Showing papers on "Maxima and minima published in 1981"

Generalized descent for global optimization

Abstract: This paper introduces a new method for the global unconstrained minimization of a differentiable objective function. The method is based on search trajectories, which are defined by a differential equation and exhibit certain similarities to the trajectories of steepest descent. The trajectories depend explicitly on the value of the objective function and aim at attaining a given target level, while rejecting all larger local minima. Convergence to the gloal minimum can be proven for a certain class of functions and appropriate setting of two parameters.

Invex functions and constrained local minima

Abstract: If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

From Local Maxima to Connected Skeletons

Abstract: In picture processing it is often convenient to deal with a stick-like version (skeleton) of binary digital images. Although skeleton connectedness is not necessary for storage and retrieval purposes, this property is desirable when a structural description of images is of interest. In this paper a parallel procedure is described which, applied to a connected image, originates a connected skeleton made by the union of simple digital arcs. The procedure involves a step by step propagation of the background over the image. At every step, contour elements either belonging to the significant convex regions of the current image or being local maxima of the original image are selected as skeleton elements. Since the final set so obtained is not ensured to be connected, the configurations in correspondence of which disconnections appear are investigated and the procedures to avoid this shortcoming are given. The presence of the whole set of local maxima among the skeleton elements ensures the possibility of recovering the original image by means of a reverse distance transform. The details of the program implementing the proposed algorithm on a parallel processor are finally included.

Convergence of a Random Optimization Method for Constrained Optimization Problems

Abstract: Matyas' random optimization method (Ref. 1) is applied to the constrained nonlinear minimization problem, and its convergence properties are studied. It is shown that the global minimum can be found with probability one, even if the performance function is multimodal (has several local minima) and even if its differentiability is not ensured.

Topography and minimization of the standard deviation in one-dimensional magnetotelluric modelling

Abstract: Summary In one-dimensional magnetotelluric modelling the standard deviation e is often used as an indicator of the degree of fit between the field measurements and the calculated model response The topography of e in the space of the model parameters has been studied and found to be rather simpler than expected The absolute minimum seems to be quite isolated from other minima In general no such other local minima were found To find the minimum it was not necessary, therefore, to look for a computing routine capable of jumping out of localized minima But the search routine had to be capable of moving along a valley with an exceedingly level floor, as the minimum is often at large distances from the initial model In this respect it was important to work with logarithmic coordinates Since the absolute minimum emin can be found, it often becomes possible to split emin into three separate components: (1) the scatter of the original field data, (2) the departure of these data from one-dimensionality and (3) a component that will occur if one attempts to model the data with a structure comprising too few layers Knowing this last contribution it becomes easy to decide which is the smallest number of layers necessary to model a given data Sci

Continuous Relaxation and Local Maxima Selection: Conditions for Equivalence

Abstract: Relaxation labeling processes are a class of iterative algorithms for using contextual information to reduce local ambiguities. This paper introduces a new perspective toward relaxation-that of considering it as a process for reordering labels attached to nodes in a graph. This new perspective is used to establish the formal equivalence between relaxation and another widely used algorithm, local maxima selection. The equivalence specifies conditions under which a family of cooperative relaxation algorithms, which generalize the well-known ones, decompose into purely local ones. Since these conditions are also sufficient for guaranteeing the convergence of relaxation processes, they serve as stopping criteria. We feel that equivalences such as these are necessary for the proper application of relaxation and maxima selection in complex speech and vision understanding systems.

Error surfaces of recursive adaptive filters

Abstract: For an adaptive filter with N adjustable coefficients or weights, the "error surface" is a plot, in N+ I dimensions, of the mean-squared error versus the N coefficient values. If the adaptive filter is nonrecursive, the error surface is a quadratic function of the coefficients. With recursive adaptive filters, the error surface is not quadratic and may even have local minima. In this correspondence we discuss the nature of the recursive error surface and give examples of conditions under which local minima may exist. We conclude with a discussion of the effects of the nonquadratic error surface on gradient-search algorithms for recursive adaptive filters.

On Local and Global Minima in Structural Optimization

Abstract: : This paper deals with convexity properties in structural optimization, and with the closely related question of local versus global optima. The problem we investigate is that of minimizing the structural weight subject to constraints on displacements, stresses and natural frequencies. It is assumed that the structure is described by a finite element model, and that the transverse sizes of the elements, e.g. thicknesses of membrane plates, are the design variables. This implies that both the objective function, i.e. the weight, and the structural stiffness matrix depend linearly on the design variables. The constraint functions, however, become nonlinear and they may in the general case give rise to a nonconvex feasible region in the design space. Then there is a risk that a local, but not global, minimum is attained when any of the various existing methods for numerically solving the problem is applied. This fact is illustrated by examples of nonconvex problems. However, there are some special cases where the feasible region always becomes convex, so that, due to the linearity of the objective function, each local optimum is in fact also a global one.

Frequency dependence of the acoustic radiation force function (Yp) for spherical targets for a wide range of materials

Abstract: In a previous communication the authors reported a procedure for calculating Yp for spherical radiometer targets over wide ranges of ka, identifying the importance of fine steps in ka for accurate representations. The present paper summarizes the results of the calculation Yp, over the ka range 0–20, for over 50 different materials which are assumed to be immersed in water. A primary grouping according to material shear velocity and density is proposed. Material density has a substantial effect upon the form of the maxima and minima of the Yp curves, a low density producing higher maxima, deeper minima, and increasing the width of these features. The present calculations do not give a shear indication of the effect of the Poisson’s ratio of the material, nor explain the extremely high maxima at low wave velocities.

Critical Paths and Passes: Application to Quantum Chemistry

Abstract: The purpose of this paper is to describe a process for finding a critical point of a function f and a path connecting this critical point to two given points (which are usually local minima of the function). This problem issues from quantum chemistry: the function f represents the energy of a molecule and, given two local minima of f (which correspond to stable molecular states), one looks for a “reaction path” connecting them. Such a path is required to make the variation of f along it the least possible; the highest point on the path being a saddle point (or pass) whose knowledge is of utmost importance reaction rates theory [1].

Nuclear Dissipation with Residual Interactions Studied by Means of the Mori Formalism

Abstract: We apply correlation function techniques to the calculation of nuclear friction within the framework of a linear response theory. We make use of the fluctuation dissipation theorem to relate the response function to the correlation function which is evaluated by exploiting projection operator techniques. We go beyond the one-body dissipation approximation in the sense that we have taken into account the decay of particle-hole excitations into more complicated configurations. A rather simple formula for the frequency and temperature dependence of the friction coefficient is derived which we have applied to the high energy fission of238U. The friction coefficients for deformations around the first and second minima of the fission barrier have been calculated using this approach.

Value on a class of non-differentiable market games

Abstract: We prove the existence of a (unique) Aumann-Shapley value on the space on non-atomic gamesQ n generated byn-handed glove games. (These are the minima ofn non-atomic mutually singular probability measures.) It is also shown that this value can be extended to a value on the smallest space containingQ n andpNA.

Lens design and local minima

Abstract: The widespread belief that local minima exist in the least squares lens-design error function is not confirmed by the Los Alamos Scientific Laboratory (LASL) optimization program. LASL finds the optimum-minimum region, which is characterized by small parameter gradients of similar size, small performance improvement per iteration, and many designs that give similar performance. Local minima and unique prescriptions have not been found in many-parameter problems. The reason for these absences is that image errors caused by a change in one parameter can be compensated by changes in the remaining parameters. False local minima have been found, and four cases are discussed.

Functions of several variables

Abstract: 1. Differentiable Functions.- 1.1 Introduction.- 1.2 Linear part of a function.- 1.3 Vector viewpoint.- 1.4 Directional derivative.- 1.5 Tangent plane to a surface.- 1.6 Vector functions.- 1.7 Functions of functions.- 2. Chain Rule and Inverse Function Theorem.- 2.1 Norms.- 2.2 Frechet derivatives.- 2.3 Chain rule.- 2.4 Inverse function theorem.- 2.5 Implicit functions.- 2.6 Functional dependence.- 2.7 Higher derivatives.- 3. Maxima and Minima.- 3.1 Extrema and stationary points.- 3.2 Constrained minima and Lagrange multipliers.- 3.3 Discriminating constrained stationary points.- 3.4 Inequality constraints.- 3.5 Discriminating maxima and minima with inequality constraints 62 Further reading.- 4. Integrating Functions of Several Variables.- 4.1 Basic ideas of integration.- 4.2 Double integrals.- 4.3 Length, area and volume.- 4.4 Integrals over curves and surfaces.- 4.5 Differential forms.- 4.6 Stokes's theorem.- Further reading.- Appendices.- A. Background required in linear algebra and elementary calculus.- B. Compact sets, continuous functions and partitions of unity.- C. Answers to selected exercises.- Index (including table of some special symbols).

The index-k-stabilizing differential equation

Abstract: This paper deals with the search for multiple local minima of a differentiable real-valued functionf off variables. Motivated by topological considerations much as Morse Theory — it makes sense to determine critical points¯x forn of index 1 (i.e. exactly one eigenvalue of the HessianD2f(¯x) is negative). For eachk ∈ 0,...,n, the gradient vectorfieldDf ofn is altered — by a partial reflection — into a new vectorfieldFK. Restricted to the critical point set off, only the critical points of indexk are attractors forFK.

A table of maxima and minima of the Bessel functions Jn(Z) for n = 0 to n = 30

Abstract: Maxima and minima are tabulated for the Bessel functions Jn(x) for values of n from 0 to 30 [i.e. zeros of J'n(x)]. The first six points are recorded for each function. The table considerably extends the range of earlier tables available in the literature. It should have applications in the interpretation of diffraction patterns from helical or wave-form structures or features, and has been used in connection with some electron microscope images.

The Stochastic Quasi-Gradient Method Applied to a Facility Location Problem

Abstract: This paper explores the computational aspects of using the stochastic quasi-gradient method (SQG) to solve some facility location problems. The problems addressed belong to a general class of resource allocation problems with random demand. An algorithm is first developed for the simplest formulation, where a convex objective function is minimized, and results are shown for the location of high schools in Turin, Italy. Fixed charges are then introduced in the objective function, giving rise to a non-convex problem possessing many local minima, and some numerical results for the same case study are reported.

The simplex method for minimisation of a general function

Abstract: The simplex method is a reasonably fast and efficient procedure for minimising a general function of any number of variables. It operates with function values at a number of points and so can more easily avoid local minima than other methods. It works by constructing a polyhedron and reflecting the point with the highest function value in the centroid of the others. If this results in a lower value, the exploration continues in the same direction; if not the highest point or its reflection is moved towards the centroid. The polyhedron thus moves gradually to lower values until it settles into a minimum. The simplex method contains procedures to deal with special cases and refinements to speed up the process. The report gives the full mathematical statement of the procedure, and the appendix describes a computer program to implement it. The method was developed as part of a research study of long term trends in transport: it is published as a contribution which may find a wider application in the field of mathematical techniques. (Author/TRRL)

Differential sensitivity theory applied to movement of maxima responses. [LMFBR]

Abstract: Differential sensitivity theory (DST) is a recently developed methodology to evaluate response derivatives dR/d..cap alpha.. by using adjoint functions which correspond to the differentiated (with respect to an arbitrary parameter ..cap alpha..) linear or nonlinear physical system of equations. However, for many problems, where responses of importance are local maxima such as peak temperature, power, or heat flux, changes in the phase space location of the peak itself are of interest. This summary will present the DST procedure for predicting phase space shifts of maxima responses as applied to the MELT-III fast reactor safety code. An FFTF protected transient involving a $.23/s ramp reactivity insertion with scram on high power was selected for investigation.

Maxima and minima

Abstract: Let U be an open set in ℝn; this means that to every point u ∈ U there corresponds a ball N(u) = {y∈ℝn: ‖y-u‖ 0, such that N(u) ⊂ U. This concept is illustrated in Fig. 3.1. Let f: U → ℝ be a (Frechet) differentiable function. (Note that the definition of f’(u) at u ∈ U requires that the domain of f contains some ball N(u). An open set U, from the definition, does not contain the points of its boundary.)

Stability Regions and Spectra of Discrete Third-Order Autoregressive Time Series

Abstract: This paper presents an analysis of third order autoregressive time series. The parameter regions, in the three dimensional parameter space, that produce the six separate types of power spectral density are analyzed. The study reveals that when a particular two dimensional cross section of the three dimensional parameter space is taken, the region of stability is always triangular. Within each triangular stability region in this two dimensional space, subregions which produce the six possible types of spectral shape are indicated. From these subregions it is possible to approximately choose the parameters necessary to model a process whose power spectral density contains at most two critical frequencies (maxima and minima).

Letters to the Editor Error Surfaces of Recursive Adaptive Filters

Abstract: A bsrracr- For an adaptive filter with N adjustable coefficients or weights, the “error surface” is a plot, in N+ 1 dimensions, of the mean-squared error versus the N coefficient values. If the adaptive filter is nonrecursive, the error surface is a quadratic function of the coefficients. With recursive adaptive filters, the error surface is not quadratic and may even have local minima. In this correspondence we discuss the nature of the recursive error surface and give examples of conditions under which local minima may exist. We conclude with a discussion of the effects of the nonquadratic error surface on gradient-search algorithms for recursive adaptive filters.