基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

本文基于分段二次多项式方程,构造了一种积极集策略的光滑化max函数.通过给出与光滑化max函数相关的分量函数指标集的直接计算方法,将分段二次多项式方程转化为一般二次多项式方程.利用二次多项式方程根的性质,给出了该光滑化max函数的稳定计算策略,证明了其具有一阶光滑性,其梯度函数具有局部Lipschitz连续性和强半光滑性.该光滑化max函数仅与函数值较大的分量函数相关,适用于含分量函数较多且复杂的max函数的问题.为了验证其效率,本文基于该函数构造了一种解含多个复杂分量函数的无约束minimax问题的光滑化算法,数值实验表明了该光滑化max函数的可行性及有效性.     

解含多个复杂分量函数无约束minimax问题的积极集光滑化算法

本文利用分段三次多项式方程构造了一种积极集策略的二次连续可微的光滑化max函数,给出积极集及稳定的光滑化max函数的计算方法.基于该光滑化max函数,结合Armijo线搜索,负梯度和牛顿方向及光滑化参数的更新策略,给出一种解含多个复杂分量函数无约束minimax问题的积极集光滑化算法.初步的数值实验表明了该算法的有效性.     

调节熵函数法

１．引言 考虑如下极小极大问题这里ｆｉ（ｘ）是Ｒｎ中连续可微的函数,ｍ≥２是正整数（Ｐ）是一类比较典型的非光滑优化问题,是许多实际问题的数学模型．同时,线性规划的 Ｋａｒｍａｒｋａｒ标准型的对偶也是（Ｐ）的形式,光滑约束优化问题的一类重要罚函数法也是将问题化为类似（Ｐ）的形式．所以,如何有效地求解（Ｐ）,是一个重要问题．近些年发展起来的嫡函数法（或称凝聚函数法）是一种较新颖而实用的方法．它借助信息论中 Ｓｈａｎｎｏｎ熵的概念,推导出一族光滑的极大熵函数Ｆｐ（ｘ）,且Ｆｐ（ｘ）一致逼近要极小化的非光…     

鲁棒稀疏重构问题的凝聚同伦算法

鲁棒稀疏重构问题是信号处理领域的重要问题,该问题的数学本质是一个NP难的数学优化问题.同伦算法是一类典型的路径跟踪算法,该算法是解非线性问题的一类成熟算法,具有全局收敛性,且易于并行实现.本文考虑同伦算法在鲁棒稀疏重构问题中的数值求解.基于l_∞范数及罚函数策略,我们首先将原始的基于l_0范数的最优化模型,转化为含参数的无约束极大极小值问题,进而构造凝聚函数光滑化模型中的极大值函数,并构造凝聚同伦算法数值求解.数值仿真实验验证了新方法的有效性,为大规模鲁棒重构问题的并行化数值求解奠定基础.     

互补问题算法的新进展

互补问题是一类重要的优化问题，在最近３０多年的时间里，人们为求解它而提出了许多算法，该文主要介绍１９９０－１９９７年之间出现的某些新算法，它们大致可归类为：（１）光滑方程法；（２）非光滑方程法；（３）可微无约束优化法；（４）ＧＬＰ投影法；（５）内点法；（６）磨光与非内点连续法，文中对每类算法及相应的收敛性结果做了描述与评论，并列出有关文献。     

求解摩擦接触问题的一个非内点光滑化算法

给出了一个求解三维弹性有摩擦接触问题的新算法,即基于NCP函数的非内点光滑化算法．首先通过参变量变分原理和参数二次规划法,将三维弹性有摩擦接触问题的分析归结为线性互补问题的求解;然后利用NCP函数,将互补问题的求解转换为非光滑方程组的求解;再用凝聚函数对其进行光滑化,最后用NEWTON法解所得到的光滑非线性方程组．方法具有易于理解及实现方便等特点．通过线性互补问题的数值算例及接触问题实例证实了该算法的可靠性与有效性．     

基于一类光滑函数法求解选址问题

选址问题是组合优化中一类有着重要理论意义和广泛实际背景的问题.在利用数学模型解决这类问题时经常会遇到非线性L_1问题,也就是不可微优化问题.为了解决这类问题,构造了适合于选址问题的一类新的光滑函数,并对这类光滑函数进行了性质描述,然后在此基础上提出了基于有效集法进行优化求解的计算步骤.最后,以实例证明了这类光滑函数应用在选址问题的优化求解上是有效的.     

一个光滑化函数的两个性质

本文考虑文[6]中提出的光滑化函数,证明了：该光滑化函数拥有两个在求解变分不等式和互补问题的非内部连续化算法的全局线性和局部超线性（或二次）收敛性分析中非常有用的两个性质。     

非线性极大极小问题一个新的QP-free算法

本文研究非线性无约束极大极小优化问题. QP-free算法是求解光滑约束优化问题的有效方法之一,但用于求解极大极小优化问题的成果甚少.基于原问题的稳定点条件,既不需含参数的指数型光滑化函数,也不要等价光滑化,提出了求解非线性极大极小问题一个新的QP-free算法.新算法在每一次迭代中,通过求解两个相同系数矩阵的线性方程组获得搜索方向.在合适的假设条件下,该算法具有全局收敛性.最后,初步的数值试验验证了算法的有效性.     

基于LBFGS的求解最小闭包球的光滑化方法

考虑在n维空间中求m个球的最小闭包球(the Smallest Enclosing Ball,SEB)问题.首先将SEB问题转化为一个含有函数max(0,z)的等价无约束非光滑凸优化问题,然后利用光滑化技巧和有限内存BFGS方法来求解高维空间中的SEB问题,并分析了方法的收敛性.数值实验结果表明文中给出的算法是有效的.     

约束Fractional Tikhonov正则化的模迭代方法

反问题是现在数学物理研究中的一个热点问题,而反问题求解面临的一个本质性困难是不适定性。求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法.如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.当前,最为流行的正则化方法有基于变分原理的Tikhonov正则化及其改进方法,此类方法是求解不适定问题的较为有效的方法,在各类反问题的研究中被广泛采用,并得到深入研究.     

Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method

In this article we study limit cycles in piecewise smooth perturbations of a linear center. In this setting it is common to adapt classical results for smooth systems, like Melnikov functions, to non-smooth ones. However, there is little justification for this procedure in the literature. By using the regularization method we give a theoretical proof that supports the use of Melnikov functions directly from the original non-smooth problem.     

A one-parametric class of smoothing functions and an improved regularization Newton method for the NCP

In this paper, we introduce a one-parametric class of smoothing functions, which enjoys some favourable properties and includes two famous smoothing functions as special cases. Based on this class of smoothing functions, we propose a regularization Newton method for solving the non-linear complementarity problem. The main feature of the proposed method is that it uses a perturbed Newton equation to obtain the direction. This not only allows our method to have global and local quadratic convergences without strict complementarity conditions, but also makes the regularization parameter converge to zero globally Q-linearly. In addition, we use a new non-monotone line search scheme to obtain the step size. Some numerical results are reported which confirm the good theoretical properties of the proposed method.     

Epsilon-Trig Regularization Method for Bang-Bang Optimal Control Problems

Bang-bang control problems have numerical issues due to discontinuities in the control structure and require smoothing when using optimal control theory that relies on derivatives. Traditional smooth regularization introduces a small error into the original problem using error controls and an error parameter to enable the construction of accurate smoothed solutions. When path constraints are introduced into the problem, the traditional smooth regularization fails to bound the error controls involved. It also introduces a dimensional inconsistency related to the error parameter. Moreover, the traditional approach solves for the error controls separately, which makes the problem formulation complicated for a large number of error controls. The proposed Epsilon-Trig regularization method was developed to address these issues by using trigonometric functions to impose implicit bounds on the controls. The system of state equations is modified such that the smoothed control is expressed in sine form, and only one of the state equations contains an error control in cosine form. Since the Epsilon-Trig method has an error parameter only in one state equation, there is no dimensional inconsistency. Moreover, the Epsilon-Trig method only requires the solution to one control, which greatly simplifies the problem formulation. Its simplicity and improved capability over the traditional smooth regularization method for a wide variety of problems including the Goddard rocket problem have been discussed in this study.     

求解最大团问题的熵正则化方法

最大团问题是组合优化的一个经典问题.在Motzkin和Straus的二次规划模型基础上,给出一种求解该问题的熵正则化算法.引进熵函数有两个目的,一是将问题的求解纳入信息论方法的框架,二是通过它的引进改善问题的凸性.几个标准考题的计算结果表明,该算法稳定有效.     

求解带线性约束的凸优化的一类自适应不定线性化增广拉格朗日方法

增广拉格朗日方法是求解带线性约束的凸优化问题的有效算法.线性化增广拉格朗日方法通过线性化增广拉格朗日函数的二次罚项并加上一个临近正则项,使得子问题容易求解,其中正则项系数的恰当选取对算法的收敛性和收敛速度至关重要.较大的系数可保证算法收敛性,但容易导致小步长.较小的系数允许迭代步长增大,但容易导致算法不收敛.本文考虑求解带线性等式或不等式约束的凸优化问题.我们利用自适应技术设计了一类不定线性化增广拉格朗日方法,即利用当前迭代点的信息自适应选取合适的正则项系数,在保证收敛性的前提下尽量使得子问题步长选择范围更大,从而提高算法收敛速度.我们从理论上证明了算法的全局收敛性,并利用数值实验说明了算法的有效性.     

On the acceleration of the double smoothing technique for unconstrained convex optimization problems

In this article, we investigate the possibilities of accelerating the double smoothing (DS) technique when solving unconstrained nondifferentiable convex optimization problems. This approach relies on the regularization in two steps of the Fenchel dual problem associated with the problem to be solved into an optimization problem having a differentiable strongly convex objective function with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method. The aim of this article is to show how the properties of the functions in the objective of the primal problem influence the implementation of the DS approach and its rate of convergence. The theoretical results are applied to linear inverse problems by making use of different regularization functionals.     

Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters

In this paper, we study the multi-parameter Tikhonov regularization method which adds multiple different penalties to exhibit multi-scale features of the solution. An optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters. Some theoretical results of the regularization solution about the dependence on regularization parameters are presented. Then, an a posteriori parameter choice, i.e., the damped Morozov discrepancy principle, is introduced to determine multiple regularization parameters. Five model functions, i.e., two hyperbolic model functions, a linear model function, an exponential model function and a logarithmic model function, are proposed to solve the damped Morozov discrepancy principle. Furthermore, four efficient model function algorithms are developed for finding reasonable multiple regularization parameters, and their convergence properties are also studied. Numerical results of several examples show that the damped discrepancy principle is competitive with the standard one, and the model function algorithms are efficient for choosing regularization parameters.     

Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography

In positron emission tomography, image data corresponds to measurements of emitted photons from a radioactive tracer in the subject. Such count data is typically modeled using a Poisson random variable, leading to the use of the negative-log Poisson likelihood fit-to-data function. Regularization is needed, however, in order to guarantee reconstructions with minimal artifacts. Given that tracer densities are primarily smoothly varying, but also contain sharp jumps (or edges), total variation regularization is a natural choice. However, the resulting computational problem is quite challenging. In this paper, we present an efficient computational method for this problem. Convergence of the method has been shown for quadratic regularization functions and here convergence is shown for total variation regularization. We also present three regularization parameter choice methods for use on total variation-regularized negative-log Poisson likelihood problems. We test the computational and regularization parameter selection methods on two synthetic data sets.     

求解离散不适定问题的正则化GMERR方法

迭代极小残差方法是求解大型线性方程组的常用方法, 通常用残差范数控制迭代过程.但对于不适定问题, 即使残差范数下降, 误差范数未必下降. 对大型离散不适定问题,组合广义最小误差(GMERR)方法和截断奇异值分解(TSVD)正则化方法, 并利用广义交叉校验准则(GCV)确定正则化参数,提出了求解大型不适定问题的正则化GMERR方法.数值结果表明, 正则化GMERR方法优于正则化GMRES方法.