziar Salahi关于线性规划预估校正算法推广到线性互补问题中,给出了算法的具体迭代步骤并讨论了算法迭代复杂性,最后证明了算法具有多项式复杂性为O(ηlog(X~0)~Ts~0/ε)。     

P*(κ)阵线性互补问题一种新的宽邻域预估-校正内点算法

基于邻近度量函数的最小值,对P*(κ)阵线性互补问题提出了一种新的宽邻域预估-校正算法,在较一般的条件下,证明了算法的迭代复杂性为O(κ+1)23n log(x0ε)Ts0.算法既可视为Miao的P*(κ)阵线性互补问题Mizuno-Todd-Ye预估-校正内点算法的一种变形,也可以视为最近Zhao提出的线性规划基于邻近度量函数最小值的宽邻域内点算法的推广.     

An infeasible interior-point arc-search algorithm for nonlinear constrained optimization

Numerical Algorithms - In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are...