Fuzzy Weirstrass theorem and convex fuzzy mappings

The convexity and continuity of fuzzy mappings are defined through a linear ordering and a metric on the set of fuzzy numbers. The local-global minimum property of real-valued convex functions is extended to convex fuzzy mappings. It is proved that a strict local minimizer of a quasiconvex fuzzy mapping is also a strict global minimizer. Characterizations for convex fuzzy mappings and quasiconvex fuzzy mappings are given. In addition, the Weirstrass theorem is extended from real-valued functions to fuzzy mappings.     

直觉模糊映射的凸分析

介绍了直觉模糊数和直觉模糊映射的凸性、凹性、拟凸性、拟凹性\上半连续性、下半连续性和正齐次性的定义.通过引入一种新的偏序关系来研究凸直觉模糊映射性质,对凸直觉模糊映射和凹直觉模糊映射分别建立了两个刻画定理.讨论了直觉模糊映射的凸性与拟凸性的关系,证明了一个凸直觉模糊映射必是一个拟凸直觉模糊映射和一个凹直觉模糊映射必是一个拟凹直觉模糊映射.考虑了直觉模糊映射的凸性与半连续性的关系,获得了直觉模糊映射的凸性与上半连续性(下半连续性)等价的条件.在直觉模糊算子方面,给出了正齐次直觉模糊映射是凸直觉模糊映射的充要条件及直觉模糊算子是凸直觉模糊映射的判别定理.拓展了经典集合上的凸函数和凸模糊映射的相关理论,使之成为直觉模糊理论的有益补充.     

Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

Convexity and semicontinuity of fuzzy mappings

In this paper, a criterion for the convex fuzzy mapping is obtained under the condition of upper and lower semicontinuity, respectively. An upper (lower) semicontinuous fuzzy mapping is proved, which convexity is equivalent to weak convexity or B-vexity satisfying a special condition.     

On properties of discretized convex curves

The connection between a continuous convex curve and its discrete image is investigated using an appropriate definition of discrete convexity. It is shown that the discrete image of continuous convex curve may not be convex; however, its deviation from convexity (in the discrete definition) is bounded by a small constant. The actual pixel patterns that are obtained by discretizing convex curves are studied. Certain constraints on the context of discrete images of continuous convex curves were discovered     

A new convexity measure for polygons

Abstract-Convexity estimators are commonly used in the analysis of shape. In this paper, we define and evaluate a new convexity measure for planar regions bounded by polygons. The new convexity measure can be understood as a "boundary-based" measure and in accordance with this it is more sensitive to measured boundary defects than the so called "area-based" convexity measures. When compared with the convexity measure defined as the ratio between the Euclidean perimeter of the convex hull of the measured shape and the Euclidean perimeter of the measured shape then the new convexity measure also shows some advantages-particularly for shapes with holes. The new convexity measure has the following desirable properties: 1) the estimated convexity is always a number from (0, 1], 2) the estimated convexity is 1 if and only if the measured shape is convex, 3) there are shapes whose estimated convexity is arbitrarily close to 0, 4) the new convexity measure is invariant under similarity transformations, and 5) there is a simple and fast procedure for computing the new convexity measure.     

Approximate global optimization with convexity estimation of response surface using Kriging method

This paper describes a new method for approximate global optimization using convexity estimation of a multi-peaked or partially non-convex response surface. This method is based on convexity estimation of a response surface and a cell-based clustering technique. Convexity of an approximated function is estimated from the Hessian matrix and its eigenvalue. For this purpose, a Kriging-based convexity estimation method is also introduced in this paper. At first, a formulation for the convexity estimation with the Kriging method is provided. The convexity of an objective function at each location is estimated without using a finite difference based technique. With using this convexity estimation and a cell-based clustering technique, convex clusters are constructed in a solution space. The global optimization is performed with iterative local optimization to the convex clusters. From the numerical results, validity and effectiveness of the proposed method are confirmed.     

On the cellular convexity of complexes

In this paper we discuss cellular convexity of complexes. A new definition of cellular convexity is given in terms of a geometric property. Then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image. Hence, the definition of cellular convexity by Sklansky [7] is equivalent to the new definition for the case of regular complexes. The definition of Minsky and Papert [4] is shown to be equivalent to our definition. Therefore, aU definitions are virtually equivalent. It is shown that a regular complex is cellularly convex if and only if its minimum-perimeter polygon does not meet the boundary of the complex. A 0(n) time algorithm is presented to determine the cellular convexity of a complex when it resides in n × m cells and is represented by the run length code.     

Linking Special Semiconvex Functions and Three- Term-Recurrent Structures

Different notions of convexity that arise in several problems within the calculus of variations are briefly revised. We derive some particular stability properties for convex functions being homogeneous of degree one. Stability is investigated in the sense of convexity being conserved when modifying the considered functions in a special analytic or algebraic way. Criteria on functions for being rank-one convex are stated and elaborated in the case of orthogonal polynomials. This leads to the observation that highly non-trivial convexity properties are to be expected in the case of special orthogonal polynomials. The obtained particular results might give further insight into connections between variational calculus and special functions.     

Bezier曲面的凸性分析及保凸拼接

通过分析Bezier曲面局部凸的充要条件与控制顶点网格形状的关系,在几何上将网格的形状与曲面的凸性之间建立联系,导出Bezier曲面局部凸的几个充要条件。将凸性分析的结论应用到Bezier曲面拼接中,得到Bezier曲面保凸拼接的条件,并且证明带有公共边界线的两个局部凸Bezier曲面的Cn(n≥1)拼接后,仍然是凸的且凸性相同。最后,给出几个三次Bezier曲面保凸拼接实例。     

Convex digital solids

A definition of convexity of digital solids is introduced. Then it is proved that a digital solid is convex if and only if it has the chordal triangle property. Other geometric properties which characterize convex digital regions are shown to be only necessary, but not sufficient, conditions for a digital solid to be convex. An efficient algorithm that determines whether or not a digital solid is convex is presented.     

Digital and cellular convexity

We introduce a new definition of cellular convexity on square mosaics. We also define digital convexity for 4-connected sets of points on a square lattice. Using these definitions we show that a cellular complex is cellularly convex if and only if the digital region determined by the complex is digitally convex. We also show that a digital region is digitally convex if and only if the minimum-perimeter polygon (MPP) enclosing the digital region contains only the digital region. This result is related to a property of the MPP of the half-cell expansion of the complex determined by the digital region.     

Convex B-spline surfaces

This paper gives a definition for the convexity of B-spline surfaces and points out the conditions, on which the convexity depends.A back shift smoothing method is introduced. This method is built on the basis of the convexity conditions. Application of this smoothing method gives a strictly convex curve.     

Hoops: 3D Curves as Conservative Occluders for Cell-Visibility

Most visibility culling algorithms require convexity of occluders. Occluder synthesis algorithms attempt to construct large convex occluders inside bulky non-convex sets. Occluder fusion algorithms generate convex occluders that are contained in the umbra cast by a group of objects given an area light. In this paper we prove that convexity requirements can be shifted from the occluders to their umbra with no loss of efficiency, and use this property to show how some special non-planar, non-convex closed polylines that we call "hoops" can be used to compute occlusion efficiently for objects that have no large interior convex sets and were thus rejected by previous approaches.     

A convex version of multivariate adaptive regression splines

Multivariate adaptive regression splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data and simultaneously conduct variable selection. In optimization, MARS has been used successfully to estimate the unknown functions in stochastic dynamic programming (SDP), stochastic programming, and a Markov decision process, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in surrogate optimization. Many optimization methods depend on convexity, but a non-convex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper a convex MARS modeling algorithm is described. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained, such that pairs of basis functions are guaranteed to jointly form convex functions and (2) the form of interaction terms is altered to eliminate the inherent non-convexity. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex. Convex-MARS is applied to inventory forecasting SDP problems with four and nine dimensions and to an air quality ground-level ozone problem.     

Digital convexity, straightness, and convex polygons

New schemes for digitizing regions and arcs are introduced. It is then shown that under these schemes, Sklansky's definition of digital convexity is equivalent to other definitions. Digital convex polygons of n vertices are defined and characterized in terms of geometric properties of digital line segments. Also, a linear time algorithm is presented that, given a digital convex region, determines the smallest integer n such that the region is a digital convex n-gon.     

B-样条曲面的保凸拼接

利用参数曲面局部凸的条件,导出了(准)均匀B-样条曲面局部凸的充分条件。证明了带有凸的平行四边形控制子网格的(准)均匀B-样条曲面片的凸的。给出了(准)均匀B-样条曲面保凸拼接的控制顶点算法,并给出了几个三次B-样条曲面保凸拼接实例。     

一类非线性细分格式的保凸与分形性质

在分析Dyn等人的经典4点线性插值离散细分格式的基础上,提出了一类函数型非线性离散细分格式,它具有保凸性质,即在满足一定条件时, 这种格式保证了对于凸数据,其每一步细分多边形都是凸的,从而极限曲线也是凸的.数值例子说明,在不光滑情况下,这种格式会产生具有分形性质的曲线.     

隐式磨光曲面的保凸性研究

对隐式代数磨光曲面的保凸性进行了研究。证明了如果两个隐式代数曲面是凸的,则它们光滑拼接后,得到的磨光曲面在一定条件下仍然是凸的。以凸三面角为例,提出了利用代数样条磨光的方法并证明了利用这种磨光方法得到的磨光曲面具有保凸性。