Showing papers on "Maxima and minima published in 1993"

The breakout method for escaping from local minima

Abstract: A number of algorithms have recently been proposed that use iterative improvement (a form of hill-climbing) to solve constraint satisfaction problems. These techniques have had dramatic success on certain problems. However, one factor limiting their wider application is the possibility of getting stuck at non-solution local minima. In this paper we describe an iterative improvement algorithm, called Breakout, that can escape from local minima. We present empirical evidence that this method is very effective in cases where previous approaches have difficulty. Although Breakout is not theoretically complete, in practice it appears to almost always find solutions, for solvable problems. We prove that an idealized (but less efficient) version of the algorithm is complete.

Weak sharp minima in mathematical programming

Abstract: The notion of a sharp, or strongly unique, minimum is extended to include the possibility of a nonunique solution set. These minima will be called weak sharp minima. Conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases. These conditions are also shown to be sufficient under the appropriate convexity hypotheses. The existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem. In particular, a result of Mangasarian and Meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty. Consequences for the convergence theory of algorithms are also examined, especially conditions yielding finite termination.

The applications of harmonic functions to robotics

Abstract: Harmonic functions are solutions to Laplace's equation. Such functions can be used to advantage for potential-field path planning because they do not exhibit spurious local minima. Harmonic functions are shown here to have a number of properties that are essential to robotics applications. Paths derived from harmonic functions are generally smooth. Harmonic functions also offer a complete path-planning algorithm. We show how a harmonic function can be used as the basis for a reactive admittance control. Such schemes allow incremental updating of the environment model. Methods for computing harmonic functions respond well to sensed changes in the environment, and can be used for control while the environment model is being updated.

Use of an eigenmode method to locate the stationary points on the potential energy surfaces of selected argon and water clusters

Abstract: The eigenmode method for locating stationary points (both minima and first-order saddle points) is illustrated for a two-dimensional model potential and is applied to the (Ar) n , n=7-13, and (H 2 O) m , m=2-5, clusters. A large number of previously unreported local minima and first-order saddle points are located

Multiple Copy Simultaneous Search and Construction of Ligands in Binding Sites: Application to Inhibitors of HIV- 1 Aspartic Proteinase

Abstract: Rational ligand design is a complex problem that can be divided into three parts: the search for optimal positions and orientations of functional groups in the binding site, the connection of such positions to form candidate ligands, and the estimation of their binding constants. Approaches for addressing the first two parts of the problem are described in the present work. They are applied to the construction of peptide ligands in the binding site of the human immunodeficiency virus 1 (HIV-1) proteinase. The primary objective is to test the method by comparison of the results with the MVT-101 complex structure for which coordinates are available; the results obtained with the liganded and unliganded proteinase structure are used to examine the utility of the latter for binding studies. A secondary objective is to show how to find new inhibitor candidates. The multiple copy simultaneous search (MCSS) method is utilized to search for optimal positions and orientations of a set of functional groups. For peptide ligands, functional groups corresponding to the protein main chain (N-methylacetamide) and to protein side chains (e.g., methanol, ethyl guanidinium) are used. The resulting N-methylacetamide minima are connected to form hexapeptide main chains with a simple pseudoenergy function that permits a complete search of all possible ways of connecting the minima. Side chains are added to the main-chain candidates by application of the same pseudoenergy function to the appropriate functional group minima. A set of 15 hexapeptides with the sequence of MVT-101 is then minimized by a Monte Carlo scheme, which allows for escape from local minima. Comparison of the MCSS results with the structure of MVT-101 in the HIV-1 binding site showed that all of its functional group positions correspond (within 2.4 A) to some (usually more than one) MCSS minima. There were also many other low-energy MCSS minima which do not appear in any known inhibitors, e.g., methyl ammonium minima in the neighborhood of the catalytic aspartates. Among the 15 lowest minima are seven hexapeptides with the same main-chain orientation as the one found by X-ray crystallography for the inhibitor MVT-101 in the binding site and eight with the main chain oriented in the opposite direction; the latter tend to be more stable. [Addendum: These results are in agreement with recent high-resolution crystallographic data provided after the study was completed.(ABSTRACT TRUNCATED AT 400 WORDS)

Some remarks on the regularity of minima of anisotropic integrals

Abstract: (1993). Some remarks on the regularity of minima of anisotropic integrals. Communications in Partial Differential Equations: Vol. 18, No. 1-2, pp. 153-167.

Earthquake hypocenter location using genetic algorithms

Abstract: A new method for hypocenter location is proposed introducing some recent developments in global optimization techniques. The approach is based on the use of genetic algorithms to minimize some misfit criteria of the data. The method does not use derivative information and therefore does not require the calculation of partial derivatives of travel times of particular phases with respect to hypocentral parameters. The method is completely independent of details of the forward modeling. The only requirement is that the misfit function can be evaluated. Consequently one may use robust error statistics, any type of velocity model (including laterally heterogeneous 3-D models), and combine any type of data that can be modeled (e.g., arrival times and waveforms) without any modification of the algorithm. The new approach is extremely efficient and is superior to previous techniques that share its advantages, in the sense that it can rapidly locate near optimal solutions without an exhaustive search of the parameter space. It achieves this by using an analogy with biological evolution to concentrate sampling in the more favorable regions of parameter space, while improving upon a group of hypocenters simultaneously. Initially, the population of hypocenters is generated randomly and at each subsequent iteration three stochastic processes are applied. The first, “reproduction”, imposes a survival of the fittest criterion to select a new population of hypocenters; the second, “crossover”, produces an efficient exchange of information between the surviving hypocenters; the third, “mutation”, introduces a purely random element that maintains diversity in the new population. Together these steps mimic an evolutionary process, allowing the algorithm to rapidly assimilate and exploit the information, gained from the group as a whole, to find better data fitting hypocenters. The algorithm is illustrated with some synthetic examples using an actual local earthquake network. It is demonstrated how the initially random cloud of hypocenters quickly shrinks and concentrates sampling near the global minimum. Some simple new improvements to the basic algorithm are proposed to assist in avoiding local minima.

Method and apparatus for detecting high-impedance faults in electrical power systems

Abstract: The present invention features a method and apparatus for detecting and enabling the clearance of high impedance faults (HIFs) in an electrical transmission or distribution system. Current in at least one phase in a distribution system is monitored in real time by sensors. Analog current signature information is then digitized for processing by a digital computer. Zero crossings are identified and current maxima and minima located. The first derivatives of the maxima and minima are computed and a modified Fast Fourier Transform (FFT) is then performed to convert time domain to frequency domain information. The transformed data is formatted and normalized and then applied to a trained neural network, which provides an output trigger signal when an HIF condition is probable. The trigger signal is made available to either a network administrator for manual intervention, or directly to switchgear to deactivate an affected portion of the network. The inventive method may be practiced using either conventional computer hardware and software or dedicated custom hardware such as a VLSI chip.

Artificial neural network method and architecture

Abstract: An architecture and data processing method for a neural network that can approximate any mapping function between the input and output vectors without the use of hidden layers. The data processing is done at the sibling nodes (second row). It is based on the orthogonal expansion of the functions that map the input vector to the output vector. Because the nodes of the second row are simply data processing stations, they remain passive during training. As a result the system is basically a single-layer linear network with a filter at its entrance. Because of this it is free from the problems of local minima. The invention also includes a method that reduces the sum of the square of errors over all the output nodes to zero (0.000000) in fewer than ten cycles. This is done by initialization of the synaptic links with the coefficients of the orthogonal expansion. This feature makes it possible to design a computer chip which can perform the training process in real time. Similarly, the ability to train in real time allows the system to retrain itself and improve its performance while executing its normal testing functions.

Locating stationary points for clusters in cartesian coordinates

Abstract: Location of minima and transition states by eigenvector following using Cartesian coordinates and a projection operator is described. Comparisons with calculations employing standard internal coordinates are made for a wide variety of model clusters. The new method, suggested by Baker and Hehre, generally produces faster converagence and solves a number of problems that are inherent when using distance, bond angle, dihedral angle internal coordinates. In particiular, eigenvector-following calculations using analytic first and second energy derivatives should now be possible for much larger systems. Some example reaction paths are illustrated, including a new facetting rearrangement of 55- and 147-atom Mackay icosahedra. The basins of attraction of minima and transition states are also calculated, that is, the regions of the potential-energy surface for which stationary-point searches converge to a given structure. The superiority of the projection operator approach is again demonstrated, and the previous observation that initial geometrical contraction is helpful in transition-state searches is confirmed.

Groundwater management using numerical simulation and the outer approximation method for global optimization

Abstract: Groundwater quantity management problems with fixed charges have been formulated in the past as mixed integer and linear programming problems. In this paper a new methodology is presented where the fixed charges are incorporated into the objective function in an exponential form and the problem is solved as a concave minimization problem. The principal difficulty in the minimization of a concave function over a linear or nonlinear set of constraints is that the local minima which are determined by the classical minimization algorithms may not be global. In an effort to circumvent this problem the outer approximation method is introduced. This method is applicable to the global minimization of a concave function over a compact set of constraints. In the present work the outer approximation is applied to concave minimization problems over a convex compact set of constraints. Two applications of the method to groundwater management problems are presented herein, and the results are compared with an existing solution obtained using a different optimization approach.

A ‘Self-Referential’ Weight Matrix

Abstract: Weight modifications in traditional neural nets are computed by hard-wired algorithms. Without exception, all previous weight change algorithms have many specific limitations. Is it (in principle) possible to overcome limitations of hard-wired algorithms by allowing neural nets to run and improve their own weight change algorithms? This paper constructively demonstrates that the answer (in principle) is ‘yes’. I derive an initial gradientbased sequence learning algorithm for a ‘self-referential’ recurrent network that can ‘speak’ about its own weight matrix in terms of activations. It uses some of its input and output units for observing its own errors and for explicitly analyzing and modifying its own weight matrix, including those parts of the weight matrix responsible for analyzing and modifying the weight matrix. The result is the first ‘introspective’ neural net with explicit potential control over all of its own adaptive parameters. A disadvantage of the algorithm is its high computational complexity per time step which is independent of the sequence length and equals O(nconnlognconn), where riconn is the number of connections. Another disadvantage is the high number of local minima of the unusually complex error surface. The purpose of this paper, however, is not to come up with the most efficient ‘introspective’ or ‘self-referential’ weight change algorithm, but to show that such algorithms are possible at all.

Successive-minima-type inequalities

Abstract: We show analogues of Minkowski's theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lovasz on covering minima.

Parts of visual form: computational aspects

Abstract: A proposed general principle of form from function motivates a particular partitioning scheme involving two types of parts, neck-based and limb-based. Neck-based parts arise from narrowings in shape, or the local minima in distance between two points on the boundary, while limb-based parts arise from a pair of negative curvature extrema which have co-circular tangents. Computational support for the limb-based and neck-based parts is presented by showing that they are invariant, robust, stable, and yield a hierarchy of parts. Examples illustrate that the resulting decompositions are robust in the presence of occlusion and noise for a range of man-made and natural objects and that they lead to natural and intuitive parts which can be used for recognition. >

Use of genetic algorithms with backpropagation in training of feedforward neural networks

Abstract: Genetic algorithms are searching strategies available for finding the globally optimal solution. The problem of genetic algorithms is that they are inherently slow. A hybrid of genetic and backpropagation algorithms (GA-BP) that should always find the correct global minima without getting stuck at local minima is presented. Various versions of the GA-BP method are presented and experimental results show that GA-BP algorithms are as fast as the backpropagation algorithm and do not get stuck at local minima. The proposed GA-BP algorithms are also not sensitive to the values of momentum and learning rate used in backpropagation and can be made independent of the learning rate and momentum. It is shown that the adaptive GA-BP algorithm can provide the optimal learning rate and better performance than simple backpropagation. >

Locating global minima in optimization problems by a random-cost approach

Abstract: OPTIMIZATION problems are common to many diverse disciplines. The classic example is the travelling-salesman problem1,2, in which the objective is to find the shortest path connecting a number of cities. Spin glasses3, meanwhile, exemplify a general class of systems subject to conflicting constraints: in this case, the impossibility of each spin in a lattice aligning favourably with all of its neighbours leads to 'frustration' and to a large number of local energy minima. The optimization problem then becomes the attempt to find the global minimum, or ground state. Predicting the conformational ground state of proteins4 is closely allied to the spin-glass problem. The chief difficulty in searching for global minima is that straightforward search algorithms1 tend to become trapped in local minima. Only a few promising approaches, such as simulated annealing2 or genetic algorithms5, exist. Here I present a general purpose Monte Carlo procedure in which local redirections of the search path are effected at all relevant length scales while enforcing a one-dimensional random walk in the function being minimized. This method yields a series of extrema, from which the global minimum can be extracted with high probability in the limit of large statistics.

Optimal steady-state temperature distribution for a phased array hyperthermia system

Abstract: A method is presented for the evaluation of optimal amplitude and phase excitations for the radiating elements of a phased array hyperthermia system, in order to achieve desired steady-state temperature distributions inside and outside of malignant tissues Use is made of a detailed electromagnetic and thermal model of the heated tissue in order to predict the steady-state temperature at any point in tissue Optimal excitations are obtained by minimizing the squared error between desired and model predicted temperatures inside the tumor volume, subject to the constraint that temperatures do not exceed an upper bound outside the tumor The penalty function technique is used to solve the constrained optimization problem Sequential unconstrained minima are obtained by a modified Newton method Numerical results for a four element phased array hyperthermia system are presented >

On global optimization in two-dimensional scaling

Abstract: We consider multidimensional scaling for embedding dimension two, which allows the detection of structures in dissimilarity data by simply drawing two-dimensional figures. The corresponding objective function, called STRESS, is generally nondifferentiable and has many local minima. In this paper we investigate several features of this function, and discuss the application of different global optimization procedures. A method based on combining a local search algorithm with an evolutionary strategy of generating new initial points is proposed, and its efficiency is investigated by numerical examples.

Performance of an iterative reconstruction algorithm for near-infrared absorption and scatter imaging

Abstract: We have developed an iterative reconstruction algorithm for TOAST, based on a finite element method (FEM) forward model that is fast and very flexible. The algorithm can be used at present with either non-time-resolved and/or time-resolved data, and can reconstruct either (mu) a and/or (mu) s parameters. An equivalent version can be formulated in terms of phase shift and modulation frequency. The basis of the algorithm is to attempt to find the minimum error norm between the measured data and the forward model acting on the trial solution, by a `classical' non-linear search in the distribution of the (mu) a and (mu) s parameters. In principle any search strategy could be used, but the advantage of our approach is that it employs analytical results for the gradient change (partial)M/(partial)(mu) , where M is the measurement. A number of factors influence the performance of the algorithm -- sampling density of the data and solution, noise in the data, accuracy of the model, and appropriate usage of a priori information. It appears that the presence of local minima of the error norm surface cannot be ignored. This paper presents an analysis of the performance of the algorithm on data generated from the FEM model, and from an independent Monte-Carlo model.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Matrix searching with the shortest path metric

Abstract: We present an O(n) time algorithm for computing row-wise maxima or minima of an implicit, totally monotone nn matrix whose entries represent shortest-path distances between pairs of vertices in a simple polygon. We apply this result to derive improved algorithms for several well- known problems in computational geometry. Most prominently, we obtain linear-time algorithms for computing the geodesic diameter, all farthest neighbors, and external farthest neighbors of a simple polygon, improving the previous best result by a factor of O(logn) in each case.

Estimation variances for estimators of product densities and pair correlation functions of planar point processes

Abstract: Approximations of the estimation variances of kernel estimators of the pair correlation function and the product density of a planar Poisson process are given. Furthermore, a heuristic approximation of the estimation variance of an estimator of the pair correlation function of a "general" planar point process is suggested. All formulae have been tested by simulation experiments. Second order characteristics play an important role in point process statistics. Usually, Ripley's K-function and the L-function (see Ripley (1981) and Stoyan st al. (1987)) are used for goodness-of-fit tests and parameter estimation, while the product density 0(t) and the pair correlation function g(t) are used in exploratory data analysis. The form of these functions helps to understand the type of interac- tion in the point pattern and to find suitable point process models. In particular, minima and maxima (if existing) of the pair correlation function may give valuable information on the strength of order. Since estimated second order characteristics deviate from their theoretical counterparts because of statistical fluctuations, error bounds for these functions are very important. For example, they are needed to distinguish between statisti- cal fluctuations in an estimated pair correlation function and peaks which are due to real properties of the point process under study. Until now, variances of esti- mation for second order characteristics are known only in particular cases. Ripley (1988) has given such variances for a series of estimators of the K-function for the Poisson process. This paper gives estimation variances for product densities and pair correla- tion functions. First, in analogy with to Ripley's (1988) calculations, estimation variances in the Poisson process case are derived. The formulae obtained are quite

Accurate computation of the energy spectrum for potentials with multiminima

Abstract: The eigenvalues of the Schrodinger equation with a polynomial potential are calculated accurately by means of the Rayleigh–Ritz variational method and a basis set of functions satisfying Dirichlet boundary conditions. The method is applied to the well potentials having one, two, and three minima. It is shown, in the entire range of coupling constants, that the basis set of trigonometric functions has the capability of yeilding the energy spectra of unbounded problems without any loss of convergence providing that the boundary value α remains greater than a critical value αcr. Only the computation of the nearly degenerate states of multiwell oscillators requires dealing with a relatively large truncation order. © 1993 John Wiley & Sons, Inc.

New approach to selection of initial values of weights in neural function approximation

Abstract: It is proven that the weights and biases generated with certain constraints based on the piecewise linear principle result in an initial neural network which is better able to form a function approximation of an arbitrary function. Use of these initial constraints greatly shortens the training time and avoids the local minima usually associated with an arbitrary random choice of initial weights.