Adaptive augmented Lagrangian methods: algorithms and practical numerical experience

In this paper, we consider augmented Lagrangian (AL) algorithms for solving large-scale nonlinear optimization problems that execute adaptive strategies for updating the penalty parameter. Our work is motivated by the recently proposed adaptive AL trust region method by Curtis et al. [An adaptive augmented Lagrangian method for large-scale constrained optimization, Math. Program. 152 (2015), pp. 201–245.]. The first focal point of this paper is a new variant of the approach that employs a line search rather than a trust region strategy, where a critical algorithmic feature for the line search strategy is the use of convexified piecewise quadratic models of the AL function for computing the search directions. We prove global convergence guarantees for our line search algorithm that are on par with those for the previously proposed trust region method. A second focal point of this paper is the practical performance of the line search and trust region algorithm variants in Matlab software, as well as that of an adaptive penalty parameter updating strategy incorporated into the Lancelot software. We test these methods on problems from the CUTEst and COPS collections, as well as on challenging test problems related to optimal power flow. Our numerical experience suggests that the adaptive algorithms outperform traditional AL methods in terms of efficiency and reliability. As with traditional AL algorithms, the adaptive methods are matrix-free and thus represent a viable option for solving large-scale problems.     

Weighted and extended total variation for image restoration and decomposition

In various information processing tasks obtaining regularized versions of a noisy or corrupted image data is often a prerequisite for successful use of classical image analysis algorithms. Image restoration and decomposition methods need to be robust if they are to be useful in practice. In particular, this property has to be verified in engineering and scientific applications. By robustness, we mean that the performance of an algorithm should not be affected significantly by small deviations from the assumed model. In image processing, total variation (TV) is a powerful tool to increase robustness. In this paper, we define several concepts that are useful in robust restoration and robust decomposition. We propose two extended total variation models, weighted total variation (WTV) and extended total variation (ETV). We state generic approaches. The idea is to replace the TV penalty term with more general terms. The motivation is to increase the robustness of ROF (Rudin, Osher, Fatemi) model and to prevent the staircasing effect due to this method. Moreover, rewriting the non-convex sublinear regularizing terms as WTV, we provide a new approach to perform minimization via the well-known Chambolle's algorithm. The implementation is then more straightforward than the half-quadratic algorithm. The behavior of image decomposition methods is also a challenging problem, which is closely related to anisotropic diffusion. ETV leads to an anisotropic decomposition close to edges improving the robustness. It allows to respect desired geometric properties during the restoration, and to control more precisely the regularization process. We also discuss why compression algorithms can be an objective method to evaluate the image decomposition quality.     

On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P$P$ with integer coefficients in k$k$ variables over the standard simplex of R^k${\mathbb{R}}^k$, assuming that P$P$ is positive on the simplex. This bound depends only on the number of variables k$k$, the degree d$d$ and the bitsize τ$\tau$ of the coefficients of P$P$ and improves all the previous bounds for arbitrary polynomials which are positive over the simplex.     

基于LASVM-NC和TF.RF的文本分类方法

非凸在线支持向量机（LASVM-NC）具有抗噪能力强和训练速度快的优点,而词频相关频率积（tf.rf）则是一种自适应能力很强、分类性能非常好的文本特征。通过把非凸在线支持向量机和词频相关频率积相结合,提出了一种新的文本分类方法,即LASVM-NC＋tf.rf。实验结果表明,这种方法在LASVM-NC与多种其他特征的结合中性能是最好的,且与SVM＋tf.rf相比,不仅所产生的分类器具有泛化能力更强、模型表达更稀疏的优点,而且在处理含噪声的数据时具有更好的鲁棒性,在处理大规模数据时具有快得多的训练速度。     

机组耗量特性的混合整数模型建立与分析

对机组耗量特性的建模是机组组合问题的一个重要组成部分.文中建立并分析了机组耗量特性的5种模型,并在IEEE RTS 96系统上,分别针对具有凸连续、非凸连续和非连续耗量特性曲线的机组,通过优化其在某日24时段的出力,比较各模型的求解效率,并根据各模型所引入变量数目对测试结果进行分析,从而为根据机组特性选择合适的模型以提高求解效率提供参考.结果表明:影响算法求解性能的主要因素是模型中整数变量的数目.对于凸连续耗量特性机组,尽量采用各区间出力叠加的建模方法,以避免引入整数变量;对于非凸和非连续耗量特性机组,引入辅助整数变量不可避免,此时则应采用连续变量较少的模型.     

考虑发电机阀点效应的改进拉格朗日松弛有功调度策略

发电机阀点效应导致的非凸非光滑特性增加了有功经济调度模型的复杂度,从而增大了求解难度.文中提出一种考虑发电机阀点效应影响的改进拉格朗日松弛有功调度算法.首先,提出一种二次多项式分段拟合算法,从而将发电机阀点效应成本分解为多个分段二次多项式形式;其次,将拟合后的表达式代入拉格朗日松弛子问题中,从而将原问题转化为分段二次规划问题,并采用比较对称轴及优化变量上下限的方法快速求解,避免了引入整数变量造成的求解效率偏低问题.同时,为防止多个局部最优解导致的优化过程不收敛问题,提出一种基于近端梯度法的增广求解策略,通过在目标函数中引入辅助变量及惩罚项来加强子问题的凸性,加快收敛速度.最后,基于40机和48机测试系统对所提方法的有效性进行了测试.     

空间目标图像的非凸稀疏正则化波后复原

现有的空间目标图像波后处理方法多直接套用自然光学图像的复原技术,效果并不理想。本文通过分析空间目标图像的近似稀疏性和灰度值服从超拉普拉斯分布的独有特点,提出了一个采用正则化方法的非凸稀疏正则化空间目标图像复原模型。在数值计算过程中,根据交替方向乘数法将复原模型分解为两个子问题,对凸优化子问题采用快速傅里叶变换求解,对非凸优化子问题采用固定点迭代方法求解。文中设计了非凸稀疏正则化空间目标图像波后复原的完整算法流程,并针对模拟图像和真实空间目标图像进行了对比验证。结果显示:相对于最近的流行算法,提出方法的最大峰值信噪比提高了2dB,最大平均结构相似度提高了0.17,最大信息熵提高了3.85,图像清晰度提高了2.65。     

基于非凸低秩约束的图像修复方法

受传输干扰或存储不当等因素的影响,现实应用中获取的某些图像通常会存在像素缺失现象,这给图像的后续分析与处理带来了一定影响.解决该问题的常用方法是对图像进行低秩修复.利用低秩特性进行修复的方法大多以秩函数建模,由于矩阵秩函数是非凸离散的,该模型的求解是一个NP难问题,所以通常利用核范数对矩阵的秩进行凸松弛.但是,基于核范...     

碳载钴镍合金纳米电极的制备表征与电催化性能

采用电化学方法在玻碳(GC)表面电沉积CoNi合金纳米粒子,成功制得碳载CoNi合金纳米电极(CoNi/GC)。SEM结果显示,CoNi粒子呈十八面体结构,粒径约100 nm,分布较均匀。选区电子衍射(SAED)结果显示,CoNi合金纳米粒子为单晶结构。XPS结果显示,金属态的Co(0)和Ni(0)占主导地位。性能测试结果表明:CoNi/GC不但对亚硝酸钠具有较好的催化性能,相对于本体Co和本体Ni,CoNi/GC的起始还原电位(E_i)正移约90 mV,还原峰电流(j_p)增大6~14倍。而且对氧还原亦有较好的电催化活性,CoNi/GC的峰电流密度(j_p)和动力电流密度(j_k)分别是GC电极的1.7和 5.2倍。