Showing papers on "Maxima and minima published in 2001"

Spectral Relaxation for K-means Clustering

Abstract: The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.

Algorithm for data clustering in pattern recognition problems based on quantum mechanics

Abstract: We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrodinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the scale over which cluster structures are searched. We demonstrate it on data analyzed in two dimensions (chosen from the eigenvectors of the correlation matrix). The method is applicable in higher dimensions by limiting the evaluation of the Schrodinger potential to the locations of data points.

Globally optimal regions and boundaries as minimum ratio weight cycles

Abstract: We describe a new form of energy functional for the modeling and identification of regions in images. The energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary (intensity gradients, etc.) and from the interior of the region (texture, homogeneity, etc.). We describe two polynomial-time digraph algorithms for finding the global minima of this energy. One of the algorithms is completely general, minimizing the functional for any choice of modeling information. It runs in a few seconds on a 256/spl times/256 image. The other algorithm applies to a subclass of functionals, but has the advantage of being extremely parallelizable. Neither algorithm requires initialization.

Method and apparatus for location estimation

Abstract: A system for locating a number of devices (112-130) by measuring signals transmitted between known location devices (112-118, 134-138, 214-218, 224-228) and unknown location devices (120-130, 222), and signals transmitted between pairs of unknown location devices (120-130,222), entering signal measurements into a graph function that includes a number of first sub-expressions, a number of which include signal measurement prediction sub-expressions, and have extrema when a predicted signal measurement is equal to an actual signal measurement, and optimizing the graph function.

Optimization Criteria and Geometric Algorithms for Motion and Structure Estimation

Abstract: Prevailing efforts to study the standard formulation of motion and structure recovery have recently been focused on issues of sensitivity and robustness of existing techniques. While many cogent observations have been made and verified experimentally, many statements do not hold in general settings and make a comparison of existing techniques difficult. With an ultimate goal of clarifying these issues, we study the main aspects of motion and structure recovery: the choice of objective function, optimization techniques and sensitivity and robustness issues in the presence of noise. We clearly reveal the relationship among different objective functions, such as “(normalized) epipolar constraints,” “reprojection error” or “triangulation,” all of which can be unified in a new “optimal triangulation” procedure. Regardless of various choices of the objective function, the optimization problems all inherit the same unknown parameter space, the so-called “essential manifold.” Based on recent developments of optimization techniques on Riemannian manifolds, in particular on Stiefel or Grassmann manifolds, we propose a Riemannian Newton algorithm to solve the motion and structure recovery problem, making use of the natural differential geometric structure of the essential manifold. We provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions. The geometric characterization of critical points and the simulation results clarify the difference between the effect of bas-relief ambiguity, rotation and translation confounding and other types of local minima. This leads to consistent interpretations of simulation results over a large range of signal-to-noise ratio and variety of configurations.

An adaptive-hybrid algorithm for geoacoustic inversion

Abstract: This paper presents an adaptive hybrid algorithm to invert ocean acoustic field measurements for seabed geoacoustic parameters. The inversion combines a global search (simulated annealing) and a local method (downhill simplex), employing an adaptive approach to control the trade off between random variation and gradient-based information in the inversion. The result is an efficient and effective algorithm that successfully navigates challenging parameter spaces including large numbers of local minima, strongly correlated parameters, and a wide range of parameter sensitivities. The algorithm is applied to a set of benchmark test cases, which includes inversion of simulated measurements with and without noise, and cases where the model parameterization is known and where the parameterization most be determined as part of the inversion. For accurate data, the adaptive inversion often produces a model with a Bartlett mismatch lower than the numerical error of the propagation model used to compute the replica fields. For noisy synthetic data, the inversion produces a model with a mismatch that is lower than that for the true parameters. Comparison with previous inversions indicates that the adaptive hybrid method provides the best results to date for the benchmark cases.

Modification of the Particle Swarm Optimizer for Locating All the Global Minima

Abstract: In many optimization applications, escaping from the local minima as well as computing all the global minima of an objective function is of vital importance. In this paper the Particle Swarm Optimization method is modified in order to locate and evaluate all the global minima of an objective function. The new approach separates the swarm properly when a candidate minimizer is detected. This technique can also be used for escaping from the local minima which is very important in neural network training.

Boundary-processing-technique in EMD method and Hilbert transform

Abstract: By virtue of neural network, a series of signals is extended forward and backward, as a result, two additional maxima and two additional minima are obtained at both ends of the original data set, with which the EMD decomposition can be exactly achieved with cubic spline interpolation. Meanwhile, by using of neural network every IMF component can also be extended forward and backward, which effectively restrains the end effect, thus the veracious Hilbert spectra are achieved. Verifications of the sample signals and the actual surface elevation of sea waves show that the present extension method is relatively accurate.

A genetic algorithm with real-value coding to optimize multimodal continuous functions

Abstract: In this paper a new Genetic Algorithm (GA) to optimize multimodal continuous functions is proposed. It is based on a splitting of the traditional GA into a sequence of three processes. The first process creates several appropriate sub-populations using the information entropy theory. The second process applies the genetic operators (selection, crossover and mutation) on every subpopulation that is so gradually enriched with better individuals. We then determine the best point s* among the best solutions issued from each of the preceding subpopulations. In the neighbourhood of this point s* is generated a population used to initialize a traditional GA in the third process. Inthis last process, the population is entirely renewed after each generation, the new population being generated in the neighborhood of the best point found. The neighborhood size is decreased after each generation. A detailedcomparison of performances with several stochastic global search methods is presented, using test functions of which local and global minima are known.

Modifications of the direct algorithm

Abstract: This work presents improvements of a global optimization method for bound constraint problems along with theoretical results. These improvements are strongly biased towards local search. The global optimization method known as DIRECT was modified specifically for small-dimensional problems with few global minima. The motivation for our work comes from our theoretical results regarding the behavior of DIRECT. Specifically, we explain how DIRECT clusters its search near a global minimizer. An additional influence is our explanation of DIRECT's behavior for both constant and linear functions. We further improved the effectiveness of both DIRECT, and our modification, by combining them with another global optimization method known as Implicit Filtering. In addition to these improvements the methods were also extended to handle problems where the objective function is defined solely on an unknown subset of the bounding box. We demonstrate the increased effectiveness and robustness of our modifications using optimization problems from the natural gas transmission industry, as well as commonly used test problems from the literature.

Finding Global Minima with a Computable Filled Function

Abstract: The Filled Function Method is an approach to finding global minima of multidimensional nonconvex functions. The traditional filled functions have features that may affect the computability when applied to numerical optimization. This paper proposes a new filled function. This function needs only one parameter and does not include exponential terms. Also, the lower bound of weight factor a is usually smaller than that of one previous formulation. Therefore, the proposed new function has better computability than the traditional ones.

Minimizing model fitting objectives that contain spurious local minima by bootstrap restarting.

Abstract: Objective functions that arise when fitting nonlinear models often contain local minima that are of little significance except for their propensity to trap minimization algorithms. The standard methods for attempting to deal with this problem treat the objective function as fixed and employ stochastic minimization approaches in the hope of randomly jumping out of local minima. This article suggests a simple trick for performing such minimizations that can be employed in conjunction with most conventional nonstochastic fitting methods. The trick is to stochastically perturb the objective function by bootstrapping the data to be fit. Each bootstrap objective shares the large-scale structure of the original objective but has different small-scale structure. Minimizations of bootstrap objective functions are alternated with minimizations of the original objective function starting from the parameter values with which minimization of the previous bootstrap objective terminated. An example is presented, fitting a nonlinear population dynamic model to population dynamic data and including a comparison of the suggested method with simulated annealing. Convergence diagnostics are discussed.

Improving the Particle Swarm Optimizer by Function “Stretching”

Abstract: In this paper a new technique, named Function “Stretching”, for the alleviation of the local minima problem is proposed. The main feature of this technique is the usage of a two—stage transformation of the objective function to eliminate local minima, while preserving the global ones. Experiments indicate that combined with the Particle Swarm Optimizer method, the new algorithm is capable of escaping from local minima and effectively locate the global ones. Our experience is that the modified algorithm behaves predictably and reliably and the results were quite satisfactory. The function “Stretching” technique provides stable convergence and thus a better probability of success to the method with which it is combined.

Gradient-Based History Matching With a Global Optimization Method

Abstract: We investigate the application of a global optimization algorithm called the Tunneling Method to the problem of history-matching of petroleum reservoirs. Results are presented for two test cases. The first is a small synthetic case in which the global minimum is known. The second is a real field example. In both cases, a series of minima was found. The computational cost of each tunneling phase is found to be comparable with the cost of each local minimization. It is concluded that the Tunneling Method may have a practical application in history-matching as an alternative to immediate reformulation of the problem if the first minimum found does not represent an acceptable match.

On approximate minima in vector optimization

Abstract: Necessary and sufficient conditions are obtained for the existence of approximate minima in vector optimization problems. The notion of approximate saddle point is introduced and the relation between approximate saddle points and the approximate minima are established.

The Step and Slide method for finding saddle points on multidimensional potential surfaces

Abstract: We present the Step and Slide method for finding saddle points between two potential-energy minima. The method is applicable when both initial and final states are known. The potential-energy surface is probed by two replicas of the system that converge to the saddle point by following isoenergetic surfaces. The value of the transition-state potential is bracketed in the process, such that a convergence criterion based on the potential can be used. We applied the method to study diffusion mechanisms of a small Ag cluster on a Ag(111) surface using an embedded-atom method potential. Our approach is comparable in efficiency to other commonly used methods.

A Stochastic Iterative Closest Point Algorithm (stochastICP)

Abstract: We present a modification to the iterative closest point algorithm which improves the algorithm's robustness and precision. At the start of each iteration, before point correspondence is calculated between the two feature sets, the algorithm randomly perturbs the point positions in one feature set. These perturbations allow the algorithm to move out of some local minima to find a minimum with a lower residual error. The size of this perturbation is reduced during the registration process. The algorithm has been tested using multiple starting positions to register three sets of data: a surface of a femur, a skull surface and a registration to hepatic vessels and a liver surface. Our results show that, if local minima are present, the stochastic ICP algorithm is more robust and is more precise than the standard ICP algorithm.

A globally convergent approach for blind MIMO adaptive deconvolution

Abstract: We discuss the blind deconvolution of multiple input/multiple output (MIMO) linear convolutional mixtures and propose a set of hierarchical criteria motivated by the maximum entropy principle. The proposed criteria are based on the constant-modulus (CM) criterion in order to guarantee that all minima achieve perfectly restoration of different sources. The approach is moreover robust to errors in channel order estimation. Practical implementation is addressed by a stochastic adaptive algorithm with a low computational cost. Complete convergence proofs, based on the characterization of all extrema, are provided. The efficiency of the proposed method is illustrated by numerical simulations.

Topological description of the aging dynamics in simple glasses.

Abstract: We numerically investigate the aging dynamics of a monatomic Lennard-Jones glass, focusing on the topology of the potential energy landscape which, to this aim, has been partitioned in basins of attraction of stationary points (saddles and minima). The analysis of the stationary points visited during the aging dynamics shows the existence of two distinct regimes: (i) at short times the system visits basins of saddles whose energies and orders decrease with t ; (ii) at long times the system mainly lies in basins pertaining to minima of slowly decreasing energy. The long time dynamics can be represented by a simple random walk on a network of minima with a jump probability proportional to the inverse of the waiting time.

A constrained global inversion method using an overparameterized scheme : Application to poststack seismic data

Abstract: A global optimization algorithm using simulated annealing has advantages over local optimization approaches in that it can escape from being trapped in local minima and it does not require a good initial model and function derivatives to find a global minimum. It is therefore more attractive and suitable for seismic waveform inversion. I adopt an improved version of a simulated annealing algorithm to invert simultaneously for acoustic impedance and layer interfaces from poststack seismic data. The earth’s subsurface is overparameterized by a series of microlayers with constant thickness in two‐way traveltime. The algorithm is constrained using the low‐frequency impedance trend and has been made computationally more efficient using this a priori information as an initial model. A search bound of each parameter, derived directly from the a priori information, reduces the nonuniqueness problem. Application of this technique to synthetic and field data examples helps one recover the true model parameters and ...

Finite Difference Approximation of Free Discontinuity Problems

Abstract: We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Parameter estimation in a trip distribution model by random perturbation of a descent method

Abstract: We consider the problem of the estimation of some parameters involved in a trip distribution model issued from the Transportation Planning. The estimators of the maximum likelihood of the model are the global minima of a non-convex functional. The numerical method must prevent convergence to local minima and we apply a new algorithm of global optimization involving random perturbations of the gradient method. Numerical experiments involving real data show that the method is effective to calculate.

Heuristic methods for randomized path planning in potential fields

Abstract: Randomized path planning driven by a potential field is a well established technique for solving complex, many degrees of freedom motion planning problems. In this technique a suitable potential field shapes the search of the path toward the goal. However, randomized path planning can become relatively inefficient when deep local minima are present in the potential field. Indeed, the algorithm usually spends most its running time trying to escape from local minima by means of uninformed random motions. In this paper we present simple yet effective heuristics for escaping local minima, with the goal of improving overall planning performance. We integrate these heuristics into a path planner without sacrificing the overall probabilistic completeness of the algorithm. Experimental results on several test cases show a remarkable performance improvement, up to a factor of 4 for complex problem instances.

Nonlinear system modeling via knot-optimizing B-spline networks

Abstract: In using the B-spline network for nonlinear system modeling, owing to a lack of suitable theoretical results, it is quite difficult to choose an appropriate set of knot points to achieve a good network structure for minimizing, say, a minimum error criterion. In this paper, a novel knot-optimizing B-spline network is proposed to approximate the general nonlinear system behavior. The knot points are considered to be independent variables in the B-spline network and are optimized together with the B-spline expansion coefficients. The simulated annealing algorithm with an appropriate search strategy is used as an optimization algorithm for the training process in order to avoid any possible local minima. Examples involving dynamic systems up to six dimensions in the input space to the network are solved by the proposed method to illustrate the effectiveness of this approach.

A conservative semi‐Lagrangian transport model for rivers with transient storage zones

Abstract: A conservative, semi-Lagrangian numerical method, domain of influence search for convective unconditional stability (DISCUS), for computing solute transport in fluvial systems with transient storage is presented. The model includes an explicit, shape-preserving cubic interpolation update for advection and fully implicit temporal treatments of dispersion and transient storage. Numerical results with this new method are compared against exact solutions in a uniform advective-dispersive flow with transient storage. Results from a conventional Eulerian numerical model are also shown. Both models are inherently stable and mass-conserving. However, the semi-Lagrangian method is shown to be accurate and free of grid-scale oscillations over a wide range of Courant and grid Peclet numbers in contrast to the Eulerian approach whose accuracy degrades at high Courant numbers and/or large-grid Peclet numbers. Some quantitative aspects of the comparative model efficiencies are discussed, with the semi-Lagrangian model being particularly attractive when large time steps are used. In contrast to earlier semi- Lagrangian approaches, DISCUS employs the use of a cumulative solute mass variable in the advective update, and the issue of its interpolation is examined in the paper. Linear interpolation is shown to guarantee bounded solutions but can be very inaccurate, with numerical errors manifesting themselves by introducing artificially enhanced longitudinal dispersion. Higher-order interpolation (cubic) increases accuracy but at the expense of introducing spurious maxima and minima which manifest themselves as small grid-scale oscillations. By implementing a flux-limiting technique, based on convexity preservation arguments, completely bounded solutions can be achieved in both spatial and temporal domains with the higher-order interpolation.

Euclidean reconstruction and auto-calibration from continuous motion

Abstract: This paper deals with the problem of incorporating natural regularity conditions on the motion in an MAP estimator for structure and motion recovery from uncalibrated image sequences. The purpose of incorporating these constraints is to increase performance and robustness. Auto-calibration and structure and motion algorithms are known to have problems with (i) the frequently occurring critical camera motions, (ii) local minima in the non-linear optimization and (iii) the high correlation between different intrinsic and extrinsic parameters of the camera, e.g. the coupling between focal length and camera position. The camera motion (both intrinsic and extrinsic parameters) is modelled as a random walk process, where the inter-frame motions are assumed to be independently normally distributed. The proposed scheme is demonstrated on both simulated and real data showing the increased performance.