对称锥规划的邻域跟踪算法

本文把艾文宝的邻域跟踪算法推广到对称锥规划, 定义中心路径的宽邻域N(τ, β), 并证明该邻域的一个重要性质, 该性质在算法的复杂性分析中起到关键作用. 取宽邻域N(τ, β) 中一点为初始点并采用Nesterov-Todd (NT) 搜索方向, 则该算法的迭代复杂界为O(√r logε^-1), 其中, r是EuclidJordan 代数的秩, ε是允许误差. 这是对称锥规划的宽邻域内点算法最好的复杂界.     

线性规划的邻域跟踪算法

提出了线性规划的邻域跟踪算法. 当这个邻域是宽邻域时，该算法就是宽邻域原始-对偶内点算法; 如果这个邻域退化成中心路径, 则算法就退化成中心路径跟踪算法. 证明了该算法具有O(nL)次迭代复杂性, 而经典的宽邻域算法是O(nL)次迭代复杂性. 也证明了该算法在非退化条件下是二次收敛的, 并给出了一些计算结果.     

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．     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

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)。     

关于特殊形式素数$p$的$\alpha p^2$ 模$1$的分布

证明了如果$0<\theta < \frac {2}{375}$, 则对于无理数$\alpha$, 存在无限个素数$p$, 使得$p+2$不超过4个素因子, 并满足不等式$\|\alpha p^2+\beta\|相似文献   

Clifford 环面 Sm(\sqrt{\frac{m}{n}})×Sn-m(\sqrt{\frac{n-m}{n}})的刚性定理

\small\zihao{-5}\begin{quote}{\heiti 摘要:} 设$M$为$n+1$维单位球面$S^{n+1}(1)$中的一个极小闭超曲面，如果 $ n \le S \le n+\frac{2}{3}$, 则有 $S=n$ 且 $M$ 与某一Clifford 环面 $S^m(\sqrt{m/n}) \times S^{n-m}(\sqrt{(n-m)/n})$等距.     

一个求解P_*(k)阵线性互补问题的渐近O((k 1)n~3L)宽域路径跟踪算法

本文通过使用高界校正技术，给出了一个求解P*(k)阵线性互补问题的宽域路径跟踪算法，其迭代复杂性为渐近O((k＋1)L)．通过使用秩-1校正技术，其每步的计算复杂性从常规的O(n3)约减到O(n2.5)；因此，算法总的计算复杂性为渐近O((k＋1)n3L)．     

A NOTE ON SOME METRICAL THEOREMS IN DIOPHANTINE APPROXIMATION

本文证明了两条关于丢番图逼近论中的测度定理.(详细叙迷见本文1)这些定理 是P.X.Gallagher定理的改进，并包有W.M.Schmidt的测度定理.还可以导出，例如： 1.对于几乎有的\[({\theta _1},...,{\theta _n}) \in {R_n}\]，适合于 \[\prod\limits_{i = 1}^n {\left\| {q{\theta _i}} \right\|} q{(\log {\kern 1pt} {\kern 1pt} {\kern 1pt} q)^n} < 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 \leqslant q \leqslant h\] 的整数q的个数为 \[\frac{{{2^n}}}{{(n - 1)!}}\log \log h + O({(\log \log h)^{1/2 + \varepsilon }}\] . 此处||X||表示实数X至最近整数的距离，\[\varepsilon \]为任意正常数，而与“О”有关的常数依赖于\[\varepsilon \] 与诸\[\theta \] 2 W. M. Schmidt与王元的转换定理中的性质2奋于几乎所有的\[({\theta _{11}},...,{\theta _{nm}}) \in {R_{nm}}\]都成立.     

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

Two wide neighborhood interior-point methods for symmetric cone optimization

In this paper, we present two primal–dual interior-point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood \(\mathcal {N}(\tau ,\,\beta )\) of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We derive that these two path-following algorithms have

$$\begin{aligned} \text{ O }\left( \sqrt{r\text{ cond }(G)}\log \varepsilon ^{-1}\right) , \text{ O }\left( \sqrt{r}\left( \text{ cond }(G)\right) ^{1/4}\log \varepsilon ^{-1}\right) \end{aligned}$$

iteration complexity bounds, respectively. The obtained complexity bounds are the best result in regard to the iteration complexity bound in the context of the path-following methods for symmetric cone optimization. Numerical results show that the algorithms are efficient for this kind of problems.

     

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

     

Neighborhood-following algorithms for linear programming

In this paper, we present neighborhood-following algorithms for linear programming. When the neighborhood is a wide neighborhood, our algorithms are wide neighborhood primal-dual interior point algorithms. If the neighborhood degenerates into the central path, our algorithms also degenerate into path-following algorithms. We prove that our algorithms maintain the O(n~(1/2)L)-iteration complexity still, while the classical wide neighborhood primal-dual interior point algorithms have only the O(nL)-iteration complexity. We also proved that the algorithms are quadratic convergence if the optimal vertex is nondegenerate. Finally, we show some computational results of our algorithms.     

An {O(\sqrt{n}L)} iteration primal-dual second-order corrector algorithm for linear programming

In this paper, we propose a primal-dual second-order corrector interior point algorithm for linear programming problems. At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction [Ai and Zhang in SIAM J Optim 16:400–417 (2005)], in an attempt to improve performance. The corrector is multiplied by the square of the step-size in the expression of the new iterate. We prove that the use of the corrector step does not cause any loss in the worst-case complexity of the algorithm. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm enjoyed the low iteration bound of O(?nL){O(\sqrt{n}L)}, the same as the best known complexity results for interior point methods.     

$P_*(\kappa)$线性互补问题的Mehrotra型预估-校正算法

Mehrotra-type predictor-corrector algorithm,as one of most efficient interior point methods,has become the backbones of most optimization packages.Salahi et al.proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice.We extend their algorithm to P_*（κ）linear complementarity problems.The way of choosing corrector direction for our algorithm is different from theirs. The new algorithm has been proved to have an ο(（1+4κ）（17+19κ） √（1+2κn）^3/2log[（x⁰）^Ts⁰/ε] worst case iteration complexity bound.An numerical experiment verifies the feasibility of the new algorithm.     

An $$O(\sqrt{n}L)$$ iteration Mehrotra-type predictor-corrector algorithm for monotone linear complementarity problem

In this paper we propose a new class of Mehrotra-type predictor-corrector algorithm for the monotone linear complementarity problems (LCPs). At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction (SIAM J Optim 16:400–417, 2005), in an attempt to improve performance. Starting with a feasible point \((x^0, s^0)\) in the wide neighborhood \(\mathcal {N}(\tau ,\beta )\), the algorithm enjoys the low iteration bound of \(O(\sqrt{n}L)\), where \(n\) is the dimension of the problem and \(L=\log \frac{(x^0)^T s^0}{\varepsilon }\) with \(\varepsilon \) the required precision. We also prove that the new algorithm can be specified into an easy implementable variant for solving the monotone LCPs, in such a way that the iteration bound is still \(O(\sqrt{n}L)\). Some preliminary numerical results are provided as well.     

A primal-dual large-update interior-point algorithm for <Emphasis Type="Italic">P</Emphasis><Subscript>*</Subscript>(<Emphasis Type="Italic">κ</Emphasis>)-LCP based on a new class of kernel functions

In this paper, we propose a large-update primal-dual interior point algorithm for P_*(κ)-linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor self-regular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has \(o\left( {(1 + 2k)p\sqrt n {{\left( {\frac{1}{p}\log n + 1} \right)}^2}\log \frac{n}{\varepsilon }} \right)\)iteration complexity which becomes \(o\left( {(1 + 2k)\sqrt n log{\kern 1pt} {\kern 1pt} n\log \frac{n}{\varepsilon }} \right)\)with special choice of the parameter p. It is matches the currently best known iteration bound for P_*(κ)-linear complementarity problem. Some computational results have been provided.     

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of ${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.     

Interior-point algorithms for P_{*}(\kappa )-LCP based on a new class of kernel functions

In this paper, we propose interior-point algorithms for $P_* (\kappa )$ -linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functions. We show that if a strictly feasible starting point is available, then the new algorithm has $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n}\log n \log \frac{n\mu ^0}{\epsilon }\bigr )$ and $\mathcal{O }\bigl ((1+2\kappa )\sqrt{n} \log \frac{n\mu ^0}{\epsilon }\bigr )$ iteration complexity for large- and small-update methods, respectively. These are the best known complexity results for such methods.     

Polynomial interior-point algorithm for P * {(\kappa)} horizontal linear complementarity problems

In this paper an interior-point algorithm for P _*(κ) horizontal linear complementarity problems is proposed that uses new search directions. The theoretical complexity of the new algorithm is calculated. It is investigated that the proposed algorithm has quadratically convergent with polynomial iteration complexity $O((1+\kappa)\sqrt{n}\log\frac{n}{\varepsilon})$ , coincide with the best known iteration bound for P _*(κ) horizontal linear complementarity problems.