Nonlinear Function Minimization Research Articles

An experimental and straightforward approach to simultaneously estimate temperature-dependent thermophysical properties of metallic materials

This article describes an experimental and straightforward technique towards the simultaneous estimation of temperature-dependent thermal conductivity and specific heat of metals. The thermal model is based on transient nonlinear one-dimensional heat conduction across a metallic sample, which is subject to a constant heat flux on the upper side, and insulated on the bottom side. The thermal analyses are performed in plates of 304 stainless steel and WC10Co cemented carbide. The Levenberg–Marquardt method is employed to provide the solution to an inverse heat conduction problem capable of simultaneously evaluating the temperature-dependent thermophysical properties using transient temperature measurements. Through sensitivity analysis, it is possible to obtain prior information about estimation feasibility and establish all experimental aspects. The imperfect contact at the heater-plate interface causes contact resistance effect, which is considered a reducing agent on heat flux. Beck's nonlinear function minimization technique is used to confirm the reliability of the inverse estimation technique by using the achieved outcomes to recover the heat flux imposed on the test plate. Furthermore, the statistical study into confidence bounds and comparison with literature reveal the robustness of the results. Finally, the accuracy of the developed approach is investigated through the analysis of the errors deriving from experimental and numerical procedures.

ALTERNATE ITERATIVE NUMERICAL ALGORITHMS FOR MINIMIZATION OF UNCONSTRAINED NONLINEAR FUNCTIONS

In this paper, we propose few alternate numerical algorithms for minimization of unconstrained non-linear functions by using modified Adomian decomposition method. Then comparative study among the new algorithms and Newton's algorithm is established by means of examples.

Camera calibration based on the back projection process

Camera calibration plays a crucial role in 3D measurement tasks of machine vision. In typical calibration processes, camera parameters are iteratively optimized in the forward imaging process (FIP). However, the results can only guarantee the minimum of 2D projection errors on the image plane, but not the minimum of 3D reconstruction errors. In this paper, we propose a universal method for camera calibration, which uses the back projection process (BPP). In our method, a forward projection model is used to obtain initial intrinsic and extrinsic parameters with a popular planar checkerboard pattern. Then, the extracted image points are projected back into 3D space and compared with the ideal point coordinates. Finally, the estimation of the camera parameters is refined by a non-linear function minimization process. The proposed method can obtain a more accurate calibration result, which is more physically useful. Simulation and practical data are given to demonstrate the accuracy of the proposed method.

SIMULATION-BASED OPTIMIZATION BY NEW STOCHASTIC APPROXIMATION ALGORITHM

This paper proposes one new stochastic approximation algorithm for solving simulation-based optimization problems. It employs a weighted combination of two independent current noisy gradient measurements as the iterative direction. It can be regarded as a stochastic approximation algorithm with a special matrix step size. The almost sure convergence and the asymptotic rate of convergence of the new algorithm are established. Our numerical experiments show that it outperforms the classical Robbins–Monro (RM) algorithm and several other existing algorithms for one noisy nonlinear function minimization problem, several unconstrained optimization problems and one typical simulation-based optimization problem, i.e., (s, S)-inventory problem.

IDENTIFICATION OF FRACTIONAL-ORDER DYNAMICAL SYSTEMS BASED ON NONLINEAR FUNCTION OPTIMIZATION

In general, real objects are fractional-order systems and also dynamical processes taking place in them are fractional-order processes, although in some types of systems the order is very close to an integer order. So we consider dynamical system whose mathematical description is a differential equation in which the orders of derivatives can be real numbers. With regard to this, in the task of identification, it is necessary to consider also the fractional order of the dynamical system. In this paper we give suitable numerical solutions of differential equations of this type and subsequently an experimental method of identification in the time domain is given. We will concentrate mainly on the identification of parameters, including the orders of derivatives, for a chosen structure of the dynamical model of the system. Under mentioned assumpReceived: September 12, 2013 c © 2013 Academic Publications, Ltd. url: www.acadpubl.eu Correspondence author 226 Ľ. Dorcak, E.A. Gonzalez, J. Terpak, J. Valsa, L. Pivka tions, we would obtain a system of nonlinear equations to identify the system. More suitable than to solve the system of nonlinear equations is to formulate the identification task as an optimization problem for nonlinear function minimization. As a criterion we have considered the sum of squares of the vertical deviations of experimental and theoretical data and the sum of squares of the corresponding orthogonal distances. The verification was performed on systems with known parameters and also on a laboratory object. AMS Subject Classification: 26A33, 37N35

Minimization of nonlinear functions with linear constraints

In this paper, some aspects of numerical implementation of algorithms from a software package for solving problems of minimization of nonlinear nonsmooth functions with linear constraints in the form of sparse matrices are considered. Examples of solving test problems are presented.

Solución de la cinemática directa de la plataforma Gough-Stewart general usando un algoritmo híbrido de optimización

The direct kinematics problem for parallel robots can be stated as follows: given values of the joint variables, the corresponding Cartesian variable values, the pose of the end-effector, must be found. Most of the times the direct kinematics problem involves the solution of a system of non-linear equations. The most efficient methods to solve such kind of equations assume convexity in a cost function which minimum is the solution of the non-linear system. In consequence, the capacity of such methods depends on the knowledge about an starting point which neighboring region is convex, hence the method can find the global minimum. This article propose a method based on probabilistic learning about an adequate starting point for the Dogleg method which assumes local convexity of the function. The proposed method efficiently avoids the local minima, without need of human intervention or apriori knowledge, thus it shows a more robust performance than the simple Dogleg method or other gradient based methods. To demonstrate the performance of the proposed hybrid method, numerical experiments and the respective discussion are presented. The proposal can be extended to other structures of closed-kinematics chains, to the general solution of systems of non-linear equations, and to the minimization of non-linear functions.

3D LIDAR–camera intrinsic and extrinsic calibration: Identifiability and analytical least-squares-based initialization

In this paper we address the problem of estimating the intrinsic parameters of a 3D LIDAR while at the same time computing its extrinsic calibration with respect to a rigidly connected camera. Existing approaches to solve this nonlinear estimation problem are based on iterative minimization of nonlinear cost functions. In such cases, the accuracy of the resulting solution hinges on the availability of a precise initial estimate, which is often not available. In order to address this issue, we divide the problem into two least-squares sub-problems, and analytically solve each one to determine a precise initial estimate for the unknown parameters. We further increase the accuracy of these initial estimates by iteratively minimizing a batch nonlinear least-squares cost function. In addition, we provide the minimal identifiability conditions, under which it is possible to accurately estimate the unknown parameters. Experimental results consisting of photorealistic 3D reconstruction of indoor and outdoor scenes, as well as standard metrics of the calibration errors, are used to assess the validity of our approach.

Numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform

In this article, we discuss a classical ill-posed problem on numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform (DCT). After implementing an eigenvalue decomposition of the second-order difference matrix in the penalty we obtain an explicit minimizer of the Tikhonov functional in a linear combination of DCT-2 components. The choice of the regularization parameter thus can be properly chosen after calling a single variable bounded nonlinear function minimization. Moreover, we propose two approaches to weaken the Gibbs phenomena appearing near the boundary of the reconstructed derivatives. Several numerical examples are provided to show the computational efficiency of our proposed algorithm. Finally, the extension to the multidimensional numerical differentiation leads the approach to a wider scope.

Estimating Parameters of Generalized Integrate-and-Fire Neurons from the Maximum Likelihood of Spike Trains

When a neuronal spike train is observed, what can we deduce from it about the properties of the neuron that generated it? A natural way to answer this question is to make an assumption about the type of neuron, select an appropriate model for this type, and then choose the model parameters as those that are most likely to generate the observed spike train. This is the maximum likelihood method. If the neuron obeys simple integrate-and-fire dynamics, Paninski, Pillow, and Simoncelli (2004) showed that its negative log-likelihood function is convex and that, at least in principle, its unique global minimum can thus be found by gradient descent techniques. Many biological neurons are, however, known to generate a richer repertoire of spiking behaviors than can be explained in a simple integrate-and-fire model. For instance, such a model retains only an implicit (through spike-induced currents), not an explicit, memory of its input; an example of a physiological situation that cannot be explained is the absence of firing if the input current is increased very slowly. Therefore, we use an expanded model (Mihalas & Niebur, 2009 ), which is capable of generating a large number of complex firing patterns while still being linear. Linearity is important because it maintains the distribution of the random variables and still allows maximum likelihood methods to be used. In this study, we show that although convexity of the negative log-likelihood function is not guaranteed for this model, the minimum of this function yields a good estimate for the model parameters, in particular if the noise level is treated as a free parameter. Furthermore, we show that a nonlinear function minimization method (r-algorithm with space dilation) usually reaches the global minimum.

Fractional Particle Swarm Optimization in Multidimensional Search Space

In this paper, we propose two novel techniques, which successfully address several major problems in the field of particle swarm optimization (PSO) and promise a significant breakthrough over complex multimodal optimization problems at high dimensions. The first one, which is the so-called multidimensional (MD) PSO, re-forms the native structure of swarm particles in such a way that they can make interdimensional passes with a dedicated dimensional PSO process. Therefore, in an MD search space, where the optimum dimension is unknown, swarm particles can seek both positional and dimensional optima. This eventually removes the necessity of setting a fixed dimension a priori, which is a common drawback for the family of swarm optimizers. Nevertheless, MD PSO is still susceptible to premature convergences due to lack of divergence. Among many PSO variants in the literature, none yields a robust solution, particularly over multimodal complex problems at high dimensions. To address this problem, we propose the fractional global best formation (FGBF) technique, which basically collects all the best dimensional components and fractionally creates an artificial global best (aGB) particle that has the potential to be a better "guide" than the PSO's native gbest particle. This way, the potential diversity that is present among the dimensions of swarm particles can be efficiently used within the aGB particle. We investigated both individual and mutual applications of the proposed techniques over the following two well-known domains: 1) nonlinear function minimization and 2) data clustering. An extensive set of experiments shows that in both application domains, MD PSO with FGBF exhibits an impressive speed gain and converges to the global optima at the true dimension regardless of the search space dimension, swarm size, and the complexity of the problem.

A family of preconditioned iteratively regularized methods for nonlinear minimization

The preconditioned iteratively regularized Gauss-Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan (2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods which includes M times iterated Tikhonov regularization with line search. Numerical schemes illustrating the theoretical results are also presented.

Actuator-Based Infield Sensor Calibration

In this paper, an online infield nonparametric calibration and error-modeling approach was developed. The approach employs a single source as the external stimulus that creates the differential sensor readings used for calibration. Under very mild assumptions imposed on the calibration functions, error model, and the environment, the technique utilizes the maximal likelihood principle and a nonlinear function minimization solver to derive both the calibration function and the error model of a specified accuracy simultaneously. The approach is intrinsically localized and presents two variants: 1) one where only pairs of neighboring sensors have to communicate in order to conduct the calibration and 2) one where probably a minimum amount of communication is achieved. In addition, the broadcasting tree problem was also formulated as an integer linear programming (ILP) instance; therefore, the broadcasts used in the second variant are optimally resolved. The techniques were evaluated using the traces from the light sensors recorded by the infield deployed sensors, and the statistical evaluations are conducted in order to obtain the interval of confidence to support all the results

New tuning rules for fractional PIα controllers

This paper describes a new tuning method for fractional PIα controllers. The main theoretical contribution of the paper is the analytical solution of a nonlinear function minimization problem, which plays a central role in deriving the tuning formulae. These formulae take advantage of the fractional order α to offer an excellent tradeoff between dynamic performances and stability robustness. Finally, a position control is implemented to compare laboratory experiments with computer simulations. The comparison results show the good performance of the tuning formulae.

Some disconjugacy criteria for differential equations with oscillatory coefficients

AbstractWe determine conditions on the coefficients of both second and fourth order differential operators which give disconjugacy. The conditions exploit the change of sign of the coefficients to show how the change of sign enhances disconjugacy. An application is given to the stability of solutions of equations with periodic coefficients. Focal point and other boundary conditions are considered as well, and sufficient conditions are given which imply the nonexistence of nontrivial solutions of the differential equations satisfying the boundary conditions. Some open problems are stated for the minimization of certain nonlinear functionals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Fractional PI? Controllers: Some Tuning Rules for Robustness to Plant Uncertainties

The objective of this work is to find out optimum settings for a fractional PIλ controller in order to fulfill three different robustness specifications of design for the compensated system, taking advantage of the fractional order, λ. Since this fractional controller has one parameter more than the conventional PI controller, one more specification can be fulfilled, improving the performance of the system and making it more robust to plant uncertainties, such as gain and time constant changes. For the tuning of the controller an iterative optimization method has been used, based on a nonlinear function minimization. Two real examples of application are presented and simulation results are shown to illustrate the effectiveness of this kind of unconventional controllers.

A successive preconditioned conjugate gradient method for the minimization of quadratic and nonlinear functions

This paper is concerned with a conjugate gradient method that utilizes a successive set of preconditioner matrices. The method is developed for the solution of min{f(x): x∈ R n} , where f is a nonlinear function that is sufficiently smooth to possess a Hessian matrix that is continuous. The theory underpinning the method is developed for quadratic polynomials of the form p(x)= 1 2 (x T Ax−2x T b) where A is an arbitrary n× n matrix that is symmetric and positive definite. The minimization of p is achieved by the successive minimization of h i ( u i )= p∘ g i ( u i ) where g i(u i)=K iu i, i=0,1,…,n−1 , and where K i are invertible matrices designed to accelerate the convergence of the conjugate gradient method. It is shown that quadratic termination is achievable for an arbitrary set, {K: i=0,1,…,n−1} , of nonsingular matrices. This remarkable property provides for a very general formulation, where the preconditioned conjugate gradient method (PCGM) and quasi-Newton method are just special cases. The method is extended to nonlinear functions by means of an inexact Newton method. The method is tested on some simple functions to demonstrate the flexibility and computational effectiveness of the new approach. Systems of nonlinear equations generated from a finite element flow formulation are tested to illustrate the practical usefulness of the method.

Incorporating a statistically based shape model into a system for computer-assisted anterior cruciate ligament surgery

This paper addresses the problem of extrapolating very sparse three-dimensional (3-D) data to obtain a complete surface representation. A new method that uses statistical shape models is proposed and its application to computer-assisted anterior cruciate ligament (ACL) reconstruction is detailed. The rupture of the ACL has become one of the most common knee injuries. One problem during reconstruction is to find the optimal attachment points for the graft. Therefore a system for computer-assisted reconstruction of the ACL has been proposed by TIMC laboratory. During surgery the surgeon collects several data points on the tibial and femoral joint surface with a 3-D localizer system. These 3-D data are used to find those attachment points resulting in a low anisometry of the graft, while preventing impingement between the graft and the femoral notch. As the collected data points only cover a small surface patch of the femur, it is desirable to extrapolate these data to also have a visualization in those areas where no data points are available. A sufficiently good approximation of the actual femur by the model would further allow us to better deal with the notch impingement problem of the graft. The chosen approach is to fit a deformable model to the data points, it can be subdivided into two steps, constructing the model and fitting this model to the data. To incorporate a priori knowledge into the model, the allowed deformations are determined by the statistics of the shape variation of a set of training objects. Matching the training objects together is obtained by elastic registration of surface points using octree splines. The fitting process of the sparse intra-operative data with the statistical model results in a non-linear multi-dimensional function minimization. A hybrid search strategy combining local and global methods is used to avoid local minima. First experimental results with a model generated from 10 femurs are presented, including fitting of the model with both simulated and real intra-operative data.

Fast flaw reconstruction from 3D eddy current data

An eddy current flaw reconstruction strategy based on the minimization of nonlinear least-squares error functionals is developed for problems with arbitrary specimen, probe and defect shapes. A fast 3D forward solver is created to rapidly predict eddy current signals in the inversion shell. The high speed of the signal evaluation comes by utilizing a reaction data set constructed before performing the inversion by a finite element electromagnetic field simulator. The same pre-calculated reaction data set supports the quick evaluation of sensitivity information, thereby ensuring the efficient implementation of an optimization algorithm. This optimization algorithm combines first-order and stochastic optimization strategies and improves the reliability of the reconstruction significantly if the observed data contain large noise components. Examples with tube specimens are presented for different 3D flaw shapes.

On the computer implementation of feasible direction interior point algorithms for nonlinear optimization

The paper discusses the computer implementation of a class of interior point algorithms for the minimization of nonlinear functions with equality and inequality constraints. These algorithms consist of fixed point iterations to solve KKT firstorder optimality conditions. At each iteration a descent direction is defined by solving a linear system. Then, the linear system is perturbed in such a way as to deflect the descent direction and obtain a feasible descent direction. A line search is finally performed to obtain a new interior point with a lower objective. Newton, quasi-Newton, or first-order versions of the algorithm can be obtained. This paper is mainly concerned with the solution of the internal linear systems, the algorithms that are employed for the constrained line search and also with the quasi-Newton matrix updating. Some numerical results obtained with a quasi Newton algorithm are also presented. A set of test problems were solved very efficiently with the same values of the internal parameters.