Global Minimization Problem Research Articles

Parameterization-free projection for geometry reconstruction

We introduce a Locally Optimal Projection operator (LOP) for surface approximation from point-set data. The operator is parameterization free, in the sense that it does not rely on estimating a local normal, fitting a local plane, or using any other local parametric representation. Therefore, it can deal with noisy data which clutters the orientation of the points. The method performs well in cases of ambiguous orientation, e.g., if two folds of a surface lie near each other, and other cases of complex geometry in which methods based upon local plane fitting may fail. Although defined by a global minimization problem, the method is effectively local, and it provides a second order approximation to smooth surfaces. Hence allowing good surface approximation without using any explicit or implicit approximation space. Furthermore, we show that LOP is highly robust to noise and outliers and demonstrate its effectiveness by applying it to raw scanned data of complex shapes.

Critical friction for wedged configurations

A wedged configuration with Coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads. We analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3-D context. The critical friction coefficient, μ w , is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses. For friction coefficients μ with μ > μ w the wedged problem has at least a solution and for μ < μ w there exits no wedged configurations. For the in-plane problem we discuss the link between the critical friction and the smallest real eigenvalue μ s which is related to the loss of uniqueness. The wedged problem is stated in a discrete framework using a mixed finite element approach and the (discrete) critical friction coefficient μ h w is introduced as the solution of a global minimization problem involving a non differentiable and non-convex functional. The existence of Φ h * , the displacement field of a critical wedged state, is proved and a specific numerical method, based on a genetic algorithm, was developed to compute the critical wedged configurations. Some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations.

Global optimization analysis of slope stability by simulated annealing with dynamic bounds and Dirac function

In slope stability analysis, the search for the minimum factor of safety is a difficult NP-hard global minimization problem as the objective function is non-smooth and non-convex and there are multiple local minima. The use of a simulated annealing method where the control variables are controlled within dynamic domains instead of the conventional static domains is proposed. A simple transformation technique for slopes with a soft band domain (equivalent to a Dirac function) is also proposed. With these improvements, the minimum factor of safety for complicated problems can be determined with high accuracy and reasonable computer time. The proposed algorithm is demonstrated to be efficient and effective for various difficult problems.

Globally concavized filled function method for the box constrained continuous global minimization problem

We present a new definition of filled function, named globally concavized filled function, for the box constrained continuous global minimization problem. A class of this kind of filled functions are constructed. These functions contain two easily determinable parameters, which are not dependent on the radius of the basin at the current local minimizer to make them be globally concavized filled functions. We design a randomized algorithm to solve the problem basing on these functions and prove that the algorithm can converge asymptotically with probability one to a global minimizer of the problem. Numerical experiments are presented to show the effectiveness and robustness of the algorithm.

A Global Minimization Algorithm for Empirical Contact Potential Functions

Global minimization algorithm is indispensable for solving protein folding problems based on thermodynamic hypothesis. A contact difference (CD) based on pseudo potential function, for simulating empirical contact potential functions and testing global minimization algorithm was proposed. The present article describes a conformational sampling and global minimization algorithm, which is called WL, based on Monte Carlo simulation and simulated annealing. It can be used to locate CD's globe minimum and refold extended protein structures, as small as 0. 03 nm, from the native structures, back to ones with root mean square distance (RMSD). These results demonstrate that the global minimization problems for empirical contact potential functions may be solvable.

Deterministic global optimization of molecular structures using interval analysis

The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance because it usually dictates both the physical and chemical properties of the molecule. The existence of an extremely large number of local minima, the number of which may increase exponentially with the size of the molecule, makes this global minimization problem extremely difficult. A new strategy is described here for solving such global minimization problems deterministically. The methodology is based on interval analysis, and provides a mathematical and computational guarantee that the molecular structure with the global minimum potential energy will be found. The technique is demonstrated using two sets of example problems. The first set involves a relatively simple potential model, and problems with up to 40 atoms. The second set involves a more realistic potential energy function, representative of those in current use, and problems with up to 11 atoms.

Global optimization of rational functions: a semidefinite programming approach

We consider the problem of global minimization of rational functions on ** (unconstrained case), and on an open, connected, semi-algebraic subset of **, or the (partial) closure of such a set (constrained case). We show that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean [16]. This extends the analogous results by Nesterov [13] for global minimization of univariate polynomials.For the bivariate case (n = 2), we obtain a fully polynomial time approximation scheme (FPTAS) for the unconstrained problem, if an a priori lower bound on the infimum is known, by using results by De Klerk and Pasechnik [1].For the NP-hard multivariate case, we discuss semidefinite programming-based relaxations for obtaining lower bounds on the infimum, by using results by Parrilo [15], and Lasserre [12].

A class of filled functions for box constrained continuous global optimization

In this paper we show that the unconstrained continuous global minimization problem cannot be solved by any algorithm, so we consider the box constrained continuous global minimization problem. We present a definition of filled function for the problem. Moreover, a new class of filled functions are constructed, which contains only one parameter. A randomized algorithm is designed to solve the box constrained continuous global minimization problem based on the filled function. Numerical experiments are presented to show the practicability of the algorithm.

Fast Road Network Extraction in Satellite Images Using Mathematical Morphology and Markov Random Fields

We present a fast method for road network extraction in satellite images. It can be seen as a transposition of the segmentation scheme "watershed transform region adjacency graph Markov random fields" to the extraction of curvilinear objects. Many road extractors which are composed of two stages can be found in the literature. The first one acts like a filter that can decide from a local analysis, at every image point, if there is a road or not. The second stage aims at obtaining the road network structure. In the method we propose to rely on a "potential" image, that is, unstructured image data that can be derived from any road extractor filter. In such a potential image, the value assigned to a point is a measure of its likelihood to be located in the middle of a road. A filtering step applied on the potential image relies on the area closing operator followed by the watershed transform to obtain a connected line which encloses the road network. Then a graph describing adjacency relationships between watershed lines is built. Defining Markov random fields upon this graph, associated with an energetic model of road networks, leads to the expression of road network extraction as a global energy minimization problem. This method can easily be adapted to other image processing fields, where the recognition of curvilinear structures is involved.

Estimating Intrinsic Camera Parameters from the Fundamental Matrix Using an Evolutionary Approach

Calibration is the process of computing the intrinsic (internal) camera parameters from a series of images. Normally calibration is done by placing predefined targets in the scene or by having special camera motions, such as rotations. If these two restrictions do not hold, then this calibration process is called autocalibration because it is done automatically, without user intervention. Using autocalibration, it is possible to create 3D reconstructions from a sequence of uncalibrated images without having to rely on a formal camera calibration process. The fundamental matrix describes the epipolar geometry between a pair of images, and it can be calculated directly from 2D image correspondences. We show that autocalibration from a set of fundamental matrices can simply be transformed into a global minimization problem utilizing a cost function. We use a stochastic optimization approach taken from the field of evolutionary computing to solve this problem. A number of experiments are performed on published and standardized data sets that show the effectiveness of the approach. The basic assumption of this method is that the internal (intrinsic) camera parameters remain constant throughout the image sequence, that is, the images are taken from the same camera without varying such quantities as the focal length. We show that for the autocalibration of the focal length and aspect ratio, the evolutionary method achieves results comparable to published methods but is simpler to implement and is efficient enough to handle larger image sequences.

Elliptic and hyperbolic quadratic eigenvalue problems and associated distance problems

Two important classes of quadratic eigenvalue problems are composed of elliptic and hyperbolic problems. In [Linear Algebra Appl., 351–352 (2002) 455], the distance to the nearest non-hyperbolic or non-elliptic quadratic eigenvalue problem is obtained using a global minimization problem. This paper proposes explicit formulas to compute these distances and the optimal perturbations. The problem of computing the nearest elliptic or hyperbolic quadratic eigenvalue problem is also solved. Numerical results are given to illustrate the theory.

Improved version of the McMurtry-Fu reinforcement learning scheme

An improved version of the reinforcement scheme originally developed by McMurtry and Fu is presented. A projection procedure as well as a regularizing parameter are introduced to ensure the probability measure and uniqueness of the solution. To prevent degenerate situations where the realization of the function to be optimized is equal to zero, an auxiliary strictly positive regularizing parameter is introduced. A vector representation and a convergence analysis of this multimodal one-dimensional search technique are derived on the basis of the traditional convergence results on Robbins-Monro type of stochastic algorithms. Global maximization and minimization problems are discussed. Finally, some simulation results illustrate the performance and the feasibility of this self-learning optimization algorithm.

A Sequential Convexification Method (SCM) for Continuous Global Optimization

A new method for continuous global minimization problems, acronymed SCM, is introduced. This method gives a simple transformation to convert the objective function to an auxiliary function with gradually `fewer' local minimizers. All Local minimizers except a prefixed one of the auxiliary function are in the region where the function value of the objective function is lower than its current minimal value. Based on this method, an algorithm is designed which uses a local optimization method to minimize the auxiliary function to find a local minimizer at which the value of the objective function is lower than its current minimal value. The algorithm converges asymptotically with probability one to a global minimizer of the objective function. Numerical experiments on a set of standard test problems with several problems' dimensions up to 50 show that the algorithm is very efficient compared with other global optimization methods.

Strain‐driven homogenization of inelastic microstructures and composites based on an incremental variational formulation

AbstractThe paper investigates computational procedures for the treatment of a homogenized macro‐continuum with locally attached micro‐structures of inelastic constituents undergoing small strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro‐structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi‐hyperelastic micro‐stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a setting of smooth single‐surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables. The existence of the quasi‐hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity. Focusing on macro‐strain‐driven micro‐structures, we develop a new incremental variational formulation of the global homogenization problem where a quasi‐hyperelastic macro‐stress potential is obtained from a global minimization problem with respect to the fine‐scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro‐structure for finite increments of time. We consider three different settings of the global variational problem for prescribed linear displacements, periodic fluctuations and constant stresses on the boundary of the micro‐structure and discuss their numerical solutions based on a spatial discretization of the fine‐scale displacement fluctuation field. The performance of the proposed methods is demonstrated for the model problem of von Mises‐type elasto‐visco‐plasticity of the constituents and applied to a comparative study of micro‐to‐macro transitions of inelastic composites. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.

Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems

An important class of generalized eigenvalue problems Ax= λBx is those in which A and B are Hermitian and some real linear combination of them is definite. For the quadratic eigenvalue problem (QEP) ( l ̷ 2A+ l ̷ B+C)x=0 with Hermitian A, B and C and positive definite A, particular interest focuses on problems in which ( x * Bx) 2−4( x * Ax)( x * Cx) is one-signed for all non-zero x—for the positive sign these problems are called hyperbolic and for the negative sign elliptic. The important class of overdamped problems arising in mechanics is a sub-class of the hyperbolic problems. For each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class. For definite pairs ( A, B) the distance is the Crawford number, and we derive bisection and level set algorithms both for testing its positivity and for computing it. Testing hyperbolicity of a QEP is shown to reduce to testing a related pair for definiteness. The distance to the nearest non-hyperbolic or non-elliptic n× n QEP is shown to be the solution of a global minimization problem with n−1 degrees of freedom. Numerical results are given to illustrate the theory and algorithms.

An evolutionary algorithm for generating planar developments of arbitrarily curved surfaces

In this paper, a new method based on constrained global optimization is proposed for generating planar developments of arbitrarily three-dimensional surfaces. With this method an initial planar development is derived which is refined in order to satisfy certain criteria. This refinement is formulated as a global minimization problem under constraints. Generally, the constraints are used to control the local accuracy in the derived planar development. The overall minimization problem is resolved using a real-coded genetic algorithms (GA), which is developed for resolving multi-dimensional minimization problems. Indicative applications are presented to illustrate the effectiveness of the proposed method.

A TOEPLITZ CHARACTERIZATION OF THE STA TICOUTPUT FEEDBACK STABILIZATION PROBLEM FOR LINEAR DISCRETE-TIME SYSTEMS

The static output feedback (SOF) stabilization problem for linear SIMO or MISO discrete-time systems is presented and characterized in terms of the properties of positiv edefinite T oeplitzmatrices. A global minimization problem in a compact sets is introduced whose solution, if any, guarantees closed-loop stability and the fulfillment of an upper bound to a suitable closed loop performance.

Synchronizing sample curves nonparametrically

More and more often, the outcome of a study is not a random variable but a noisy function for each experimental unit, resulting in a sample of curves. Typically, the individual curves vary not only in amplitude or intensity, but also with respect to the time axis: different subjects experience certain events sooner or later. Analyzing such data involves finding out the time changes (or curve registration) among curves. Following our previous work where modified dynamic time warping is applied to align two curves, we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve. Algorithms for solving the minimization problem are presented and tested with simulated and real data. The test results are promising. The method, which involves kernel smoothing of regression functions, estimates the time changes and the average of the aligned curves from noisy data. Large sample asymptotics is derived.

Iron shielded MRI optimization

The design of the main current systems of an actively shielded and of an iron shielded MRI device for nuclear resonance imaging, is considered. The model for the analysis of the magnetic induction produced by the current system, is based on the combination of a Boundary Element technique and of the integration of two Fredholm integral equations of the first and the second kind. The equivalent current magnetization model is used for the calculation of the magnetization produced by the iron shield. High field uniformity in a spherical region inside the device, and a low stray field in the neighborhood of the device are required. In order to meet the design requirements a multi-objective global minimization problem is solved. The minimization method is based on the combination of the filled function technique and the (1+1) evolution strategy algorithm. The multi-objective problem is treated by means of a penalty method. The actively shielded MRI system results to utilize larger amount of conductor and produce higher magnetic energy than the iron shield device.