Nonlinear optimization and support vector machines

Support Vector Machine (SVM) is one of the most important class of machine learning models and algorithms, and has been successfully applied in various fields. Nonlinear optimization plays a crucial role in SVM methodology, both in defining the machine learning models and in designing convergent and efficient algorithms for large-scale training problems. In this paper we present the convex programming problems underlying SVM focusing on supervised binary classification. We analyze the most important and used optimization methods for SVM training problems, and we discuss how the properties of these problems can be incorporated in designing useful algorithms.     

Parametric Ghost Image Processes for Fixed-Charge Problems: A Study of Transportation Networks

We present a parametric approach for solving fixed-charge problems first sketched in Glover (1994). Our implementation is specialized to handle the most prominently occurring types of fixed-charge problems, which arise in the area of network applications. The network models treated by our method include the most general members of the network flow class, consisting of generalized networks that accommodate flows with gains and losses. Our new parametric method is evaluated by reference to transportation networks, which are the network structures most extensively examined, and for which the most thorough comparative testing has been performed. The test set of fixed-charge transportation problems used in our study constitutes the most comprehensive randomly generated collection available in the literature. Computational comparisons reveal that our approach performs exceedingly well. On a set of a dozen small problems we obtain ten solutions that match or beat solutions found by CPLEX 9.0 and that beat the solutions found by the previously best heuristic on 11 out of 12 problems. On a more challenging set of 120 larger problems we uniformly obtain solutions superior to those found by CPLEX 9.0 and, in 114 out of 120 instances, superior to those found by the previously best approach. At the same time, our method finds these solutions while on average consuming 100 to 250 times less CPU time than CPLEX 9.0 and a roughly equivalent amount of CPU time as taken by the previously best method.     

Hopfield神经网络中的数学问题

Ｈｏｐｆｉｅｌｄ人工神经网络和多层网络模型中有许多基本的数学问题，其中，最重要的是存储容量的问题，即吸引子的个数，在以往的文章中，大多数都以概率模型为背景，本文从组合论的角度提供了几个问题，我们把Ｈｏｐｆｉｅｌｄ网络中的状态按连接矩阵所构成的变换分类，网络的容量就是类的个数，我们得出了一些基本结果，还有一些有意义的问题，如多层网络中的结果是否可交换等问题，还有待进一步研究。     

On the existence of solutions to quasivariational inclusion problems

We establish sufficient existence conditions for general quasivariational inclusion problems, which contain most of variational inclusion problems and quasiequilibrium problems considered in the literature. These conditions are shown to extend recent existing results and sharpen some of them even for particular cases.     

几类考虑有限理性平衡问题解的稳定性

对Ky Fan点问题定义了理性函数,证明了大多数的KyFan点问题(在Baire分类意义上)都是结构稳定的,对$\varepsilon$-平衡也都是鲁棒的.作为应用,还给出了Nash平衡问题和变分不等式问题的稳定性结论.     

Robustness analysis in Multi-Objective Mathematical Programming using Monte Carlo simulation

In most multi-objective optimization problems we aim at selecting the most preferred among the generated Pareto optimal solutions (a subjective selection among objectively determined solutions). In this paper we consider the robustness of the selected Pareto optimal solution in relation to perturbations within weights of the objective functions. For this task we design an integrated approach that can be used in multi-objective discrete and continuous problems using a combination of Monte Carlo simulation and optimization. In the proposed method we introduce measures of robustness for Pareto optimal solutions. In this way we can compare them according to their robustness, introducing one more characteristic for the Pareto optimal solution quality. In addition, especially in multi-objective discrete problems, we can detect the most robust Pareto optimal solution among neighboring ones. A computational experiment is designed in order to illustrate the method and its advantages. It is noteworthy that the Augmented Weighted Tchebycheff proved to be much more reliable than the conventional weighted sum method in discrete problems, due to the existence of unsupported Pareto optimal solutions.     

局部G-凸空间内的广义矢量拟平衡问题

在局部G-凸空间内引入和研究了几类广义矢量拟平衡问题(GVQEP)．包含了大多数广义矢量平衡问题,广义矢量变分不等式问题,拟平衡问题和拟变分不等式问题作为特殊情形．首先在局部G-凸空间内对一人对策证明了一个平衡存在性定理．作为应用,在非紧局部G-凸空间内对GVQEP的解建立了某些新的存在定理．这些结果和论证方法与最近文献中的结果和论证方法相比较是新的和完全不同的．     

数学规划的求解与广义Hopfield神经网络

神经网络技术最为成功的应用领域之一是用于求解优化问题,本文就近年来的求解优化问题的神经网络方法进行了综述     

Generic uniqueness theorems with some applications

We first present a proper condition under which the image of a set-valued mapping becomes a singleton and then obtain several generic uniqueness theorems which can be applied to study the uniqueness of the solutions for nonlinear problems. As applications, we prove that, in the sense of Baire category, most optimization problems (respectively, saddle point problems and variational inequality problems) have unique solution.     

FC-空间内的广义向量变分型不等式

在FC-空间中引入和研究了一类广义向量变分型不等式(GVVTIP),包含了大多数向量平衡问题,向量变分不等式问题,广义向量平衡问题和广义向量变分不等式问题作为特殊情况．利用F-KKM定理,在非紧FC-空间中,建立了关于GVVTIP解的某些新的存在定理．这些定理统一、改进和推广了文献中的一些重要的已知结果．     

A comparison of analytic series methods for Laplacian free boundary problems

Solutions to Laplace's equation are required for a wide range of problems. Arguably, the most difficult class of problems involves a “free” boundary, where the location of one (or more) of the boundaries is initially unknown. Analytical solutions for these problems were restricted to regular boundary geometries. However, recently the classical series method has been modified, to cater for arbitrary boundary geometries, using least squares methods. For free boundary problems, solutions can be obtained by solving a sequence of known boundary problems—at each iteration, the series coefficients can be estimated. Efficient calculation of the series coefficients becomes very important, particularly when the number of iterations is relatively high. In this paper, three methods for estimating the series coefficients will be described, in the context of a free boundary problem. The computational cost of each method will be analysed and compared, and the most appropriate method for this class of problem is indicated.     

广义向量似变分不等式解集的通有稳定性及本质连通区的存在性

本文研究广义向量似变分不等式解集的稳定性．证明了在满足一定的连续性和凸 性条件的广义向量似变分不等式问题构成的空间Ｍ中，大多数（在Ｂａｉｒｅ分类意下）广 义向量似变分不等式问题的解集是稳定的，并证明了Ｍ中的每个广义向量似变分不等式 的解集至少存在一个本质连通区。     

Generalized Vector Equilibrium Problems in Generalized Convex Spaces

In this paper, we introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.     

Genetic clustering algorithms

This study employs genetic algorithms to solve clustering problems. Three models, SICM, STCM, CSPM, are developed according to different coding/decoding techniques. The effectiveness and efficiency of these models under varying problem sizes are analyzed in comparison to a conventional statistics clustering method (the agglomerative hierarchical clustering method). The results for small scale problems (10–50 objects) indicate that CSPM is the most effective but least efficient method, STCM is second most effective and efficient, SICM is least effective because of its long chromosome. The results for medium-to-large scale problems (50–200 objects) indicate that CSPM is still the most effective method. Furthermore, we have applied CSPM to solve an exemplified p-Median problem. The good results demonstrate that CSPM is usefully applicable.     

Robustness in multi-objective optimization using evolutionary algorithms

This work discusses robustness assessment during multi-objective optimization with a Multi-Objective Evolutionary Algorithm (MOEA) using a combination of two types of robustness measures. Expectation quantifies simultaneously fitness and robustness, while variance assesses the deviation of the original fitness in the neighborhood of the solution. Possible equations for each type are assessed via application to several benchmark problems and the selection of the most adequate is carried out. Diverse combinations of expectation and variance measures are then linked to a specific MOEA proposed by the authors, their selection being done on the basis of the results produced for various multi-objective benchmark problems. Finally, the combination preferred plus the same MOEA are used successfully to obtain the fittest and most robust Pareto optimal frontiers for a few more complex multi-criteria optimization problems.     

Error bounds of two smoothing approximations for semi-infinite minimax problems

In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.     

A simple direct cosine simplex algorithm

Linear programming (LP) is the core model of constrained optimization. The Simplex method (Simplex in short) has been proven in practice to perform very well in small- or medium-sized LP problems. A new algorithm called the direct cosine Simplex algorithm (DCA) is presented here to improve upon Simplex and to solve LP problems. The proposed DCA implements a specific cosine criterion to choose the entering variable instead of the traditional most negative rule used in Simplex. Three examples are given to illustrate the implementation of the proposed DCA to improve Simplex and to serve as the optimization tool. The utility of the proposed approach is evident from the extensive computational results on test problems adapted from NETLIB. DCA reduced the number of iterations of Simplex in most cases in our computational experiment. Preliminary results for medium-sized problems are encouraging.     

A globally most violated cutting plane method for complex minimax problems with application to digital filter design

In [4,6], the authors have presented a numerical method for the solution of complex minimax problems, which implicitly solves discretized versions of the equivalent semi-infinite programming problem on increasingly finer grids. While this method only requires the most violated constraint at the current iterate on a finite subset of the infinitely many constraints of the problem, we consider here a related and more direct approach (applicable to general convex semi-infinite programming problems) which makes use of the globally most violated constraint. Numerical examples with up to 500 unknowns, which partially originate from digital filter design problems, are discussed.     

On the number of iterations of local improvement algorithms

A general model of local improvement algorithms in combinatorial optimization accurately confirms performance characteristics often observed in individual cases. The model predicts exponentially bad worst case and low order polynomial average run times for single optimum problems including some linear complementarity problems and linear programming. For problems with multiple local optima, most notably those that are NP-complete, average speed is linearly bounded but accuracy is poor.