A class of test functions for global optimization

We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.This paper was written while the second author was a visiting Professor at Aachen University of Technology, funded by the Deutsche Forschungsgemeinschaft.     

Microlocal analysis of edge flatness through directional multiscale representations

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using ‘higher-than-parabolic’ scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points.     

On global optimization in two-dimensional scaling

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.This paper was written while the second author was visiting Professor at Aachen University of Technology, funded by the Deutsche Forschungsgemeinschaft.     

An analysis of neighborhood functions on generic solution spaces

The effectiveness of local search algorithms on discrete optimization problems is influenced by the choice of the neighborhood function. A neighborhood function that results in all local minima being global minima is said to have zero L-locals. A polynomially sized neighborhood function with zero L-locals would ensure that at each iteration, a local search algorithm would be able to find an improving solution or conclude that the current solution is a global minimum. This paper presents a recursive relationship for computing the number of neighborhood functions over a generic solution space that results in zero L-locals. Expressions are also given for the number of tree neighborhood functions with zero L-locals. These results provide a first step towards developing expressions that are applicable to discrete optimization problems, as well as providing results that add to the collection of solved graphical enumeration problems.     

Analysis of stock market indices through multidimensional scaling

We propose a graphical method to visualize possible time-varying correlations between fifteen stock market values. The method is useful for observing stable or emerging clusters of stock markets with similar behaviour. The graphs, originated from applying multidimensional scaling techniques (MDS), may also guide the construction of multivariate econometric models.     

Visualizing parameters from loglinear models

Summary  This paper presents a graphical display for the parameters resulting from loglinear models. Loglinear models provide a method for analyzing associations between two or several categorical variables and have become widely accepted as a tool for researchers during the last two decades. An important part of the output of any computer program focused on loglinear models is that devoted to estimation of parameters in the model. Traditionally, this output has been presented using tables that indicate the values of the coefficients, the associated standard errors and other related information. Evaluation of these tables can be rather tedious because of the number of values shown as well as their rather complicated structure, mainly when the analyst needs to consider several models before reaching a model with a good fit. Therefore, a graphical display summarizing tables of parameters could be of great help in this situation. In this paper we put forward an interactive dynamic graphical display that could be used in such fashion.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

Parallel branch and bound for multidimensional scaling with city-block distances

Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a multimodal function with unfavorable properties like invariants and non-differentiability. Recently a branch and bound algorithm for multidimensional scaling with city-block distances has been proposed for solution of medium-size problems exactly. The algorithm exploits piecewise quadratic structure of the objective function. In this paper a parallel version of the branch and bound algorithm for multidimensional scaling with city-block distances has been proposed and investigated. Parallel computing enabled solution of larger problems what was not feasible with the sequential version.     

An exemplified semi-analytical approach to the transient sensitivity of nonlinear systems

The internal logic of a model formulated as coupled differential equations is severely challenged in cases where the value of a poorly determined parameter greatly affects the solution of the model. Thus, the sensitivity to parameter changes is essential in the verification stage of the modelling process. Since a sensitivity analysis points out where the influence of flaws in present knowledge has important consequences, a sensitivity analysis will also act as a guideline for future research.In the literature known to the author, the approach to sensitivity has been either by parameter changes or local linearization. However, neither of these approaches will indicate the role played by parameter interactions in sensitivity unless a prohibitively large number of simulations are carried out.The present paper shows a known analytical expression for the sensitivity to parameter changes in nonlinear systems. A simplified version of the expression is solved for a model of the biological processes in activated sludge with 10 state-variables and 31 parameters leading to the presentation of simulated sensitivity of sensitivity, i.e. to an estimate of parameter interactions in sensitivity. The result of the approach is demonstrated through a comparison of actual and predicted changes in solution. Finally, a proposal of the extent of a complete sensitivity analysis is given.     

Rotation Techniques in Asymmetric Multidimensional Scaling

Multidimensional scaling (MDS) is a set of techniques, used especially in behavioral and social sciences, that enable a researcher to visualize proximity data in a multidimensional space. This article focuses on a particular class of MDS models proposed to deal with proximities which describe asymmetric relationships (i.e., trade indices for a set of countries, brand switching data, occupational mobility tables, and so on). They are based on the decomposition of the relationships into a symmetric and a skew-symmetric part. In this way the objects are represented as points in a multidimensional space and the intensity of their relationships as scalar products (symmetry) or triangle areas (skew-symmetry). These models are seen as special cases of a general model and their rotational indeterminacy is investigated. The aim is to propose a rotation method that makes easier the visual inspection of the graphical representation, highlighting the simple structure of the data. In particular an orthomax-like family of rotation methods and a general algorithm are proposed. Advantages of the proposal are illustrated by analysis of import-export data.     

A search technique for global optimization in a chaotic environment

We describe a new algorithm which uses the trajectories of a discrete dynamical system to sample the domain of an unconstrained objective function in search of global minima. The algorithm is unusually adept at avoiding nonoptimal local minima and successfully converging to a global minimum. Trajectories generated by the algorithm for objective functions with many local minima exhibit chaotic behavior, in the sense that they are extremely sensitive to changes in initial conditions and system parameters. In this context, chaos seems to have a beneficial effect: failure to converge to a global minimum from a given initial point can often be rectified by making arbitrarily small changes in the system parameters.     

Stability Index Method for Global Minimization

The Stability Index Method (SIM) combines stochastic and deterministic algorithms to find global minima of multidimensional functions. The functions may be nonsmooth and may have multiple local minima. The method examines the change of the diameters of the minimizing sets for its stopping criterion. At first, the algorithm uses the uniform random distribution in the admissible set. Then normal random distributions of decreasing variation are used to focus on probable global minimizers. To test the method, it is applied to seven standard test functions of several variables. The computational results show that the SIM is efficient, reliable and robust.The authors thank the referees for valuable suggestions.     

A global optimization algorithm using adaptive random search

A new random-search global optimization is described in which the variance of the step-size distribution is periodically optimized. By searching over a variance range of 8 to 10 decades, the algorithm finds the step-size distribution that yields the best local improvement in the criterion function. The variance search is then followed by a specified number of iterations of local random search where the step-size variance remains fixed. Periodic wide-range searches are introduced to ensure that the process does not stop at a local minimum. The sensitivity of the complete algorithm to various search parameters is investigated experimentally for a specific test problem. The ability of the method to locate global minima is illustrated by an example. The method also displays considerable problem independence, as demonstrated by two large and realistic example problems: (1) the identification of 25 parameters in a nonlinear model of a five-degrees-of-freedom mechanical dynamic system and (2) solution of a 24-parameter inverse problem required to identify a pulse train whose frequency spectrum matched a desired reference spectrum.     

Tactical design of rail freight networks. Part II: Local search methods with statistical analysis

This paper deals with the statistical study of the local search methods which we used in Part I of this work. In that part, a tactical planning model of the rail freight transportation was defined as a network design model. Different local search methods have been used to solve it: Simulated Annealing, Tabu Search and a ‘Descent’ method. The solution and method convergence depends on the initial feasible solution and the convexity of the feasible region, so the comparison among them will be made with the help of statistical theory. Assuming the hypothesis that the distribution of local minima can be represented by the Weibull distribution, it is possible to obtain an approach to the global minimum and a confidence interval of this. The global minimum estimation has been used to compare the heuristic methods and the parameters for a given heuristic, and to obtain a stopping criterion.     

Fractal-based exponential distribution of urban density and self-affine fractal forms of cities

Urban population density always follows the exponential distribution and can be described with Clark’s model. Because of this, the spatial distribution of urban population used to be regarded as non-fractal pattern. However, Clark’s model differs from the exponential function in mathematics because that urban population is distributed on the fractal support of landform and land-use form. By using mathematical transform and empirical evidence, we argue that there are self-affine scaling relations and local power laws behind the exponential distribution of urban density. The scale parameter of Clark’s model indicating the characteristic radius of cities is not a real constant, but depends on the urban field we defined. So the exponential model suggests local fractal structure with two kinds of fractal parameters. The parameters can be used to characterize urban space filling, spatial correlation, self-affine properties, and self-organized evolution. The case study of the city of Hangzhou, China, is employed to verify the theoretical inference. Based on the empirical analysis, a three-ring model of cities is presented and a city is conceptually divided into three layers from core to periphery. The scaling region and non-scaling region appear alternately in the city. This model may be helpful for future urban studies and city planning.     

Note on sensitivity analysis of inventory model with partial backorders

In reality, most of the values used for estimating future inventory are parameters derived from the present inventory model. Therefore, researchers must consider how changes in related environments will impact these parameters. A sensitivity analysis explores the impact on the optimal solution when the parameter values vary. This note extends Chu and Chung’s [European Journal of Operational Research 152 (2004) 289–295] paper for the sensitivity analysis of the inventory model with partial backorders. We found the criterion for determining when to run the inventory system. This new discovery deserves careful examination. Numerical examples are provided to demonstrate our findings.     

Identification of mechanical properties of arteries with certification of global optimality

In this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

     

Energy Minimization and Flux Domain Structure in the Intermediate State of a Type-I Superconductor

The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting plate in a transverse magnetic field. Our methods include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical. In these regimes there are two small parameters, and the ground state patterns depend on the relation between them.     

Existence of stationary solutions and local minima for 2D models of fine structure dynamics

In this paper we study two-dimensional models for the motion of a viscoelastic material with a non-monotone stress-strain relationship. We prove existence of infinitely many stationary solutions to two model problems. This is achieved by constructing sequences of increasingly oscillatory functions, whose limit is a stationary solution. These equilibria may have arbitrarily small energy. We also prove that it is always possible to construct paths in phase space that strictly decrease the energy. This result negates the existence of local minima for the energy and asymptotically stable equilibria. These results are important first steps towards understanding the dynamics of fine structure in more than one dimension.     

LOCAL CLASSICAL SOLUTION OF FREE BOUNDARY PROBLEM FOR A COUPLED SYSTEM

This paper considers a two-phase free boundary problem for coupled system including one parabolic equation and two elliptic equations. The problem comes from the discussion of a growth model of self-lnaintaining protocell in multidimensional case. The local classical solution of the problem with free boundary F : y = g(x,t) between two domains is being seeked. The local existence and uniqueness of the problem will be proved in multidimensional case.