Structural eigenvalue analysis under the constraint of a fuzzy convex set model

In small-sample problems, determining and controlling the errors of ordinary rigid convex set models are difficult. Therefore, a new uncertainty model called the fuzzy convex set（FCS） model is built and investigated in detail. An approach was developed to analyze the fuzzy properties of the structural eigenvalues with FCS constraints. Through this method, the approximate possibility distribution of the structural eigenvalue can be obtained. Furthermore, based on the symmetric F-programming theory, the conditional maximum and minimum values for the structural eigenvalue are presented, which can serve as nonfuzzy quantitative indicators for fuzzy problems. A practical application is provided to demonstrate the practicability and effectiveness of the proposed methods.

Vertical derivative conversion of irregular-range potential fi eld data based on improved projection onto convex sets method

Gravity and magnetic exploration areas are usually irregular,and there is some data defi ciency.Missing data must be interpolated before the vertical derivative conversion in the wavenumber domain.Meanwhile,for improved processing precision,the data need to be edge-padded to the length required by the fast Fourier transform algorithm.For conventional vertical derivative conversion of potential fi eld data(PFD),only vertical derivative conversion is considered,or interpolation,border padding,and vertical derivative conversion are executed independently.In this paper,these three steps are considered uniformly,and a vertical derivative conversion method for irregular-range PFD based on an improved projection onto convex sets method is proposed.The cutoff wavenumber of the filter used in the proposed method is determined by fractal model fi tting of the radial average power spectrum(RAPS)of the potential fi eld.Theoretical gravity models and real aeromagnetic data show the following:(1)The fitting of the RAPS with a fractal model can separate useful signals and noise reasonably.(2)The proposed iterative method has a clear physical sense,and its interpolation,border padding error,and running time are much smaller than those of the conventional kriging and minimum curvature methods.

Convex Set Theory for Reliability Assessment of Steel Beam with Bounded Uncertainty

Probabilistic reliability model established by insufficient data is inaccessible. The convex model was applied to model the uncertainties of variables. A new non-probabilistic reliability model was proposed based on the robustness of system to uncertainty. The non-probabilistic reliability model,the infinite norm model,and the probabilistic model were used to assess the reliability of a steel beam,respectively. The results show that the resistance is allowed to couple with the action effect in the non-probabilistic reliability model. Additionally,the non-probabilistic reliability model becomes the same accurate as probabilistic model with the increase of the bounded uncertain information. The model is decided by the available data and information.

Optimal Separation of Twin Convex Sets under Externalities

This paper studies the outcomes of independent and interdependent pair-wise contests between economic agents subject to an optimal external decision problem for each pair. The external decision maker like the government or regulator is faced with the problem of how to devise rules and regulations regarding contests. In this paper, a decision problem is faced under negative and positive externalities. A pair of entities is represented by disjoint convex sets in a small area in a neighborhood. I assume that each entity imposes an equal externality on the other (and the other only) and thus they can be considered to be twins. Among the group of twins in any neighborhood, there is a set of twin pairs such that, for each pair in the set, each twin can impose a strictly negative externality on the other (and the other only), and this is a potential welfare loss which concerns the decision maker. A separating hyper-plane can block the negative externalities between any pair of twins given convexity. However, this can be costly if positive externality from the neighborhood is also blocked by the separation technology. Thus, this paper compares the pair-wise utility from separation to that of non-separation. A simple representation of the decision problem is developed with respect to a single and isolated neighborhood. A complete characterization of the decision problem is obtained with a large number of pair-wise intersecting neighborhoods.

Martingales and Super-Martingales Relative to a Convex Set of Equivalent Measures

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.

On Drop and Weak Drop Properties for Bounded Closed Convex Sets*

In this paper we prove three equivalent conditions of bounded closed convexset K in Banach space to have the drop and weak drop properties. We also give fourequivalent conditions of Banach space and its dual space to have the drop and weak dropproperties.

Non-deterministic fatigue life analysis using convex set models

The non-probabilistic approach to fatigue life analysis was studied using the convex models-interval, ellipsoidal and multiconvex models. The lower and upper bounds of the fatigue life were obtained by using the second-order Taylor series and Lagrange multiplier method. The solving process for derivatives of the implicit life function was presented. Moreover, a median ellipsoidal model was proposed which can take into account the sample blind zone and almost impossibility of concurrence of some small probability events. The Monte Carlo method for multi-convex model was presented, an important alternative when the analytical method does not work. A project example was given. The feasibility and rationality of the presented approach were verified. It is also revealed that the proposed method is conservative compared to the traditional probabilistic method, but it is a useful complement when it is difficult to obtain the accurate probability densities of parameters.

On Proximinality of Convex Sets in Superspaces

Abstract In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal （proximinal, resp.） in every Banach space isometrically containing it if and only if C is locally （weakly, resp.） compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

Lifts of Non-Compact Convex Sets and Cone Factorizations

This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional,whether it is a translated cone and whether it contains lines.The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone,but also its recession cone is the image of the linear slice of the closed convex cone.The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

Calculus of Generalized Quasi-differentiable Functions I: Some Results on the Space of Pairs of Convex-Set Collections

A Riesz space K1 whose elements are pairs of convex-set collections is presented for the study on the calculus of generalized quasi-differentiable functions. The space K1 is constructed by introducing a well-defined equivalence relation among pairs of collections of convex sets. Some important properties on the norm and operations in K1 are given.

Homotopy Method for Non-convex Programming in Unbonded Set

In the past few years, much and much attention has been paid to the method for solving non-convex programming. Many convergence results are obtained for bounded sets. In this paper, we get global convergence results for non-convex programming in unbounded sets under suitable conditions.

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation:The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method. © 2014 Chinese Association of Automation.

MODELLING RANDOM CONVEX SETS

A model is developed for convex-set valued data,which are the Minkowski sum of aconvex parameter set and a convex noise set.This model is generalized to include location and scaleparameters.A further generalization results in a model for convex sets which is analogous to aone-way analysis of variance models.In each case,estimators of the parameters are given which,under certain conditions,are shown to be consistent.Data generated from the model and theestimators are studied.An outlier detection method is examined.

Optimality Conditions of the Set-valued Optimization Problem with Generalized Cone Convex Set-valued Maps Characterized by Contingent Epiderivative

In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.

Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

一种结构不确定分析的改进多维平行六面体模型

多维平行六面体模型是一种能够同时考虑相关变量和独立变量的非概率凸集模型,更适用于工程结构中常见的多源不确定性问题。为更加有效合理地度量结构不确定性,提出一种改进多维平行六面体模型。通过定义区间变量的相关角和边缘区间,给出模型不确定域的显式表达式,进而给出依据实验样本点构建多维平行六面体模型的方法。3个算例分析结果表明,改进多维平行六面体模型能够较好地反映区间变量之间的相关性,是一种比传统多维平行六面体模型更为紧凑合理的模型。

基于虚拟样本伪标签生成的高光谱图像分类

半监督高光谱图像分类的精度一般随着标记像素数的增加而提高.然而,标签样本的获得费时费力,且依赖于专家知识.针对这个问题,提出了一种通过少量标签样本生成具有伪标签的虚拟样本新方法.基于数学中的凸集理论,所提出的方法利用少量的训练样本可以生成任意多的带有伪标签的虚拟样本,有效地扩大了训练样本集,明显改善了半监督分类器的分类结果.为了验证所提方法的有效性,在Indian Pines和Pavia University两个常用的实际高光谱数据集上进行了广泛测试.实验结果表明,利用所提出的方法在分类具有少量标签样本的高光谱图像时,3个评价分类结果的指标值均有明显提升.

二层多目标随机规划逼近弱有效解集的上半收敛性

为研究二层多目标随机规划逼近问题的弱有效解与精确的弱有效解之间的逼近收敛性,针对上、下层都带有约束条件的一类多目标二层随机规划的逼近问题,构建了二层多目标随机规划逼近问题的弱有效解集的上半收敛性理论框架。即在假设下层反馈到上层的最优解集函数为凸函数的前提下,借助严格凸函数的性质,利用多目标随机规划的弱有效解可以表示成相应的单目标随机规划最优解集交集的结构特征,建立了二层多目标随机规划逼近弱有效解集的上半收敛性,提供了逼近方法求解二层多目标随机规划弱有效解集可以近似替代精确的弱有效解集的理论依据。

图-凸集值映射的性质

借助回收映射和余切导数得到了图-凸集值映射的两个性质.

脱萼剂对“浙梨6号”果实品质与着果率的影响

针对“浙梨6号”因部分萼片宿存或残存、果顶尖凸等而影响果实品相问题,选择PBO、氟硅唑、丙环唑3种脱萼剂,以清水为对照进行试验。结果表明:与对照相比,3种脱萼剂处理果实的脱萼率提高23.27~26.01百分点,果顶尖凸率降低24.55~26.44百分点,扁圆果占比、单果质量、可溶性固形物含量分别增加33.19~37.66百分点、37.9~57.1 g和0.68~0.94百分点,但果实偏斜度、全锈果率分别偏高0.54~0.74和18.36~19.23百分点,端正果率、着果率分别下降15.89~21.07和5.43~6.62百分点,差异达显著或极显著水平。上述3种脱萼剂分别选择4000、333.3、444.4 mg·L^(-1)和花序分离期进行喷施,在保证脱萼防凸效果的同时,可有效降低对着果率及果形端正度等的不利影响。