Showing papers on "Maxima and minima published in 1969"

Optimization of Microwave Networks by Razor Search

Abstract: A new optimization method called razor search is presented. The method, which is based on pattern search, was specifically developed for the automatic optimization by computer of networks for which the objective is to minimize the maximum deviation of some response from a desired ideal response specification. Minimax response objectives, which can lead to equal-ripple optima, will in general give rise to discontinuous partial derivatives of the objective function with respect to the network parameters. Otherwise efficient optimization methods may slow down or even fail to reach an optimum in such circumstances, particularly when the response hypersurface has a narrow curved valley along which the path of discontinuous derivatives lies. Another direct search method called ripple search is also presented. This method was developed to locate the extrema of multimodal functions of one variable in an efficient manner, and is used to determine the maximum deviation of the response from the desired response. Sufficiently detailed flow diagrams are available so that the methods can be readily programmed. The razor search strategy (with ripple search) has been successfully applied to the optimization of inhomogeneous waveguide transformers. It is illustrated in this paper by examples of cascaded commensurate and noncoummensurate transmission lines acting as impedance transformers for which the optima are known.

Multivariate Maxima and Minima with Matrix Derivatives

Abstract: The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems.

Lattice Approximations to the Minima of Functions of Several Variables

Abstract: A computer-oriented method is developed for determining relative minima of functions of several variables. No derivatives (or approximations) are required and the process always converges to a relative minimum no matter which initial point is used. Numerical examples using test functions suggested in the literature are included to illustrate the effectiveness of the algorithms. Modifications can easily be incorporated which permit the inclusion of constraints or integer-valued variables.

Determination of crustal thickness from spectral behavior of SH waves

Abstract: The amplitude spectrum obtained from Haskell9s matrix formulation for body waves travelling through a horizontally layered crustal model shows a sequence of minima and maxima. It is known that multiple reflections within the crustal layers produce constructive and destructive interferences, which are shown as maxima and minima in the amplitude spectrum. Analysis of the minima in the amplitude spectra, which correspond to zero phase in the phase spectra, enables us to determine the thickness of the crust, provided the ratio of wave velocity in the crust to velocity under the Moho is known.

Minimum bandwidth of M signals with specified code or correlation matrix

Abstract: The minimum bandwidth of M real equi-energy signals is derived for the cases of either a specified code or a specified correlation matrix. The definition of bandwidth is in terms of the ratio of the energy passed by a linear time-invariant filter to the input energy, and incorporates a great variety of definitions. In one instance, the signals are time limited; in a second, a weighted integral over the infinite time interval is restrained. The extremum of the energy ratio depends only on the nonzero characteristic numbers of the correlation matrix of rank r and on the first r characteristic numbers of a kernel related to the filter used in the bandwidth definition. The optimum signal waveforms also depend on the characteristic vectors of the correlation matrix and are given by linear sums of the first r characteristic functions of the kernel. Bounds on the extrema are given, approximations in narrow-band situations are developed, and several useful examples are presented and solved.

A Computationally Compact Sufficient Condition for Constrained Minima

Abstract: This paper derives a second-order sufficient condition in differential form for a local minimum of the nonconvex nonlinear programming problem under the requirement of twice continuous differentiability. The condition is complementary to the differential form of the Kuhn-Tucker necessary conditions and involves the positive definiteness of a matrix of constrained second derivatives of rank and order equal to the number of variables having vanishing first constrained derivatives. By using linear information, the condition avoids consideration of the entire second-order portion of the Taylor expansion of the Lagrangian. An example illustrating the test is included.

Low-Cycle Random Fatigue of Steel (1st report)

Abstract: An apparatus controlling load maxima and minima according to the random load sequence simulated with the aid of an electronic digital computer was made and used to a preliminary investigation of low cycle random fatigue of steel.Fatigue lives showed considerable scatter comparing with the constant load test results, which was seemingly due to the randomness of load sequences.The effect of applied load lower than the load to which the constant load fatigue life was over 107 cycles was remarkable.Palmgren-Miner hypothesis overestimated the life of each specimen and the longer the average life is, the more discrepancy between the predicted and the actual life.