Optimization of conditions in static headspace GC

The precision and sensitivity of head-space GC can be improved by careful selection of the operating conditions: partitioning in the vapor phase can be promoted by altering the sample matrix; the ratio of the volumes of the gaseous and liquid phases, the sample temperature, the equilibration time, and the effect of sample mixing should also be considered. Most of the principles covered here can be applied to either manual headspace GC or to headspace GC utilizing an automated sampler.     

Mechanisms of Diazo Coupling Reactions. Part XXXII. The Diazoamino Rearrangement in 20% Acetonitrile/Aqueous Buffers

The rearrangements of 4′-methoxy-N-methyl- and N-methyl-4′-nitro-diatzoaminobenzene have been studied in 20% acetonitrile/aqueous buffers. The reactions are specifically acid catalyzed and involve pre-equilibrium formation of amine and diazonium salt followed by rate-limiting attack of the diazonium ion at a C-atom (C-coupling) to give the corresponding aminoazo compounds. There is no evidence to suggest that, under the present conditions, mechanisms other than the established Friswell-Green mechanism occur. The traditional two-stage synthesis of aminoazo compounds via isolated diazoamino compounds can therefore be replaced by a one stage process for amines which undergo initial attack at a N-atom by diazonium ion (N-coupling).     

Variational optimization via chattering

We develop a chattering approach to solve variational problems that lack traditional properties such as differentiable everywhere and convexity conditions. We prove that our chattering approximation approaches the true relaxed solution as the intervals get smaller. Our chattering approach suggests a nonlinear optimization problem that can be easily solved to recover the optimal trajectory. A numerical example demonstrates our approach.     

Stochastic Methods for Practical Global Optimization

Engineering design problems often involve global optimization of functions that are supplied as black box functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.     

Determination of residual solvent in pharmaceutical preparations by static headspace GC

Static headspace GC, a simple, clean technique which is easily automated, appears to be a good approach to the determination of solvent residues in pharmaceutical preparations. The feasibility of this approach has been studied with an automated system. Data is presented for the solvents designated as impurities in pharmaceutical preparations by the United States Pharmacopeia. It was found that the static headspace technique meets the United States Pharmacopeia criteria for sensitivity. The absolute area count precision was <5% relative standard deviation and correlation coefficients to a linear response were >0.999. It was concluded that the technique is viable for this application.     

Improving Hit-and-Run for global optimization

Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n^5/2) wheren is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum.Paper presented at the II. IIASA-Workshop on Global Optimization, December 9–14, 1990, Sopron (Hungary).     

A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems

There is a need for a methodology to fairly compare and present evaluation study results of stochastic global optimization algorithms. This need raises two important questions of (i) an appropriate set of benchmark test problems that the algorithms may be tested upon and (ii) a methodology to compactly and completely present the results. To address the first question, we compiled a collection of test problems, some are better known than others. Although the compilation is not exhaustive, it provides an easily accessible collection of standard test problems for continuous global optimization. Five different stochastic global optimization algorithms have been tested on these problems and a performance profile plot based on the improvement of objective function values is constructed to investigate the macroscopic behavior of the algorithms. The paper also investigates the microscopic behavior of the algorithms through quartile sequential plots, and contrasts the information gained from these two kinds of plots. The effect of the length of run is explored by using three maximum numbers of function evaluations and it is shown to significantly impact the behavior of the algorithms.     

An analytically derived cooling schedule for simulated annealing

We present an analytically derived cooling schedule for a simulated annealing algorithm applicable to both continuous and discrete global optimization problems. An adaptive search algorithm is used to model an idealized version of simulated annealing which is viewed as consisting of a series of Boltzmann distributed sample points. Our choice of cooling schedule ensures linearity in the expected number of sample points needed to become arbitrarily close to a global optimum.     

Pattern discrete and mixed Hit-and-Run for global optimization

We develop new Markov chain Monte Carlo samplers for neighborhood generation in global optimization algorithms based on Hit-and-Run. The success of Hit-and-Run as a sampler on continuous domains motivated Discrete Hit-and-Run with random biwalk for discrete domains. However, the potential for efficiencies in the implementation, which requires a randomization at each move to create the biwalk, lead us to a different approach that uses fixed patterns in generating the biwalks. We define Sphere and Box Biwalks that are pattern-based and easily implemented for discrete and mixed continuous/discrete domains. The pattern-based Hit-and-Run Markov chains preserve the convergence properties of Hit-and-Run to a target distribution. They also converge to continuous Hit-and-Run as the mesh of the discretized variables becomes finer, approaching a continuum. Moreover, we provide bounds on the finite time performance for the discrete cases of Sphere and Box Biwalks. We embed our samplers in an Improving Hit-and-Run global optimization algorithm and test their performance on a number of global optimization test problems.