Showing papers on "Maxima and minima published in 1979"

An improved algorithm for decentralized extrema-finding in circular configurations of processes

Abstract: This note presents an improvement to LeLann's algorithm for finding the largest (or smallest) of a set of uniquely numbered processes arranged in a circle, in which no central controller exists and the number of processes is not known a priori. This decentralized algorithm uses a technique of selective message extinction in order to achieve an average number of message passes of order (n log n) rather than O(n2).

Global Optimization Using Interval Analysis: The One-Dimensional Case

Abstract: We show how interval analysis can be used to compute the minimum value of a twice continuously differentiable function of one variable over a closed interval. When both the first and second deriva- tives of the function have a finite number of isolated zeros, our method never fails to find the global minimum. Consider a function f(x) in C 2. We shall describe a method for computing the minimum value of f(x) on a closed interval (a, b). We shall see that, if f'(x) and f"(x) have only a finite number of isolated zeros, our method always converges. In a subsequent paper, we shall show how the method can be extended to the case in which x is a vector of more than one variable. Moreover, it will be extended to the constrained case, and a modified method will remove the differentiability condition. The present paper serves to introduce the necessary ideas. In practice, we can only compute minima in a bounded interval. Hence, it is no (additional) restriction to confine our attention to a closed interval. The term global minimum used herein refers to the fact that we find the smallest value of f(x) throughout (a, b). We shall not mistake a local minimum for the global one. Indeed, our method will usually not find local minima, unless forced to do so. Its efficiency would then be degraded if it did. In our method, we iteratively delete subintervals of (a, b) until the remaining set is sufficiently small. These subintervals consist of points at which either f(x) is proved to exceed the minimum in value or else the derivative is proved to be nonzero.

An interval arithmetic method for global optimization

Abstract: An interval arithmetic method is described for finding the global maxima or minima of multivariable functions. The original domain of variables is divided successively, and the lower and the upper bounds of the interval expression of the function are estimated on each subregion. By discarding subregions where the global solution can not exist, one can always find the solution with rigorous error bounds. The convergence can be made fast by Newton's method after subregions are grouped. Further, constrained optimization can be treated using a special transformation or the Lagrange-multiplier technique.

Maxima and Minima of Stationary Sequences

Abstract: We show that the asymptotic behavior of the normalized maxima of a stationary sequence satisfying a weak distributional mixing and bivariate condition is completely determined by the marginal distribution of the process. Sufficient conditions are given in order for the maxima and minima to be asymptotically independent. An example of a 1-dependent sequence where the maxima and minima are not asymptotically independent is also provided.

Variational Formulation of Budyko-Sellers Climate Models

Abstract: A class of simple climate models including those of the Budyko-Sellers type are formulated from a variational principle. A functional is constructed for the zonally averaged mean annual temperature field such that extrema of the functional occur when the climate satisfies the usual energy-balance equation. Local minima of the functional correspond to stable solutions while saddle points correspond to unstable solutions. The technique can be used to construct approximate solutions from trial functions and to carry out finite-amplitude stability analyses. A spectral example is given in explicit detail.

Multiple-Doppler Radar Network Design

Abstract: Observing programs utilizing Doppler radar must have them deployed in optimum locations to best satisfy experimental objectives and maximize economies. One wishes to determine the coordinate triples (xi, yi, zi), where i equals the number of radars, which maximize the value of the data to be collected. The optimum location is governed by a value or objective function. Here, possible networks of two to nine radars are given for two different error specifications. The objective functions with both error distributions maximize the quantity (AREAL COVERAGE/ERROR). The procedure is to search the finite number of local maxima for the global maximum in the value of the objective function. This is done by employing a searching algorithm at each of a number of starting vectors which are close enough to the local maxima to converge to the desired local maxima. In all cases, the network obtained by considering all radars simultaneously is superior to that obtained by combining optimum smaller sub-networks. ...

Singularity‐free static lattice energy minimization

Abstract: Comparative analyses are made of static lattice energy minimization procedures based on Newton or quasi‐Newton algorithms with rigid body orientations being parametrized in terms of Euler angles or quaternion parameters. It is shown that potential functions in terms of Euler angles can have artificial saddle points and artificial minima in the plane where the azimuthal angle ϑ (the angle between the body and laboratory z axes) is zero. Quaternions are found to form the only numerically acceptable parametrization of orientations in less than five variables and lead to a singularity‐free energy function which can be minimized efficiently by any routine for unconstrained optimization.

Some remarks on the symmetric rank-one update

Abstract: We consider the symmetric rank-one, quasi-Newton formula. The hereditary properties of this formula do not require quasi-Newton directions of search. Therefore, this formula is easy to use in constrained optimization algorithms; no explicit projections of either the Hessian approximations or the parameter changes are required. Moreover, the entire Hessian approximation is available at each iteration for determining the direction of search, which need not be a quasi-Newton direction. Theoretical difficulties, however, exist. Even for a positive-definite, quadratic function with no constraints, it is possible that the symmetric rank-one update may not be defined at some iteration. In this paper, we first demonstrate that such failures of definition correspond to either losses of independence in the directions of search being generated or to near-singularity of the Hessian approximation being generated. We then describe a procedure that guarantees that these updates are well-defined for any nonsingular quadratic function. This procedure has been incorporated into an algorithm for minimizing a function subject to box constraints. Box constraints arise naturally in the minimization of a function with many minima or a function that is defined only in some subregion of the space.

The Post Processing Functions of a Database Computer.

Abstract: : DBC is a specialized back-end computer which is capable of managing database of 10 to the 10th power bytes in size and supporting known data models such as relational, network, hierarchical and attribute-based This report deals with the post processing functions of DBC A description of some known methods for performing natural and implicit joins is first given It then goes on to show how both natural and implicit joins are performed by the post processor (PP) of DBC utilizing the parallelism of PP This report show that the algorithm for performing joins is of O(N) time, where N is the number of records to be joined Algorithms necessary for performing set functions such as maxima, minima, average, sum and count are given The time complexities of these algorithms are also calculated Finally, it is shown how to implement the set inclusion operator Given a set of values for a particular attribute and a number of retrieved records, this operator can select those records whose values for the attribute are the values in the set

A new mini-max, constrained optimization method for solving worst case problems

Abstract: A new worst case design procedure is described. This method employs Powell's new constrained optimization procedure and is at least superlinearly convergent. A novel function splitting scheme is described to avoid singularity problems inherent in some previously reported methods.

New techniques for approximate realisation

Abstract: The problem of approximate realisation is described and various methods are discussed. A new method is then given which directly identifies the system poles from pulse response data by finding the local minima of a real function of a complex variable. These estimates of the poles are then refined using a new iterative nonlinear least-squares algorithm. Finally, these methods are applied to a ‘seismic wavelet’, and are shown to give good qualitative and quantitative information on the system being modelled.

Measurement Of Complex Elastic Modulus By Holography

Abstract: This article describes a technique for measuring complex elastic modulus E* by holographic interferometry. In this technique used are the characteristics of the amplitude distribution in viscoelastic vibration which can be obtained from time-averaged holography. The dynamic elastic modulus can be evaluated from the dimensions, the density, the frequency and the distances between the positions of the maxima (or minima), and tans can be measured from the calculated relations between tans and the ratios of the minima to their neighbor maxima. By this technique, E* can be measured at the arbitrary frequency without changing the dimensions of specimen. The frequency range and the measuring range of tanδ are higher than those in conventional techniques.© (1979) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.