Active set algorithms for isotonic regression; A unifying framework

In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.In this paper we consider the isotonic regression problem with respect to a complete order where eachw _i is strictly positive and eachy _i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n ²). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.This work was supported by the National Science and Engineering Research Council of Canada under Research Grant A8189 and an Ontario Graduate Scholarship.     

Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

Duality in parameter space and approximation of measures for mixing repellers

For one-dimensional expanding mapsT with an invariant measure we consider, in a parameter space, the envelope _n of the real lines associated to any couple of points of the orbit, connected byn iterations ofT. If the map hass inverses and is piecewise linear, then the sets _n are just the union ofs ⁿ points and converge to the invariant Cantor set ofT. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ₁ to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and hass inverses, the sets _n are homeomorphic to the Cantor set; they converge to the Cantor set ofT and their measures converge to the measure of the Cantor set whenn. The correspondence between the sets _n allows one to define converging approximation schemes for the map an its measure: one replaces each of thes ⁿ disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ₁. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.     

Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

导电性聚呋喃的氧化合成及应用

总结了芳香族聚呋喃的氧化聚合方法,系统论述了聚合过程中影响聚呋喃性质的各种因素,推测了聚呋喃的链结构,介绍了聚呋喃的各种改进方法和效果。指出聚呋喃及其共聚物是一种电导率在10-11到10-1S.cm-1范围内有机半导体材料,其电聚合沉积膜具有奇特的电活性、优良的热加工性和特殊的表面形貌,可望在金属离子和有机物质的吸附与富集、传感探测等领域发挥作用。     

泡沫铅对VRLA电池负极活性物质结构及性能影响

0引言随着36V/42V汽车电源系统的提出,新一轮汽车用电池的竞争不断加剧。从目前情况看,铅酸电池由于具有成本低廉,使用可靠,原材料来源丰富,铅回收率可高达98%等优点,因此成为电动车电源最实际的选择之一[1]。但作为电动车用电池,需要克服其比能量低、充电接受能力差和负极硫酸盐化等缺点。为此各国科学家开展了大量的研究工作。最近报道用铸造多孔体作为敞口铅酸电池的集流体,它的比表面积为14cm2·cm-3,正极活性物质利用率上升到50%,远高于传统的铸造板栅[2]。在铅酸电池集流体研究领域里另一个重大进步,就是以R V C(R etic-ulated V i…     

光学活性化合物的工业合成

本文介绍了光学活性化合物研究现状及发展趋势。并从技术和经济角度论述了工业合成光学活性化合物的可能性     

New ferrocene-containing structures: Poly(silyl ester)s

Two new polymeric structures containing ferrocene units along the chains, namely poly(silyl ester)s, have been synthesized and characterized: a geminal poly(silyl ester) (g-PSE) and one having a disiloxane spacer between the silyl ester groups (s-PSE). The condensation polymerization of AA/BB monomer systems in solution was used in both cases as preparation method involving a silicon-containing diol, (diphenylsilane diol or 1,3-bis(hydroxy)-tetramethyldisiloxane), and 1,1′-di(chlorocarbonyl)ferrocene. The polymers were investigated by differential pulse voltammetry in order to evaluate the redox behavior. Due to the presence of the silyl ester groups in the chain, these polymers are hydrolytically degradable.