Global Minimum Value Research Articles

Uniqueness of Solution of Cylindricity Objective Function

Based on the radial deviation measurement, the unconstrained optimization model is established for assessing cylindricity errors by the minimum zone method. The properties of the objective function concerned are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the established cylindricity objective function is a continuous, non-differentiable and convex function defined on the four-dimensional Euclidean space R4. Therefore, the global minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Thus, any existing optimization algorithm, as long as it is convergent, can be applied to solve the objective function to get the reliable minimum zone cylindricity errors. An example is given to verify the theoretical results presented.

Surficial properties of landslides and surrounding units in Ophir Chasma, Mars

We mapped the surface layers of the Ophir Chasma region using observations made by the Mars Global Surveyor's Thermal Emission Spectrometer (TES) and the Mars Odyssey's Thermal Emission Imaging System (THEMIS). Our goal is to understand the surface physical characteristics and how they relate to the local geology. TES observations indicate that Ophir Chasma possesses some of the most diverse thermal properties on Mars where the approximate global maximum and minimum values of the derived thermal inertia have been detected within ∼100 km of each other. The thermal maps, used in conjunction with elevation data, meteorological modeling and high‐resolution images, have led to many insights on how the thermal inertia relates to the local morphology and mineralogy. The landslide units on the northern margin of Ophir Chasma have a diverse range of characteristics. A fairly linear relation between elevation and thermal inertia was found that is consistent with the global trend. Recent and/or ongoing aeolian processes are likely closely linked to the current thermal properties of the canyon. These aeolian processes, in turn, are affected by the topography of not only Ophir, but Hebes and Candor Chasmata and neighboring regions. This study has confirmed previous geological interpretations as well as allowed new insights of how the thermal properties of a surface relates to its geology.

Typical phases of pre-failure damage in granitic rocks under differential compression

Abstract The evolution of pre-failure damage in brittle rock samples subjected to differential compression has been investigated by means of acoustic emission (AE) records. The experimental results show that the damaging process is characterized by three typical phases of microcracking activity: primary, secondary, and nucleation. The primary phase reflects the initial activity of pre-existing microcracks, and is characterized by an increase, with increasing stress, both in event rate and b value. The secondary phase involves subcritical growth of the microcrack population, revealed by an event rate increase and a dramatic decrease of the b value. The nucleation phase corresponds to initiation and accelerated growth of the ultimate macroscopic fracture along one or more incipient fracture planes. During the nucleation phase, the b value decreases rapidly to the global minimum value around 0.5. The temporal variation of b in every phase clearly correlates with grain size of the test sample, hence indicating that a comparatively larger grain size results in a lower b value. In order to investigate the fracture mechanism of each phase, a damage model was tested by employing the constitutive laws of subcritical crack growth of crack populations with a fractal size distribution.

Generalized Subinterval Selection Criteria for Interval Global Optimization

The convergence properties of interval global optimization algorithms are studied which select the next subinterval to be subdivided with the largest value of the indicator pf(f k ,X)=(f k −\(\underline F \)(X))/(\(\overline F \)(X)−\(\underline F \)(X)). This time the more general case is investigated, when the global minimum value is unknown, and thus its estimation f k in the iteration k has an important role. A sharp necessary and sufficient condition is given on the f k values approximating the global minimum value that ensure convergence of the optimization algorithm. The new theoretical result enables new, more efficient implementations that utilize the advantages of the pf* based interval selection rule, even for the more general case when no reliable estimation of the global minimum value is available.

Bose-Einstein condensation of helium and hydrogen inside bundles of carbon nanotubes

Helium atoms or hydrogen molecules are believed to be strongly bound within the interstitial channels (between three carbon nanotubes) within a bundle of many nanotubes. The effects on adsorption of a nonuniform distribution of tubes are evaluated. The energy of a single-particle state is the sum of a discrete transverse energy ${E}_{t}$ (that depends on the radii of neighboring tubes) and a quasicontinuous energy ${E}_{z}$ of relatively free motion parallel to the axis of the tubes. At low temperature, the particles occupy the lowest-energy states, the focus of this study. The transverse energy attains a global minimum value $({E}_{t}={E}_{\mathit{min}})$ for radii near ${R}_{\mathit{min}}=9.95\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ for ${\mathrm{H}}_{2}$ and $8.48\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ for $^{4}\mathrm{He}$. The density of states $N(E)$ near the lowest energy is found to vary linearly above this threshold value, i.e., $N(E)$ is proportional to $(E\ensuremath{-}{E}_{\mathit{min}})$. As a result, there occurs a Bose-Einstein condensation of the molecules into the channel with the lowest transverse energy. The transition is characterized approximately as that of a four-dimensional gas, neglecting the interactions between the adsorbed particles. The phenomenon is observable, in principle, from a singular heat capacity. The existence of this transition depends on the sample having a relatively broad distribution of radii values that include some near ${R}_{\mathit{min}}$.

Local density fluctuations, hyperuniformity, and order metrics.

Questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges. The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d. Our specific objectives are to study the variance in the number of points contained within a regularly shaped window Omega of arbitrary size, and to further illuminate our understanding of hyperuniform systems, i.e., point patterns that do not possess infinite-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area (rather than the volume) of the window. We derive two formulations for the number variance: (i) an ensemble-average formulation, which is valid for statistically homogeneous systems, and (ii) a volume-average formulation, applicable to a single realization of a general point pattern in the large-system limit. The ensemble-average formulation (which includes both real-space and Fourier representations) enables us to show that a homogeneous point pattern in a hyperuniform state is at a "critical point" of a type with appropriate scaling laws and critical exponents, but one in which the direct correlation function (rather than the pair correlation function) is long ranged. We also prove that the non-negativity of the local number variance does not add a new realizability condition on the pair correlation. The volume-average formulation is superior for certain computational purposes, including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance. We prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one-dimensional hyperuniform patterns. We also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one, two, and three dimensions for spherical windows, enabling us to rank-order the spatial patterns. Our results suggest that the local variance may serve as a useful order metric for general point patterns. Contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension, we show that for d=3, the body-centered cubic lattice has a smaller variance than the face-centered cubic lattice. Finally, for certain hyperuniform disordered point patterns, we evaluate the direct correlation function, structure factor, and associated critical exponents exactly.

Band-target entropy minimization. A robust algorithm for pure component spectral recovery. Application to complex randomized mixtures of six components.

A newly developed self-modeling curve resolution method, band-target entropy minimization (BTEM), is described. This method starts with the data decomposition of a set of spectroscopic mixture data using singular value decomposition. It is followed by the transformation of the orthonormal basis vectors/loading vectors into individual pure component spectra one at a time. The transformation is based in part on some seminal ideas borrowed from information entropy theory with the desire to maximize the simplicity of the recovered pure component spectrum. Thus, the proper estimate is obtained via minimization of the proposed information entropy function or via minimization of derivative and area of the spectral estimate. Nonnegativity constraints are also imposed on the recovered pure component spectral estimate and its corresponding concentrations. As its name suggests, in this method, one targets a spectral feature readily observed in loading vectors to retain, and then combinations of the loading vectors are searched to achieve the global minimum value of an appropriate objective function. The major advantage of this method is its one spectrum at a time approach and its capability of recovering minor components having low spectroscopic signals. To illustrate the application of BTEM, spectral resolution was performed on FT-IR measurements of very highly overlapping mixture spectra containing six organic species with a two-component background interference (air). BTEM estimates were also compared with the estimates obtained using other self-modeling curve resolution techniques, i.e., SIMPLISMA, IPCA, OPA-ALS, and SIMPLISMA-ALS.

Heuristic Rejection in Interval Global Optimization

Based on the investigation carried out in Ref. 1, this paper incorporates new studies about the properties of inclusion functions on subintervals while a branch-and-bound algorithm is solving global optimization problems. It is found that the relative place of the global minimum value within the inclusion function value of the objective function at the current interval indicates mostly whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a considerable amount of computation. Illustrative examples are discussed and an extended numerical study shows the advantages of the new approach.

Pseudorandom lattices for global optimization

Rationalization processes are proposed to improve uniformity in small samples for pseudorandom lattices in (0,1) n constructed from sequences produced by random number generators. On this basis, the space filtration and space contraction algorithms are developed for the solution of multimodal global optimization problems. Strong convergence to the global minimum value and convergence in measure onto the set of all global minimizers are proved. Numerical experiments are presented to illustrate a better uniformity provided by a rationalization process and the use of the space filtration algorithm for global optimization.

Improved first-order second moment method for uncertainty estimation in flood forecasting

The first-order second moment (FOSM) method is widely used in uncertainty analysis. This method uses a linearization of the function that relates the input variables and parameters to the output variables. This simplification occasionally leads to problems when the mean value of the input variable is close to a local or global maximum or minimum value of the function. In this case, the FOSM computes artificially a zero uncertainty because the first derivative of the function is equal to zero. An improvement to the FOSM is proposed, whereby a parabolic reconstruction is used instead of a linear one. The improved FOSM method is applied to a flood forecasting model on the Loire River (France). Verification of the method using the Monte Carlo technique shows that the improved FOSM allows the accuracy of the uncertainty assessment to be increased substantially, without adding a significant burden in computation. The sensitivity of the results to the size of the perturbation is also analysed.

Simulated Annealing as a Method for the Determination of the Spatial Distribution of a Condensable Adsorbate in Mesoporous Materials

The simulated annealing algorithm is employed for the determination of the spatial distribution of a condensable adsorbate in reconstructed digital domains of mesoporous materials. The optimum configuration for a given degree of saturation is derived assuming that in equilibrium the total interfacial free energy reaches a global minimum value. The suggested approach is applied to reconstructed matrixes of Vycor and random sphere packs. Furthermore, the effective diffusivities of inert gases in the Knudsen regime are calculated in the resulting “dry” and “wet” images with a step-by-step random walk method. The predicted gas relative permeability curves are in satisfactory agreement with experimental results, confirming that the suggested combination of digital reconstruction and simulated annealing is a useful tool for the determination of the spatial distribution of a condensable sorbed phase in mesoporous solids.

Numerical Experiences with a New Generalized Subinterval Selection Criterion for Interval Global Optimization

The convergence properties are studied for interval global optimization algorithms that select the next subinterval to be subdivided with the largest value of the indicator pf(f k, X) = $$\frac{{f_k - \underline F \left( X \right)}}{{\overline F \left( X \right) - \underline F \left( X \right)}}$$ . In contrast to previous work, here the more general case is investigated, when the global minimum value is unknown, and thus its estimation f k in the iteration k has an important role. Extensive numerical tests on 40 problems confirm that substantial improvements can be achieved both on simple and sophisticated algorithms by the new method (not utilizing the minimum value).

Global Minimization of a Multivariate Polynomial using Matrix Methods

The problem of minimizing a polynomial function in several variables over Rn is considered and an algorithm is given. When the polynomial has a minimum the algorithm returns the global minimal value and finds at least one point in every connected component of the set of minimizers. A characterization of such points is given. When the polynomial does not have a minimum the algorithm computes its infimum. No assumption is made on the polynomial.

Optimization of spot-welded structures

This paper addresses the optimization problem of a spot-welded structure under fatigue life consideration. The so-called umbrella model of spot welds and the radial stresses around a spot weld were introduced into the optimization process. The idea was to study the optimization problem using the type of analysis which is now considered to be valid for fatigue life prediction. Two formations were considered: one maximizes the stiffness of a structure under fatigue life constraints and the other seeks the maximum fatigue life of the structure. The optimization results show that the optimization process has successfully found the global minimum values of the corresponding objective functions for both the formations.

A heuristic rejection criterion in interval global optimization algorithms

This paper investigates the properties of the inclusion functions on subintervals while a Branch-and-Bound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective function at the actual interval mostly indicates whether the given interval is close to minimizer point. This information is used in a heuristic interval rejection rule that can save a big amount of computation. Illustrative examples are discussed and a numerical study completes the investigation.

X-ray phase determination by the principle of minimum charge.

When the electron density in a crystal or a quasicrystal is reconstructed from its Fourier modes, the global minimum value of the density is sensitively dependent on the relative phases of the modes. The set of phases that maximizes the value of the global minimum corresponds, by positivity of the density, to the density having the minimum total charge that is consistent with the measured Fourier amplitudes. Phases that minimize the total electronic charge (i.e. the average electron density) have the additional property that the lowest minima of the electron density become exactly degenerate and proliferate within the unit cell. The large number of degenerate minima have the effect that density maxima are forced to occupy ever smaller regions of the unit cell. Thus, by minimization of the electronic charge, the atomicity of the electron density is enhanced as well. Charge minimization applied to simulated crystalline and quasicrystalline diffraction data successfully reproduces the correct phases starting from random initial phases.

Speed-up of the Monte Carlo method by using a physical model of the Dempster-Shafer theory

By using the Monte Carlo method, we can obtain the minimum value of a function V(r) that is generally associated with the potential energy. In this paper we present a method that makes it possible to speed up the classical Monte Carlo method. The new method is based on the observation that the Bolzmann transition probability and the concept of local thermodynamical equilibrium give rise to an initial state of maximum entropy, which is subsequently modified by using the information on the internal structure of the system. The classical thermodynamic model does not take into account any structures inside the system, and therefore in many cases does not accurately model the system itself. In an attempt to take into account the internal structure of the system, we propose a physical model of the belief measure as defined in the Dempster-Shafer theory. The recent discovery by Resconi of an algorithm to calculate the probability distribution that has previously been developed by Harmanec and Klir, and which is consistent with the belief measure, opens the way to utilizing the Bolzmann distribution not only with a uniform distribution of probability, but instead with an arbitrary distribution of probability to guide the Monte Carlo iterative method to obtain the global minimum value of the potential energy. Starting from local thermodynamic equilibrium (i.e., local symmetry), the algorithm computes a new distribution over subsystems, resulting in a nonuniform distribution and in symmetry breaking. In the general case one can start with a different initial distribution induced by other local symmetries, corresponding to specific differential equations (e.g., the Fokker-Planck equation) and calculate from this the global distribution corresponding to the breaking of local symmetry. © 1998 John Wiley & Sons, Inc.

Speed‐up of the Monte Carlo method by using a physical model of the Dempster–Shafer theory

By using the Monte Carlo method, we can obtain the minimum value of a function V(r) that is generally associated with the potential energy. In this paper we present a method that makes it possible to speed up the classical Monte Carlo method. The new method is based on the observation that the Bolzmann transition probability and the concept of local thermodynamical equilibrium give rise to an initial state of maximum entropy, which is subsequently modified by using the information on the internal structure of the system. The classical thermodynamic model does not take into account any structures inside the system, and therefore in many cases does not accurately model the system itself. In an attempt to take into account the internal structure of the system, we propose a physical model of the belief measure as defined in the Dempster–Shafer theory. The recent discovery by Resconi of an algorithm to calculate the probability distribution that has previously been developed by Harmanec and Klir, and which is consistent with the belief measure, opens the way to utilizing the Bolzmann distribution not only with a uniform distribution of probability, but instead with an arbitrary distribution of probability to guide the Monte Carlo iterative method to obtain the global minimum value of the potential energy. Starting from local thermodynamic equilibrium (i.e., local symmetry), the algorithm computes a new distribution over subsystems, resulting in a nonuniform distribution and in symmetry breaking. In the general case one can start with a different initial distribution induced by other local symmetries, corresponding to specific differential equations (e.g., the Fokker–Planck equation) and calculate from this the global distribution corresponding to the breaking of local symmetry. © 1998 John Wiley & Sons, Inc.13: 221–242, 1998

On the efficiency of a continuous version of the simulated annealing algorithm

The efficiency of a continuous version of the simulated annealing algorithm is evaluated by the expected number of steps to reach an ϵ-neighborhood of the global minimum value of the objective function. Different sampling procedures are analyzed and compared to the pure random search procedure. And as a side result, we obtain simpler proofs for some asymptotic results of Dekkers and Aarts (1991).

A branch and bound algorithm for bound constrained optimization problems without derivatives

In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most “uninteresting” parts of the search domain and concentrates on its “promising” subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.