Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves

Some powerful algorithms for multi-extremal non-convex-constrained optimization problems are based on reducing these multi-dimensional problems to those of one dimension by applying Peano-type space-filling curves mapping a unit interval on the real axis onto a multi-dimensional hypercube. Here is presented and substantiated a new scheme simultaneously employing several joint Peano-type scannings which conducts the property of nearness of points in many dimensions to a property of nearness of pre-images of these points in one dimension significantly better than in the case of a scheme with a single space-filling curve. Sufficient conditions of global convergence for the new scheme are investigated.This paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Irregularly shaped space-filling truncated octahedra

For any parent tetrahedron ABCD, centroids of selected sub-tetrahedra form the vertices of an irregularly shaped space-filling truncated octahedron. To reflect these properties, such a figure will be called an ISTO. Each edge of the ISTO is parallel to and one-eighth the length of one of the edges of tetrahedron ABCD and the volume of the ISTO is 3/16-th the volume of the tetrahedron. The ISTO is symmetric about the centroid of tetrahedron ABCD and each face is symmetric about a centre and has an opposite face that is parallel and congruent. The area of the faces of the ISTO is not proportional to that of the generating tetrahedron.     

Approximate Solutions of Continuous Dispersion Problems

The problem of positioning p points so as to maximize the minimum distance between them has been studied in both location theory (as the continuous p-dispersion problem) and the design of computer experiments (as the maximin distance design problem). This problem can be formulated as a nonlinear program, either exactly or approximately. We consider formulations of both types and demonstrate that, as p increases, it becomes dramatically more expensive to compute solutions of the exact formulation than to compute solutions of the approximate formulation.     

Sigma–delta quantization errors and the traveling salesman problem

In transmission, storaging and coding of digital signals we frequently perform A/D conversion using quantization. In this paper we study the maximal and mean square errors as a result of quantization. We focus on the sigma–delta modulation quantization scheme in the finite frame expansion setting. We show that this problem is related to the classical Traveling Salesman Problem (TSP) in the Euclidean space. It is known [Benedetto et al., Sigma–delta () quantization and finite frames, IEEE Trans. Inform. Theory 52, 1990–2005 (2006)] that the error bounds from the sigma–delta scheme depends on the ordering of the frame elements. By examining a priori bounds for the Euclidean TSP we show that error bounds in the sigma–delta scheme is superior to those from the pulse code modulation (PCM) scheme in general. We also give a recursive algorithm for finding an ordering of the frame elements that will lead to good maximal error and mean square error. Supported in part by the National Science Foundation grant DMS-0139261.     

混料切片试验设计

混料试验设计在众多领域中都有广泛的应用,有时试验者不仅仅需要考虑各混料成分所占比例对响应变量的影响,同时还关心其它被称为过程变量的因素.在实际中,对于这类问题通常使用的设计方案是混料设计和因子设计的组合设计.这种组合设计在过程变量的不同水平组合下,使用的是相同的设计阵,因此空间填充性较差.基于混料球体堆积设计,文章提出了一类新的混料设计,称之为混料切片设计,它的整体设计和所有子设计(过程变量的每一水平组合对应的混料设计)都具有很好的空间填充性,从而比组合设计有更好的模型稳健性.基于同余子群的陪集分解方法,针对过程变量水平组合数的不同情况提出了相应的简单快速的构造算法,文章最后的数值例子解释了算法的可行性和设计的有效性.     

Multifractality of space-filling bearings and Apollonian packings

We study the structure of space-filling bearings (SFB) and Apollonian packings (AP) by determining the scaling behavior of the density distribution of points where the circles touch each other. Application of the sand box method to the average of theqth power of the number of touching points reveals the multifractal aspects of the structure of SFB and AP.