Showing papers on "Maxima and minima published in 1986"
TL;DR: It is shown that the Gaussian probability density function is the only kernel in a broad class for which first-order maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased.
Abstract: Scale-space filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernal containing a scale or bandwidth parameter. It is shown that the Gaussian probability density function is the only kernel in a broad class for which first-order maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased. The consequences of this result are explored when the signal?or its image by a linear differential operator?is analyzed in terms of zero-crossing contours of the transform in scale-space.
852 citations
TL;DR: In this article, a scalar description of gravitational lensing based on Fermat's principle is described, where the lensing mass is assumed to be confined to a single plane between the source and the observer and a time delay is associated with each position in the sky of a potential image.
Abstract: A scalar description of gravitational lensing based on Fermat's principle is described. The lensing mass is assumed to be confined to a single plane between the source and the observer, and a time delay is associated with each position in the sky of a potential image. The extrema of this time surface then give the true positions of the images. A topological classification of image configurations is presented, and the results are generalized to cases of three and five-image lensing geometries. A computer-graphical approach to the study of lensing by model galaxies and clusters is described, and the design of a simple optical apparatus which could be used for fast modelling of image geometries is outlined. The connection between the Fermat approach and the classical theory of caustics and the more recent general theory of catastrophies is developed. The extension of the results to multiple scattering is considered. 42 references.
403 citations
TL;DR: In this article, a theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented and a bound on the departure of the probability distribution of the state at finite time from the optimum is obtained.
Abstract: Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.
341 citations
TL;DR: In this paper, the authors considered the mathematical programming problem of finding a subset of a finite-dimensional space where f is an extended real-valued function, and C is an arbitrary subset of the space, and gave necessary and sufficient optimality conditions.
Abstract: We consider the mathematical programming problem: find $\inf \{ f(x)\mid x \in C\} $ where f is an arbitrary extended-real-valued function, and C a subset of a finite dimensional space. We give necessary and sufficient optimality conditions for this problem, generalizing previous results of A. Auslendex (this Journal, 22 (1984), pp. 239–254).
142 citations
TL;DR: Klee and Laskowski's O ( n log 2 n ) algorithm for finding all minimal area triangles enclosing a given convex polygon of n vertices is improved to Θ ( n), which is shown to be optimal both forFinding all minima and for finding just one.
129 citations
TL;DR: In this paper, an expression for an approximate Hessian matrix, or matrix of the second derivatives of the SCF energy with respect to nuclear coordinates that can be used to search the potential energy surface of a molecule for minima or saddle points was developed.
95 citations
TL;DR: In this article, the simulated annealing method is used to guide a search towards the absolute minimum of a certain multidimensional function, which is illustrated on a problem recently discussed in this journal, namely the minimum energy configuration of equal charges confined to a sphere.
Abstract: Many problems in physics1, chemistry, biology2 and mathematics3 involve the determination of the absolute minimum of a certain multidimensional function. In most cases of practical interest this is a complicated matter, owing to the presence of local minima, and even more so because the number of local minima often increases exponentially with the problem size. Standard techniques apply local optimizers to many random initial configurations, but soon become intractable as the dimensionality increases. Here I show how the simulated annealing method can be used to guide a search towards the absolute minimum. The method is illustrated on a problem recently discussed in this journal, namely the minimum-energy configuration of equal charges confined to a sphere. This problem, although easy to visualize, can be used to simulate much of the complexity of the above problems by considering a large number of particles. In this limit several minima have been found that have previously been missed by other authors using classical techniques.
94 citations
TL;DR: A formal proof that all trajectories of the BSB algorithm in state vector space approach the set of system equilibrium points, under certain specific conditions, is presented.
81 citations
TL;DR: This finding is consistent with the proposal that orientation discrimination is determined by the relative activity of broadly-tuned, orientation-sensitive neural elements, and that only a small number of elements are effective in any small retinal region.
72 citations
TL;DR: An algorithm that uses the replica of a predetermined replica to determine the time shifts and amplitudes for each path in an n-dimensional matched filter algorithm that is more robust and efficient than others currently available.
Abstract: A transmitted signal can arrive at a receiver via several refracted Fermat paths. If the paths are independent in the Fresnel sense, then the received signal can be modeled as the sum of amplitude scaled and time shifted copies of a predetermined replica plus white noise. We present an algorithm that uses the replica to determine the time shifts and amplitudes for each path. It is referred to as an n-dimensional matched filter algorithm by analogy with the well-known matched filter algorithm. The cross correlation between the received signal and the replica oscillates near the center frequency of the transmitted signal. This causes the n-dimensional matched filter output to have many local maxima that are not globally optimal. The time shifts and amplitude scalings for the Fermat paths are determined by maximizing the output of the n-dimensional matched filter. The algorithm is more robust and efficient than others currently available. Simulated realizations of received signals were generated with multipath and noise characteristics similar to an ocean acoustic transmission case. These realizations were then separated into arrival times and corresponding amplitudes by the algorithm. The results of these tests and the general limitations of the algorithm are discussed.
59 citations
TL;DR: Variation diminishing splines provide an effective tool for modeling active elements in circuit simulation using quadratic tensor product splines and maintaining uniform sampling at the boundary by linear extension of the data.
Abstract: Variation diminishing splines provide an effective tool for modeling active elements in circuit simulation. Using quadratic tensor product splines and maintaining uniform sampling at the boundary by linear extension of the data yields an algorithm that is smooth (unlike simple table lookup), shape preserving (unlike simple interpolation), and efficient (30 microseconds to evaluate on a Cray-IA). The rate of convergence to function and derivative values and to the location of minima is $O(h^2 )$.
TL;DR: It turns out that strong stability in the sense of Kojima in the first phase is a natural assumption for the iterated local minima of the parametric problem and a generalized version of a positive definiteness criterion of Fujiwara-Han-Mangasarian is used.
Abstract: In dynamic programming and decomposition methods one often applies an iterated minimization procedure. The problem variables are partitioned into several blocks, say x and y. Treating y as a parameter, the first phase consists of minimization with respect to the variable x. In a second phase the minimization of the resulting optimal value function depending on y is considered. In this paper we treat this basic idea on a local level. It turns out that strong stability in the sense of Kojima in the first phase is a natural assumption. In order to show that the iterated local minima of the parametric problem lead to a local minimum for the whole problem, we use a generalized version of a positive definiteness criterion of Fujiwara-Han-Mangasarian.
01 Dec 1986
TL;DR: A novel tracking algorithm based on a global approach utilizing containment regions approximated by four-dimensional polytopes is presented in the context of bearings-only tracking, the various tradeoffs with conventional techniques are examined, and potential applications discussed.
Abstract: Standard target tracking techniques such as Kalman filters or maximum liklihood estimators approach nonlinearities from an essentially local point of view; that is, they determine a single solution even though the problem may admit more than one. This lack of uniqueness may be due to the absence of global observability as in Doppler tracking where several isolated solutions can occur, or a result of imperfect measurements producing multiple minima in a cost function. The latter is particulary significant since noisy data often produce situations in which local minima abound, trapping a maximum liklihood steepest descent search or causing an extended Kalman filter to diverge. This paper introduces a novel tracking algorithm based on a global approach utilizing containment regions approximated by four-dimensional polytopes. The algorithm is presented in the context of bearings-only tracking, the various tradeoffs with conventional techniques are examined, and potential applications discussed.
TL;DR: The efficiency of an important class of Newton methods for solving overdetermined sets of nonlinear equations is tested in finding the solution to the two-dimensional phase problem and it is seen that the nonlinearity and number of local minima of the cost function increases dramatically with the size of the object array.
Abstract: The efficiency of an important class of Newton methods (the Levenberg-Marquardt algorithm) for solving overdetermined sets of nonlinear equations is tested in finding the solution to the two-dimensional phase problem. It is seen that the nonlinearity and number of local minima of the cost function increases dramatically with the size of the object array, making these methods of little practical use for sizes greater than 6 2 6.
TL;DR: In this paper, sufficient and necessary conditions for the persistence and the adherence of minima in general convergence spaces are provided and specialized in the cases of topological and sequential convergences, as well as in the case of local compactifications.
Abstract: Sufficient and necessary conditions for the persistence and the adherence of minima in general convergence spaces are provided and specialized in the cases of topological and sequential convergences, as well as in the case of local compactifications. In terms of multifunctions, the inquired stability properties of minima amount to lower semicontinuity (persistence) and to graph-closedness (adherence).
TL;DR: In this paper, a constraint function approach is presented for finding design changes that remove natural frequencies from undesirable frequency bands for lightly damped structures, which requires the minimization of a function which becomes smaller when (1) natural frequencies clear out of undesirable bands, and (2) design changes become small.
Abstract: A constraint function approach is presented for finding design changes that remove natural frequencies from undesirable frequency bands for lightly damped structures. The technique requires the minimization of a function which becomes smaller when (1) natural frequencies clear out of undesirable bands, and (2) design changes become small. Useful forms of these functions are defined, and the number of possible minima is explored. Graphical intepretations of the constraint functions are given, and an example is included which shows the effects of the parameters which weight these two functions.
TL;DR: In this paper, three techniques for efficiently and accurately identifying critical points for space and time-dependent parametric constraints are described for a helicopter tail-boom structure subjected to transient loading.
01 Jan 1986
TL;DR: In this article, the authors give a necessary and sufficient condition on the cooling schedule for the simulated annealing algorithm to converge in probability to the set of globally minimum cost states.
Abstract: A Monte Carlo optimization technique called "simulated annealing" is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima. The level of randomization is determined by a control parameter T, called temperature, which tends to zero according to a deterministic "cooling schedule" . We give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states ? In the special case that the cooling schedule has parametric form T^ c/log(l+k), the condition for convergence is that c be greater than or equal to the depth, suitably defined, of the deepest local minimum which is not a global minimum state?
TL;DR: In this article, a clustering method is used to concentrate sampled points around the local minima so that they can be recognized by a cluster analysis technique, and more work is spent on the global exploration thereby increasing the probability that the global minimum will be found.
Journal Article•
TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.unipd.org/legal. php) of the agreement with the Rendiconti del Seminario Matematico della Università di Padova are discussed.
Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In contrast to estimation by ordinary least squares, estimation by total least squares has much less favourable properties as far as the existence and uniqueness of local minima is concerned as discussed by the authors, and the distribution of critical points of least-squares problems is remarkably well behaved.
Abstract: In contrast to estimation by ordinary least squares, estimation by total least squares has much less favourable properties as far as existence and uniqueness of local minima is concerned. Indeed, as elementary examples show, and contrary to intuition gleaned from the Gauss-Markov theorem for ordinary least squares, for certain data sets this problem can have nonisolated local minima and local maxima. Using Morse theory and the Lie theory of coadjoint orbits, we show that, despite this apparent degeneracy, the distribution of critical points of least-squares problems is remarkably well behaved. For arbitrary data, the least-squares function is perfect in the sense of the Morse-Bott theory. In particular, the set of local minima always forms a connected manifold while there exists a unique minimum value.
TL;DR: In this article, the notion de minimum dintegrales variationnelles en liaison avec les solutions d'equations elliptiques and de systemes sous forme divergence is considered.
Abstract: On considere la notion de minimum d'integrales variationnelles en liaison avec les solutions d'equations elliptiques et de systemes sous forme divergence
TL;DR: An NP-complete problem which is not a spin-glass is exhibited, which shows how sculpting of the energy surfaces of continuous analog systems to remove hills may fail to aid solution of embedded combinatorial optimization problems.
Abstract: An NP-complete problem which is not a spin-glass is exhibited The NP-complete problem 3-satisfiability is also embedded into a continuous analog system with no hills in the energy landscape obstructing solution of the problem There is, however, a large flat plateau This shows how sculpting of the energy surfaces of continuous analog systems to remove hills may fail to aid solution of embedded combinatorial optimization problems
TL;DR: An algorithm which computes an interval of length t in which a minimizer (or a maximizer) of a periodical bimodal function h is located using a minimal number of evaluations of the function h using a dynamic programming approach.
TL;DR: In this paper, the frequency estimates are obtained by searching for minima of the inverse input power spectrum, which is estimated at each input sample from the weights of an adaptive linear predictor which uses the LMS (least mean square) algorithm to update its weights.
Abstract: A computationally efficient scheme for estimating the digital instantaneous frequencies of narrowband inputs is introduced. The frequency estimates are obtained by searching for minima of the inverse input power spectrum. This spectrum is estimated at each input sample from the weights of an adaptive linear predictor which uses the LMS (least mean square) algorithm to update its weights. The related minima are sought via an iterative search algorithm, referred to as the iterative frequency estimator. This algorithm is computationally more efficient than available methods, and also provides a higher resolution. Simulation results are included; these include tracking of random message sequences in FM signals, and the formant frequency estimation of speech.
TL;DR: The mean-square error (MSE) of two iterates of Newton's method is practically equal to the MSE computed from the probability density of the local maxima of the cross correlator (via Rice's theorem).
Abstract: A fast algorithm for the local maximum likelihood determination of the difference of arrival time of a common signal at two spatially separated sensors with uncorrelated noise is given. The fast algorithm consists of locally maximizing the cross-correlation function from the two wide-band signals by using Newton's method for finding the root of an equation. The probability density function of one iteration of Newton's method is explicitly computed in terms of exponential and error functions. Using a theorem by Rice on the probability density of local maxima of Gaussian processes, the probability density of the local maxima of the cross correlator is obtained. These results are new. When both the signal and the noises have flat power spectral densities, the mean-square error (MSE) of two iterates of Newton's method is practically equal to the MSE computed from the probability density of the local maxima of the cross correlator (via Rice's theorem). The above holds if the starting point used in Newton's method is within a quarter signal resolution binwidth from the true delay and the signal-to-noise ratio (SNR) at the cross-correlator output is 15 dB or higher. The MSE of the local maximum estimator obtained from Rice's theorem is almost equal to the Cramer-Rao bound even for low SNR, i.e., 5 dB.
Journal Article•
TL;DR: In this paper, the concepts of robust set and a class of discontinuous functions are introduced and the global optimality contions and algorithm for finding global minima of this kind of functions are given.
Abstract: In this letter the concepts of robust set and a class of discontinuous functions are introduced. The global optimality contions and algorithm for finding global minima of this kind of functions are given.
TL;DR: In this article, the classical formula on ruin probabilities is applied to determine certain joint distributions for unconditional and conditional random walk concerning maxima, minima and their indices, and further application concerns the so-called narrowest tube problem.
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TL;DR: In this article, a variational principle that governs the equilibria of materials with non-convex Helmholtz free energy is presented. But the authors do not consider the problem of finding the minimum of the free energy.
Abstract: The modelling of twins in crystals with strain gradient theories provides interesting problems both in thermodynamics and in the calculus of variations. Here, Dunn and Serrin's thermomechanical theory of interstitial working is used to obtain a variational principle that governs the equilibria of materials with non-convex Helmholtz free energy. In some geometries, this principle reduces to a novel calculus of variations problem; an example is described in which symmetry-related uniform equilibrium states can be connected by nonconstant extremals which realiselocal minima of the free energy. Certain implications of different definitions of local minima are also discussed.
TL;DR: In this paper, a nontrivial function having the properties described in the title is given explicitly, and the properties of the function are described in terms of a nonparametric function.
Abstract: A nontrivial function having the properties described in the title is given explicitly.