A trajectory-following method for unconstrained optimization

A trajectory-following method with interesting properties is considered for solving unconstrained nonlinear programming problems. The trajectory is defined by a special system of ordinary differential equations. This system uses only the gradient of the objective function. Numerical examples are given.The work of the second author was supported by the DFG Schwerpunkt Anwendungs-bezogene Optimierung and Steuerung.     

On consistency of bounding operations in deterministic global optimization

This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence _q in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.     

Generalized Brownian motion and elasticity

We introduce a family of stochastic processes which are a natural extension of Brownian motion to a tensor form. This allows us to solve a Dirichlet problem of linear elasticity obeying Lamé's equation, [1–(d– 1)]²V(x)+ [·V(x)]=0.     

Cyclizations via Palladium-Catalyzed Allylic Alkylations [New Synthetic Methods (79)]

The history of ring systems in organic chemistry parallels their synthetic accessibility. Transition-metal-catalyzed cyclizations offer a new opportunity to create carbo- and heterocyclic compounds with great facility. Among these methods, allylic alkylations catalyzed by palladium have proven unusually productive because of the extraordinary chemo-, regio-, and diastereoselectivity and the continuing possibility for the development of enantioselectivity. The rules for ring closure differ from those for non-transition-metal-catalyzed reactions. A major benefit is the ability to generate medium (eight-, nine-, ten-, and eleven-membered) and large rings in preference to normal (five-, six- and seven-membered) rings. With the appropriate substrate, efficient macrocyclizations are possible under conditions of normal concentrations. A second major benefit derives from the complementary stereochemistry of the metalcatalyzed substitution (net retention of configuration) compared to non-metal-catalyzed reactions (inversion of configuration). Further, the requirement for the substrate to conform to the transition-metal template may impose a stereochemical preference in the intermediate that ultimately translates into the thermodynamically less stable organic product regardless of the stereochemistry of the starting material. While more work has focused on carbocyclic synthesis, the possibilities for heterocyclic synthesis are just beginning to be tapped. In addition to forming heterocycles by C? C bond formation, use of a heteroatom as a nucleophile has already proven effective for oxygen and nitrogen, with other nucleophiles awaiting investigation. New dimensions for cyclization via allylic alkylation arise by generating the requisite π-allylpalladium intermediates by methods other than palladium(0)-initiated allylic ionizations. In addition, metals other than palladium will clearly expand the possibilities, but as yet remain untapped.     

Structure analytical methods for quantitative reference applications

The analytical methods mass spectrometry, UV/Vis, IR, Raman, Fluorometry, XRD, Mössbauer, and NMR used to elucidate chemical structure are evaluated regarding their capabilities to be used as primary analytical techniques in quantitative measurements, considering the criteria in the CCQM definition of primary methods. This includes a review of the respective measurement equations, the evaluation of the measurement uncertainty, and a discussion of evidence for the “highest metrological level”, as obtained from intercomparisons in contest with other methods. It is shown that only few methods fulfill the CCQM criteria. Quantitative NMR spectroscopy is one of them and may be considered as a potential primary method as recommended by CCQM because of being free of empirical factors in the uncertainty budget.     

Recent Applications of α-Metalated Isocyanides in Organic Synthesis

Being both nucleophilic and electrophilic, α-metalated isocyanides can add to polar double bonds, forming heterocycles. They are also synthons for α-metalated primary amines. This article describes recent or improved procedures for their use in organic synthesis: (1) In heterocyclic syntheses to give 2-oxazolines, 2-imidazolines, 2-thiazolines, oxazoles and oligooxazoles, thiazoles, triazoles, imidazolinones, pyrroles, 5,6-dihydro-1,3-oxazines and -thiazines, and (via cycloaddition with nitrones) 2-imidazolidinones. (2) In the field of formylaminomethylenation, for example transformation of estrone methyl ether and a keto sugar into the corresponding α-formylaminoacrylic esters, and the conversion of aldehydes and ketones by 3- and 4-pyridyl-methyl isocyanides into N-(1-pyridyl-1-alkenyl)formamides and their hydrolysis to 3- and 4-acylpyridines. (3) In connection with the use of α-metalated isocyanides as synthons for α-metalated primary amines, the author demonstrates how they may be used for preparation of 1,2- and 1,3-amino alcohols, 1,2-diamines, 2,3-diaminoalkanoic acids and for synthesis of higher amino acids starting from simple amino acids.     

Comparison of path-complete Lyapunov functions via template-dependent lifts

This paper investigates, in the context of discrete-time switched systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the ordering relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switched systems. These tools are finally illustrated with the aim of numerical examples.     

Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

Potential reduction method for harmonically convex programming

In this paper, we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and them constraint functions are allk-harmonically convex in the feasible set, then the number of iterations needed to find an -optimal solution is bounded by a polynomial inm, k, and log(1/). The method requires either the optimal objective value of the problem or an upper bound of the harmonic constantk as a working parameter. Moreover, we discuss the relation between the harmonic convexity condition used in this paper and some other convexity and smoothness conditions used in the literature.The authors like to thank Dr. Hans Nieuwenhuis for carefully reading this paper and the anonymous referees for the worthy suggestions.