Fast algorithm for the cutting angle method of global optimization

The cutting angle method for global optimization was proposed in 1999 by Andramonov et al. (Appl. Math. Lett. 12 (1999) 95). Computer implementation of the resulting algorithm indicates that running time could be improved with appropriate modifications to the underlying mathematical description. In this article, we describe the initial algorithm and introduce a new one which we prove is significantly faster at each stage. Results of numerical experiments performed on a Pentium III 750 Mhz processor are presented.

The generation of new high-angle boundaries in aluminium during hot torsion

The crystallographic rotation field for deformation in torsion is such that it is possible for orientations close to stable orientations to rotate away from the stable orientation. A Taylor type model was used to demonstrate that this phenomenon has the potential to transform randomly generated low-angle boundaries into high-angle boundaries. After imposing an equivalent strain of 1.2, up to 40% of the simulated boundaries displayed a disorientation in excess of 15°. These high-angle boundaries were characterised by a disorientation axis close to parallel with the sample radial direction. A series of hot torsion tests was carried out on 1050 aluminium to seek evidence for boundaries formed by this mechanism. A number of deformation-induced high-angle boundaries were identified. Many of these boundaries showed disorientation axes and rotation senses similar to those seen in the simulations. Between 10% and 25% of all the high-angle boundary present in samples twisted to equivalent strains between 2 and 7 could be attributed to the present mechanism.

Predicting molecular structures: an application of the cutting angle method

The ability to predict molecular geometries has important applications in chemistry. Specific examples include the areas of protein space structure elucidation, the investigation of host–guest interactions, the understanding of properties of superconductors and of zeolites. This prediction of molecular geometries often depends on finding the global minimum or maximum of a function such as the potential energy. In this paper, we consider several well-known molecular conformation problems to which we apply a new method of deterministic global optimization called the cutting angle method. We demonstrate that this method is competitive with other global optimization techniques for these molecular conformation problems.

Geometry and combinatorics of the cutting angle method

Lower approximation of Lipschitz functions plays an important role in deterministic global optimization. This article examines in detail the lower piecewise linear approximation which arises in the cutting angle method. All its local minima can be explicitly enumerated, and a special data structure was designed to process them very efficiently, improving previous results by several orders of magnitude. Further, some geometrical properties of the lower approximation have been studied, and regions on which this function is linear have been identified explicitly. Connection to a special distance function and Voronoi diagrams was established. An application of these results is a black-box multivariate random number generator, based on acceptance-rejection approach.

Cutting angle method - a tool for constrained global optimization

Cutting angle method (CAM) is a deterministic global optimization technique applicable to Lipschitz functions f: Rn → R. The method builds a sequence of piecewise linear lower approximations to the objective function f. The sequence of solutions to these relaxed problems converges to the global minimum of f. This article adapts CAM to the case of linear constraints on the feasible domain. We show how the relaxed problems are modified, and how the numerical efficiency of solving these problems can be preserved. A number of numerical experiments confirms the improved numerical efficiency.

Miscibility, crystallization κinetics and real-time small-Angle x-ray scattering investigation of the semicrystalline morphology in thermosetting polymer blends of epoxy resin and poly(ethylene oxide)

Thermosetting polymer blends of poly(ethylene oxide) (PEO) and bisphenol-A-type epoxy resin (ER) were prepared using 4,4′-methylenebis(3-chloro-2,6-diethylaniline) (MCDEA) as curing agent. The miscibility and crystallization behavior of MCDEA-cured ER/PEO blends were investigated by differential scanning calorimetry (DSC). The existence of a single composition-dependent glass transition temperature (Tg) indicates that PEO is completely miscible with MCDEA-cured ER in the melt and in the amorphous state over the entire composition range. Fourier-transform infrared (FTIR) investigations indicated hydrogen-bonding interaction between the hydroxyl groups of MCDEA-cured ER and the ether oxygens of PEO in the blends, which is an important driving force for the miscibility of the blends. The average strength of the hydrogen bond in the cured ER/PEO blends is higher than in the pure MCDEA-cured ER. Crystallization kinetics of PEO from the melt is strongly influenced by the blend composition and the crystallization temperature. At high conversion, the time dependence of the relative degree of crystallinity deviated from the Avrami equation. The addition of a non-crystallizable ER component into PEO causes a depression of both the overall crystallization rate and the melting temperature. The surface free energy of folding σe displays a minimum with variation of composition. The spherulitic morphology of PEO in the ER/PEO blends exhibits typical characteristics of miscible crystalline/amorphous blends, and the PEO spherulites in the blends are always completely volume-filling. Real-time small-angle X-ray scattering (SAXS) experiments reveal that the long period L increases drastically with increasing ER content at the same temperatures. The amorphous cured ER component segregates interlamellarly during the crystallization process of PEO because of the low chain mobility of the cured ER. A model describing the semicrystalline morphology of MCDEA-cured ER/PEO blends is proposed based on the SAXS results. The semicrystalline morphology is a stack of crystalline lamellae; the amorphous fraction of PEO, the branched ER chains and imperfect ER network are located between PEO lamellae.

Efficient serial and parallel implementations of the cutting angle global optimisation technique

We examine efficient computer implementation of one method of deterministic global optimisation, the cutting angle method. In this method the objective function is approximated from values below the function with a piecewise linear auxiliary function. The global minimum of the objective function is approximated from the sequence of minima of this auxiliary function. Computing the minima of the auxiliary function is a combinatorial problem, and we show that it can be effectively parallelised. We discuss the improvements made to the serial implementation of the cutting angle method, and ways of distributing computations across multiple processors on parallel and cluster computers.

A new method for locating the global optimum : application of the cutting angle method to molecular structure prediction

Many problems in chemistry depend on the ability to identify the global minimum or maximum of a function. Examples include applications in chemometrics, optimization of reaction or operating conditions, and non-linear least-squares analysis. This paper presents the results of the application of a new method of deterministic global optimization, called the cutting angle method (CAM), as applied to the prediction of molecular geometries. CAM is shown to be competitive with other global optimization techniques for several benchmark molecular conformation problem. CAM is a general method that can also be applied to other computational problems involving global minima, global maxima or finding the roots of nonlinear equations.

Active temperature compensation for an accelerometer based angle measuring device

An angle measuring device using a high performance and very compact accelerometer provides a new and exciting method for producing highly compact and accurate angle measuring devices. Accelerometers are micro-machined and are able to measure acceleration to a very high accuracy. By using gravity as a reference these compact devices can also be used for measuring angles of rotation. The inherent problem with these devices is that their response characteristic changes with temperature which is detrimental to measurement accuracy. This paper describes an effective method to overcome this problem using a temperature sensor and intelligent software to compensate for this drift characteristic. In order to demonstrate the effectiveness of this work, experiments have been developed and conducted with the results and analysis provided at the end
of this paper for discussion.

A discussion on passive location discovery in emitter networks using angle-only measurements

In this paper we discuss the ghost node problem found when triangulation of 2 or more nodes is required. We present and discuss a simple algorithm, termed ABLE (Angle Based Location Estimation), that will position randomly placed emitters in a wireless sensor network using a mobile antenna array. The individual nodes in the network are relieved of the localization task by the mobile antenna system and require no modifications to account for location determination. Furthermore, no beacon nodes (i.e. nodes that know their own position) are required. We provide analysis that indicates a reasonably small number of measurements are required to guarantee the successful
localization of the emitting nodes and demonstrate our results through simulation.

Fuzzy terminal attitude guidance controller with angle only measurements

This paper introduces the concept of terminal attitude guidance as an alternative to precision guidance and uses fuzzy control ideas in designing a control strategy for a pursuer in countering a manoeuvreing target. The fuzzy controller uses only angle measurements in the control strategy and produces satisfactory results in comparison to the LQR or H∞ type guidance controllers, although they were addressed in a precision guidance context. Both 2D and 3D cases have been considered.

Passive angle measurement based localization consistency via geometric constraints

In this paper, we examine the geometric relations between various measured parameters and their corresponding errors in angle-measurement based emitter localization scenarios. We derive a geometric constraint formulating the relationship among the measurement errors in such a scenario. Using this constraint, we formulate the localization task as a constrained optimization problem that can be performed on the measurements in order to provide the optimal values such that the solution is consistent with the underlying geometry.

The generation of new high angle boundaries during the hot torsional deformation of aluminium

The crystallographic rotation field for deformation in torsion is such that it is possible for orientations close to stable orientations to rotate away from the stable orientation. A Taylor type model was used to demonstrate that this phenomenon has the potential to transform randomly generated low-angle boundaries into high-angle boundaries. After imposing an equivalent strain of 1.2, up to 40% of the simulated boundaries displayed a disorientation in excess of 15°. These high-angle boundaries were characterised by a disorientation axis close to parallel with the sample radial direction. A series of hot torsion tests was carried out on 1050 aluminium to seek evidence for boundaries formed by this mechanism. A number of deformation-induced high-angle boundaries were identified. Many of these boundaries showed disorientation axes and rotation senses similar to those seen in the simulations. Between 10% and 25% of all the high-angle boundary present in samples twisted to equivalent strains between 2 and 7 could be attributed to the present mechanism.

Review of applications of the Cutting Angle methods

The theory of abstract convexity provides us with the necessary tools for building accurate one-sided approximations of functions. Cutting angle methods have recently emerged as a tool for global optimization of families of abstract convex functions. Their applicability have been subsequently extended to other problems, such as scattered data interpolation. This paper reviews three different applications of cutting angle methods, namely global optimization, generation of nonuniform random variates and multivatiate interpolation.