Numerical experience with a polyhedral-norm CDT trust-region algorithm

In this paper, we study a modification of the Celis-Dennis-Tapia trust-region subproblem, which is obtained by replacing thel ₂-norm with a polyhedral norm. The polyhedral norm Celis-Dennis-Tapia (CDT) subproblem can be solved using a standard quadratic programming code.We include computational results which compare the performance of the polyhedral-norm CDT trust-region algorithm with the performance of existing codes. The numerical results validate the effectiveness of the approach. These results show that there is not much loss of robustness or speed and suggest that the polyhedral-norm CDT algorithm may be a viable alternative. The topic merits further investigation.The first author was supported in part by the REDI foundation and State of Texas Award, Contract 1059 as Visiting Member of the Center for Research on Parallel Computation, Rice University, Houston, Texas, He thanks Rice University for the congenial scientific atmosphere provided. The second author was supported in part by the National Science Foundation, Cooperative Agreement CCR-88-09615, Air Force Office of Scientific Research Grant 89-0363, and Department of Energy Contract DEFG05-86-ER25017.     

Local Behavior of the Newton Method on Two Equivalent Systems from Linear Programming

Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce significantly different behavior when applied to equivalent systems, i.e., problems with the same solution but different mathematical formulations. In this paper, we investigate differences in the local behavior of Newton's method when applied to two different but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more effectively on the latter system.     

The bond force constants of graphene and benzene calculated by density functional theory

Stretching (k_r) and bending (k_θ) bond force constants appropriate to describe the bond stiffness of graphene and benzene are calculated using density functional theory. The effect of employing different exchange-correlation functionals for the calculation of k_r and k_θ is discussed using the generalised gradient approximation (GGA) and the local density approximation (LDA). For benzene, k_r = 7.93 mdyn Å^-1 and k_θ = 0.859 mdyn Å rad^-2 using LDA, while k_r = 7.67 mdyn Å^-1 and k_θ = 0.875 mdyn Å rad^-2 using GGA. For graphene, k_r = 7.40 mdyn Å^-1 and k_θ = 0.769 mdyn Å rad^-2 using LDA, while k_r = 6.88 mdyn Å^-1 and k_θ = 0.776 mdyn Å rad^-2 using GGA. This means the difference between the bond force constants for benzene and graphene can be as large as ～12%. The comparison between these two systems allows for elucidation of the effect of periodicity and substitution of carbon atoms by hydrogen in the stiffness of C–C bonds. This effect can be explained by a different redistribution of the charge density when the systems are subjected to strain. The parameters k_r and k_θ computed here can serve as an input to molecular mechanics or finite element codes of larger carbon molecules, which in the past had frequently assumed the same bond force constants for graphene, benzene or carbon nanotubes.     

The Molecular Weaponry Produced by the Bacterium Hafnia alvei in Foods

Hafnia alvei is receiving increasing attention from both a medical and veterinary point of view, but the diversity of molecules it produces has made the interest in this bacterium extend to the field of probiotics, the microbiota, and above all, to its presence and action on consumer foods. The production of Acyl Homoserine Lactones (AHLs), a type of quorum-sensing (QS) signaling molecule, is the most often-studied chemical signaling molecule in Gram-negative bacteria. H. alvei can use this communication mechanism to promote the expression of certain enzymatic activities in fermented foods, where this bacterium is frequently present. H. alvei also produces a series of molecules involved in the modification of the organoleptic properties of different products, especially cheeses, where it shares space with other microorganisms. Although some strains of this species are implicated in infections in humans, many produce antibacterial compounds, such as bacteriocins, that inhibit the growth of true pathogens, so the characterization of these molecules could be very interesting from the point of view of clinical medicine and the food industry. Lastly, in some cases, H. alvei is responsible for the production of biogenic amines or other compounds of special interest in food health. In this article, we will review the most interesting molecules that produce the H. alvei strains and will discuss some of their properties, both from the point of view of their biological activity on other microorganisms and the properties of different food matrices in which this bacterium usually thrives.     

Discrete time Galerkin approximations to the nonlinear filtering solution

This paper concerns discrete time Galerkin approximations to the solution of the filtering problem for diffusions. Two families of schemes approximating the unnormalized conditional density, respectively, in an “average” and in a “pathwise” sense, are presented. L² error estimates are derived and it is shown that the rate of convergence is linear in the time increment or linear in the modulus of continuity of the sample path.     

Bentonitic earth catalyzed rearrangement of aryl 1,1-dimethylpropargyl ethers. Synthesis of 2,2-dimethyl-2H-1-benzopyrans

Mexican Bentonitic earth (Tonsil) catalyzed the Claisen rearrangement of aryl 1,1-dimethylpropargyl ethers under mild conditions to provide 2,2-dimethyl-2H-1-benzopyrans. The synthesis of encecalin 2f and desmethoxyencecalin 2i , two biologically active products among other natural products ( 2b, 2e ) was performed by this procedure.     

Chapter 6 On the construction of strong complementarity slackness solutions for DEA linear programming problems using a primal-dual interior-point method

A novel approach for solving the DEA linear programming problems using a primaldual interior-point method is presented. The solution found by this method satisfies the Strong Complementarity Slackness Condition (SCSC) and maximizes the product of the positive components among all SCSC solutions. The first property is critical in the use of DEA and the second one contributes significantly to the reliability of the solution.This research was partially supported by NSF Cooperative Agreement No. CCR-88-09615, ARO Grant 9DAAL03-90-G-0093, DOE Grant DEFG05-86-ER25017, and AFOSR Grant 89-0363.Partially supported by Fulbright/LASPAU.     

Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.     

A simple protocol to help calculate saddle points. Transition-state structures for the Meyer—Schuster reaction in non-aqueous media: An ab initio MO study

A simple procedure to help calculate transition-state (TS) structures of complex chemical systems is proposed and tested. For the cases studied in this laboratory, substantial gains in computer CPU time have been obtained. As an example, the TS structures for the intramolecular mechanism corresponding to the 1,3-hydroxyl shift occurring in (CH₃)₂C(OH) CCH is calculated. STO-3G, 4-31G and 4-21G basis sets have been used to represent the wavefunction of the ground electronic state. Invariance of the present TS structure to changes in basis set has been found.