A full-step interior-point algorithm for second-order cone optimization based on a simple locally kernel function

In this paper, we investigate the properties of a simple locally kernel function. As an application, we present a full-step interior-point algorithm for second-order cone optimization (SOCO). The algorithm uses the simple locally kernel function to determine the search direction and define the neighbourhood of central path. The full step used in the algorithm has local quadratic convergence property according to the proximity function which is constructed by the simple locally kernel function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bound for SOCO.     

基于动量粒子群的混合核SVM参数优化方法

支持向量机（SVM）可以很好地用来解决分类问题,参数优化尤其重要。混合核函数的引入,使得SVM又多了一个可调参数。针对该参数用人工或经验的方法获取具有局限性,采用动量粒子群（MPSO）对SVM基本参数、混合可调核参数进行综合寻优,来寻找最佳参数组合。通过UCI数据仿真,对比结果表明：所提优化方法能够快速有效地提取最佳参数组合,所得SVM性能明显提高,分类效果更好。     

Learning general Gaussian kernel hyperparameters of SVMs using optimization on symmetric positive-definite matrices manifold

We propose a new method for general Gaussian kernel hyperparameter optimization for support vector machines classification. The hyperparameters are constrained to lie on a differentiable manifold. The proposed optimization technique is based on a gradient-like descent algorithm adapted to the geometrical structure of the manifold of symmetric positive-definite matrices. We compare the performance of our approach with the classical support vector machine for classification and with other methods of the state of the art on toy data and on real world data sets.     

Global optimization in the class of stochastically unimodal functions

Generalization of the global optimization problem based on a stochastic approach is considered. The concept of a stochastic (or unimodal in the mean) fuction is introduced. To find the optimal solution, a heuristic self-organization procedure is proposed. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 157–166, January–February, 2000.     

基于PSO_KFCM的医学图像分割

在核模糊聚类算法(KFCM)的基础上,提出了一种新的PSO KFCM聚类算法.新算法利用高斯核函数,把输入空间的样本映射到高维特征空间,利用微粒群算法的全局搜索、快速收敛的特点,代替KFCM算法逐次迭代的过程,在特征空间中进行聚类,克服了KFCM对初始值和噪声数据敏感、易陷入局部最优的缺点.通过对医学图像进行分割,仿真实验结果表明,新算法在性能上比KFCM聚类算法有较大改进,具有更好的聚类效果,且算法能够很快地收敛.     

The privacy of dense symmetric functions

Ann argument function,f, is calledt-private if there exists a distributed protocol for computingf so that no coalition of at mostt processors can infer any additional information from the execution of the protocol. It is known that every function defined over a finite domain is [(n–1)/2]-private. The general question oft-privacy (fort[n/2]) is still unresolved.In this work, we relate the question of [n/2]-privacy for the class of symmetric functions of Boolean argumentsf: {0, 1} ⁿ {0, 1,...,n} to the structure of Hamming weights inf ^–1(b) (b{0, 1, ...,n}). We show that iff is [n/2]-private, then every set of Hamming weightsf ^–1(b) must be an arithmetic progression. For the class ofdense symmetric functions (defined in the sequel), we refine this to the following necessary and sufficient condition for [n/2]-privacy off: Every collection of such arithmetic progressions must yield non-identical remainders, when computed modulo the greatest common divisor of their differences. This condition is used to show that for dense symmetric functions, [n/2]-privacy impliesn-privacy.     

Gaussian kernel optimization for pattern classification

This paper presents a novel algorithm to optimize the Gaussian kernel for pattern classification tasks, where it is desirable to have well-separated samples in the kernel feature space. We propose to optimize the Gaussian kernel parameters by maximizing a classical class separability criterion, and the problem is solved through a quasi-Newton algorithm by making use of a recently proposed decomposition of the objective criterion. The proposed method is evaluated on five data sets with two kernel-based learning algorithms. The experimental results indicate that it achieves the best overall classification performance, compared with three competing solutions. In particular, the proposed method provides a valuable kernel optimization solution in the severe small sample size scenario.     

A constructive count of rotation symmetric functions

In this paper we present a constructive detection of minimal monomials in the algebraic normal form of rotation symmetric Boolean functions (immune to circular translation of indices). This helps in constructing rotation symmetric Boolean functions by respecting the rules we present here.     

Efficient hybrid methods for global continuous optimization based on simulated annealing

We introduce several hybrid methods for global continuous optimization. They combine simulated annealing and a local proximal bundle method. Traditionally, the simplest hybrid of a global and a local solver is to call the local solver after the global one, but this does not necessarily produce good results. Besides, using efficient gradient-based local solvers implies that the hybrid can only be applied to differentiable problems. We show several ways how to integrate the local solver as a genuine part of simulated annealing to enable both efficient and reliable solution processes. When using the proximal bundle method as a local solver, it is possible to solve even nondifferentiable problems. The numerical tests show that the hybridization can improve both the efficiency and the reliability of simulated annealing.     

最大代数免疫度的偶变元对称函数的性质

代数免疫度达到最大的偶变元对称布尔函数的特征仍然是个公开问题。结合组合数学和数论的相关结论研究这类函数的性质,得到了此类函数值向量的几个特征。最后,对于变元个数为两类特殊偶数的情况,得到了代数免疫度达到最大的对称函数的一个特征。     

Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

In this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.     

Maple code of gamma algorithm for global optimization of uncertain functions over compact robust sets

Problems with uncertainties are ubiquitous in many areas of science and technology. Due to imprecision of measurements (Heisenberg’s relation), such problems are normal in nuclear physics. Due to fluidity of media, ships at sea and planes in the air have to deal with instability of the currents they move in. Yields in agriculture depend on the whims of weather. Due to the lack of information in economy and finance, problems with uncertainties (stock prices, marketing problems, inflation, unemployment) are commonplace. In such situations, it is necessary to make a choice of better parameters that produce finite intervals of possible values of a given uncertain function at each point of the parameter space. The gamma algorithm [E.A. Galperin, Global optimization in problems with uncertainties, Journal of Nonlinear Analysis 47 (2001) 941–952; E.A. Galperin, Global optimization in problems with uncertainties. The gamma algorithm. Computer and Mathematics with Applications 44 (2002) 853–862] presents a method to make that choice. A new variant of the gamma algorithm based on the beta algorithm is presented for global optimization of uncertain functions over compact robust sets in . The set-monotonic algorithm contains a block for problems with equality constraints, and operates within the unit cube [0,1]ⁿ for all problems. On this basis, a MAPLE code of modular structure is developed for full global optimization of functions of n variables. The code does not create ill-conditioned situations. Graphics are included, and the solution set can be visualized in plane projections and sections. The code is ready for engineering applications. The results of numerical experiments are presented, with graphs, to illustrate the use of the code.     

基于核模糊聚类的动态多子群协作骨干粒子群优化

针对骨干粒子群优化（BBPSO）算法易陷入局部最优、收敛速度低等问题,提出了基于核模糊聚类的动态多子群协作骨干粒子群优化（KFC-MSBPSO）算法。该算法在标准骨干粒子群算法的基础上,首先,采用核模糊聚类方法将主群分割为多个子群,令各个子群协同寻优,提高了算法的搜索效率。然后,引入非线性动态变异因子,根据子群内粒子数以及收敛情况动态调节子群粒子变异概率,通过变异的方式使子群粒子重新回到主群,提高了算法的探索能力;进一步采用主群粒子吸收策略与子群合并策略加强了主群与子群之间、子群与子群之间的信息交流,提高了算法的稳定性。最后,利用子群重建策略,结合主群与子群搜索到的最优解,调节子群重建的间隔代数。通过Sphere等6个标准测试函数进行对比实验,结果表明,KFC-MSBPSO算法和经典BBPSO算法以及反向骨干粒子群优化（OBBPSO）算法等改进算法相比寻优准确率至少提高了约11.1%,在高维解空间内测试结果的最佳均值占到83.33%并且具有更高的收敛速度。这说明KFC-MSBPSO算法具有良好的搜索性能与鲁棒性,可应用于高维复杂函数的优化问题中。     

A new infeasible interior-point algorithm with full step for linear optimization based on a simple function

In this paper, we design and analyse an infeasible interior-point algorithm based on a simple function for linear optimization. The infeasible algorithm contains two types of search directions: the feasibility search direction and the centrality search direction. Both of the directions are determined by the simple function. The algorithm uses full step, thus no need to perform the line-search procedure. Although the proposed function is simple, as it will be shown, the induced infeasible algorithm enjoys the best-known iteration complexity for infeasible interior-point algorithm.     

Structured symmetric rank-one method for unconstrained optimization

In this paper, we investigate a symmetric rank-one (SR1) quasi-Newton (QN) formula in which the Hessian of the objective function has some special structure. Instead of approximating the whole Hessian via the SR1 formula, we consider an approach which only approximates part of the Hessian matrix that is not easily acquired. Although the SR1 update possesses desirable features, it is unstable in the sense that, it may not retain positive definiteness and may become undefined. Therefore, we describe some safeguards to overcome these difficulties. Since the structured SR1 method provides a more accurate Hessian approximation, therefore the proposed method reduces significantly the computational efforts needed in solving a problem. The results of a series of experiments on a typical set of standard unconstrained optimization problems are reported, which show that the structured SR1 method exhibits a clear improvement in numerical performance over some existing QN algorithms.     

An invariance kernel representation of ISDS Lyapunov functions

We apply set valued analysis techniques in order to characterize the input-to-state dynamical stability (ISDS) property, a variant of the well known input-to-state stability (ISS) property. Using a suitable augmented differential inclusion we are able to characterize the epigraphs of minimal ISDS Lyapunov functions as invariance kernels. This characterization gives new insight into local ISDS properties and provides a basis for a numerical approximation of ISDS and ISS Lyapunov functions via set oriented numerical methods.     

基于粒子群的粗糙核聚类算法

针对K-means聚类算法容易陷入局部最优、不能处理边界对象及线性不可分的缺点,提出一种基于粒子群的粗糙核聚类算法。该算法通过Mercer核将输入样本空间中的样本映射到高维空间,使样本变得线性可分,并结合粗糙集的思想,通过动态改变上下近似集的权重因子对边界对象进行有效处理,同时采用reliefF方法对样本属性进行加权处理,以解决混合数据的聚类问题,最后利用粒子群算法防止算法陷入局部最优。仿真实验表明,相对于其他改进算法,该算法具有较高的正确率和较短的收敛时间,并进一步验证了该算法的鲁棒性和稳定性,具有一定的实用价值。     

The effect of global extrapolation on the phase-lag of symmetric methods for solving periodic initial value problems

The effects are discussed of two-grid global extrapolation procedures on the phase-lags of convergent numerical methods for solving periodic initial value problems.

The procedure is tested on two methods applied to two problems from the literature, one nonlinear the other linear, and the effects of the extrapolation are examined by comparing corresponding zeros of the waves generated by the theoretical and computed solutions.

Extensions to three- and four-grid extrapolation procedures are outlined in an Appendix.