A New Modified Conjugate Gradient for Nonlinear Minimization Problems

In this paper, a proposed method for hiding secret data in audio file (.WAV) is introduced.Genetic algorithm has been hybridized with conjugate gradient algorithm which is one of the methods for solving nonlinear unconstrained minimization problem, the hybrid genetic algorithm is used to determine the optimum positions for Least significant bit (LSB) by substituting the least significant bit of each sampling point with a binary message.Numerical results for the proposed hybrid genetic algorithm on three test function were reported and compared with the results of genetic algorithm, also two measures were used such as BER (Bit Error Rate) and MSE (Mean Square Error) in order to access the performance of the proposed steganography method.(.Wav) Least Significant Bit (LSB) . Mean Square Error ( MSE )Error Rate (BER), .

The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples.

The problem of numerical solution of systems of nonlinear algebraic equations with even powers is studied in this paper. Nondifferentiable (nonsmooth) unconstrained minimization problems are associated with the system under consideration, with objective functions, based on discrete l 1 - and l ∞ - norm, respectively, by using these two norms as proximity criteria, and the considered system is solved by minimizing the sum of absolute values of residuals, in the case of l 1 - norm, or by minimizing the maximal residual, in the case of l ∞ - norm, respectively. Also, a “differentiable” unconstrained minimization problem (least squares problem) is associated with the considered system of nonlinear equations, with objective function, based on the discrete l 2 -norm, and the considered system is solved by minimizing the sum of squares of the residuals. Subgradients of the objective functions of unconstrained minimization problems are calculated, and a subgradient method for solving the associated problems is presented. The iterative gradient method for solving the corresponding “differentiable” discrete least squares problem, based on l 2 - norm, is also outlined, and gradients of the objective function of the corresponding “differentiable” unconstrained minimization problems are calculated.

An algorithm is being presented for a special class of unconstrained minimization problems. The algorithm exploits the special structure of the Hessian in the problems under consideration. It is based on applying Bertsekas' [1] Scaled Partial Conjugate Gradient method with respect to a metric that is updated by the Rank One update, using gradients obtained in the preceeding steps. Two classes of problems are presented having the structure assumed in designing the proposed algorithm. In both cases the algorithm uses only first derivative information. Furthermore, it possesses quadratic termination in considerably fewer steps than the number of variables.

In this paper, a novel nonparallel hyperplane based classifier termed as “angle-based twin support vector machine” (ATWSVM) has been proposed which is motivated by the concept of twin support vector machine (TWSVM). TWSVM obtains two nonparallel hyperplanes by solving a pair of quadratic programming problems (QPPs). ATWSVM presents a generic classification model, where the first problem can be formulated using a TWSVM-based classifier and the second problem is an unconstrained minimization problem (UMP) which is reduced to solving a system of linear equations. The second hyperplane is determined so that it is proximal to its own class and the angle between the normals to the two hyperplanes is maximized. The notion of angle has been introduced to have maximum separation between the two hyperplanes. In this work, we have presented two versions of ATWSVM: one that solves a QPP and a UMP; second which formulates both the problems as UMPs. The training time of ATWSVM is much less than that of TWSVM because ATWSVM solves the second problem as UMP instead of QPP. To test the efficacy of the proposed classifier, experiments have been conducted on synthetic and benchmark datasets and it is observed that the proposed classifier achieves classification accuracy comparable or better than that of TWSVM. This work also proposes application of ATWSVM for color image segmentation.

A modified Polak–Ribière–Polyak conjugate gradient algorithm for nonsmooth convex programs

In a neutron scattering application, an unconstrained nonlinear minimization problem is used for the fitting of model parameters to experimental data. Automatic differentiation enables, in a completely mechanical fashion, algorithmic changes by switching from a quasi-Newton method, where first order derivatives are approximated by finite differences, to a modified Gauss-Newton method using exact first order derivatives. Compared to the original code, the code generated by this black box approach produces reliable results rather than results of dubious quality. This approach also is faster in terms of execution time. Its performance is improved further by replacing the most time-consuming subroutine involved in the derivative evaluation by a surprisingly simple, hand-coded implementation of the corresponding analytic expression.

In unconstrained minimization, trust region algorithms use directions that are a combination of the quasi-Newton direction and the steepest descent direction, depending on the fit between the quadratic approximation of the function and the function itself.Algorithms for nonlinear constrained minimization problems usually determine a quasi-Newton direction and use a line search technique to determine the step. Since trust region strategies have proved to be successful in unconstrained minimization, we develop a new trust region strategy for equality constrained minimization. This algorithm is analyzed and global as well as local superlinear convergence theorems are proved for various versions.We demonstrate how to implement this algorithm in a numerically stable way. A computer program based on this algorithm has performed very satisfactorily on test problems; numerical results are provided.

Theoretical and Computational Aspects of Nonlinear Regression

The fuzzy support vector machine (FSVM) assigns each sample a fuzzy membership value based on its relevance, making it less sensitive to noise or outliers in the data. Although FSVM has had some success in avoiding the negative effects of noise, it uses hinge loss, which maximizes the shortest distance between two classes and is ineffective in dealing with feature noise near the decision boundary. Furthermore, whereas FSVM concentrates on misclassification errors, it neglects to consider the critical within-class scatter minimization. We present a Fuzzy support vector machine with pinball loss (FPin-SVM), which is a fuzzy extension of a reformulation of a recently proposed support vector machine with pinball loss (Pin-SVM) with several significant improvements, to improve the performance of FSVM. First, because we used the squared L2- norm of errors variables instead of the L1 norm, our FPin-SVM is a strongly convex minimization problem; second, to speed up the training procedure, solutions of the proposed FPin-SVM, as an unconstrained minimization problem, are obtained using the functional iterative and Newton methods. Third, it is proposed to solve the minimization problem directly in primal. Unlike FSVM and Pin-SVM, our FPin-SVM does not require a toolbox for optimization. We dig deeper into the features of FPin-SVM, such as noise insensitivity and within-class scatter minimization. We conducted experiments on synthetic and real-world datasets with various sounds to validate the usefulness of the suggested approach. Compared to the SVM, FSVM, and Pin-SVM, the presented approaches demonstrate equivalent or superior generalization performance in less training time.

In this article, we try to proposed a new conjugate gradient method for solving unconstrained optimization problems, we focus on conjugate gradient methods applied to the non-linear unconstrained optimization problems, the positive step size is obtained by a line search and the new scalar to the new direction for the conjugate gradient method is derived from the quadratic function and Taylor series and by using quasi newton condition and Newton direction while deriving the new formulae. We also prove that the search direction of the new conjugate gradient method satisfies the sufficient descent and all assumptions of the global convergence property are considered and proved .in order to complete the benefit of our research we should take into account studied the numerical results which are written in FORTRAN language when the objective function is compared our new algorithm with HS and PRP methods on the similar set of unconstrained optimization test problems which is very efficient and encouragement numerical results.

In this paper, a novel unconstrained convex minimization problem formulation for the Lagrangian dual of the recently introduced twin support vector machine (TWSVM) in simpler form is proposed for constructing binary classifiers. Since the objective functions of the modified minimization problems contain non-smooth `plus' function, we solve them by Newton iterative method either by considering their generalized Hessian matrices or replacing the `plus' function by a smooth approximation function. Numerical experiments were performed on a number of interesting real-world benchmark data sets. Computational results clearly illustrates the effectiveness and the applicability of the proposed approach as comparable or better generalization performance with faster learning speed is obtained in comparison with SVM, least squares TWSVM (LS-TWSVM) and TWSVM.

A Liu-Storey-type conjugate gradient method for unconstrained minimization problem with application in motion control

In the image reconstruction of the electrical capacitance tomography (ECT) system, the application of the total least squares theory transforms the ill-posed problem into a nonlinear unconstrained minimization problem, which avoids calculating the matrix inversion. But in the iterative process of the coefficient matrix, the ill-posed problem is also produced. For the effect on the final image reconstruction accuracy of this problem, combined with the principle of the ECT system, the coefficient matrix is targeted and updated in the overall least squares iteration process. The new coefficient matrix is calculated, and then, the regularization matrix is corrected according to the adaptive targeting singular value, which can reduce the ill-posed effect. In this study, the total least squares iterative method is improved by introducing the mathematical model of EIV to deal with the errors in the measured capacitance data and coefficient matrix. The effect of noise interference on the measurement capacitance data is reduced, and finally, the high-quality reconstructed images are calculated iteratively.

Employing the methodology of operations research, a mathematical model is formulated to schedule the development drilling of an oil field. The objective of the model is to minimize the total discounted cost of the development operation, and the decision variable is the number of wells drilled as a function of time. The model is analogous to an inventory problem in which the penalty assessed for failing problem in which the penalty assessed for failing to meet the demand for crude oil is balanced against the cost of drilling, completing, and equipping wells and the cost of transporting rigs and personnel to and from the development area. Because of the requirement that a single decision maker controls the scheduling, the model is particularly applicable to the development of fields in foreign countries. Since the production rates from future wells are unknown and some uncertainty exists in the projected crude oil demand, neither the field producing capacity nor the demand can be treated deterministically, but must be considered as random variables. Assuming that the production rates are independent, log-normally distributed random variables and the demand is normally distributed; the expected shortage is derived as a function of the number of wells drilled and completed. An option is also provided to include a uniformly distributed demand. provided to include a uniformly distributed demand. Once the field is placed on production, the mean and variance of the production rates will decline in proportion to the remaining reservoir energy. The proportion to the remaining reservoir energy. The model is structured so that the mean and variance of the production rates can be adjusted for any type of producing mechanism; therefore, the drilling schedule is planned not only to satisfy a projected demand rate but also to offset the loss in producing capacity when the field is placed on production. The total cost function results in a nonlinear unconstrained minimization problem, which is solved by the pattern search technique of Hooke and Jeeves. The solution to the model demonstrates the sensitivity of the drilling policy to the unit shortage cost and shows that there exists a unit shortage cost below which the field cannot be developed economically. Introduction The exploration and production of crude oil and natural gas are classic examples of decision making under uncertainty. irrespective of the technological advances in the fields of geology, geophysics, geochemistry, and reservoir engineering, the presence of commercial hydrocarbon deposits can presence of commercial hydrocarbon deposits can only be proved by a substantial investment in the drilling of one of more wells. Geologists have long recognized the stochastic nature of hydrocarbon deposits and several interesting prospecting models have been derived. Dowds prospecting models have been derived. Dowds proposed that the rock-hydrocarbon distribution proposed that the rock-hydrocarbon distribution within a geographical area could be studied as a Markov chain. using this approach, he found that the Markov chains from well-to well and area-to area tended toward statistical equilibrium. By contouring the statistical entropy, he was able to isolate areas with the highest probability of finding oil. An interesting application of discriminant analysis was presented by Wignall, in which well and field data were analyzed in order to define a discriminant function that could be used to differentiate between producing and nonproducing wells. producing and nonproducing wells. Grayson, in a famous application of utility theory, proposed that alternative drilling locations be examined on the basis of their expected utility. Kaufmann later improved on this work by showing that, within a given sedimentary basin, reserves were log-normally distributed. Once oil has been discovered and responsibility for developing these reserves has been transferred to the production department, petroleum engineers historically have ignored risk and uncertainty and have considered the development drilling problem as a deterministic one. The first approach toward scheduling the development drilling of a field was given by Aronofsky and Williams. They formulated the problem as a linear programming model with the objective of maximizing the present worth of future net income.