Convex approximations in stochastic programming by semidefinite programming

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in L ₁, L _?? and L ₂ norm. Extensive numerical experiments to investigate the behavior of the proposed methods are also performed.     

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid \(\{t_n\}_{n=0}^N\) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., \(t_n = {\varPhi }(\tau _n)\), where \(\tau _n = n/N\) and the map \({\varPhi }\) is monotonically increasing with \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines \({\varPhi }\), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an \(N^*\) such that the method is zero stable for \(N>N^*\), provided that \({\varPhi }\in C^2[0,1]\). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy \(h_n/h_{n-1} = 1 + {\mathrm {O}}(N^{-1})\) as \(N\rightarrow \infty \). The results are exemplified for BDF-type methods.     

Classification of edge‐transitive rose window graphs

Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}$ and edge set $\{\{{{x}}_{{i}},{{x}}_{{{i+1}}}\} \mid {{i}}\in {\mathbb{Z}}_n \} \cup \{\{{{y}}_{{{i}}},{{y}}_{{{i+r}}}\}\mid {{i}} \in{\mathbb{Z}}_{{n}}\}\cup \{\{{{x}}_{{{i}}},{{y}}_{{{i}}}\} \mid {{i}}\in {\mathbb{Z}}_{{{n}}}\}\cup \{\{{{x}}_{{{i+a}}},{{y}}_{{{i}}}\} \mid{{i}} \in {\mathbb{Z}}_{{{n}}}\}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010     

Book Reviews

Structural Chemistry -     

Book Reviews

Structural Chemistry -     

Creating Advanced Bases For Large Scale Linear Programs Exploiting Embedded Network Structure

In this paper, we investigate how an embedded pure network structure arising in many linear programming (LP) problems can be exploited to create improved sparse simplex solution algorithms. The original coefficient matrix is partitioned into network and non-network parts. For this partitioning, a decomposition technique can be applied. The embedded network flow problem can be solved to optimality using a fast network flow algorithm. We investigate two alternative decompositions namely, Lagrangean and Benders. In the Lagrangean approach, the optimal solution of a network flow problem and in Benders the combined solution of the master and the subproblem are used to compute good (near optimal and near feasible) solutions for a given LP problem. In both cases, we terminate the decomposition algorithms after a preset number of passes and active variables identified by this procedure are then used to create an advanced basis for the original LP problem. We present comparisons with unit basis and a well established crash procedure. We find that the computational results of applying these techniques to a selection of Netlib models are promising enough to encourage further research in this area.     

A practical anti-degeneracy row selection technique in network linear programming

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.This research was supported in part by Hungarian Research Fund OTKA 2587, and by DAAD 314 108 060 0 while the author was at Universität Heidelberg, Germany, October, 1990.     

Synthesis of Vinca Alkaloids and Related Compounds. PART LVIII. A Novel Formal Synthesis of (−)-Criocerine from (+)-Vincamine

Abstract

18-lodo criocerine (3), a synthetic precursor to (?)-criocerine (1), was prepared from (+)-vincamine (4) via a new one-step procedure. Full ¹H and ¹³C NMR assignments for 3 are also given.     

Characterization and control of the aggregation behavior of cyclodextrins

Photon correlation spectroscopy has been employed for the purpose of characterizing the aggregation behavior of cyclodextrin molecules in aqueous solutions. This optical method is generally intended to study particle size distribution of colloidal particles, associates and macromolecules. Herein we report on some general methodological issues of photon correlation spectroscopy aiming to illustrate aggregated and non-aggregated state of parent cyclodextrins and cyclodextrin derivatives, such as (2-hydroxy)propyl-β-cyclodextrin and tetraamino rhodaminyl (2-hydroxypropyl)-β-cyclodextrin in different aqueous media. Based on particle size analysis data we have demonstrated that the tendency of cyclodextrin molecules to form aggregates may be controlled by temperature and by various additives, e.g. urea, citric acid and polyvinylpyrrolidone. In the case of (2-hydroxypropyl)-β-cyclodextrin the effect of degree of substitution was also studied.