Existence and global higher integrability of quasiminimizers among minimizing sequences of variational integrals

Quasiminimizers can be viewed as the perturbations of minimizers of variational integrals. We first establish the existence of good minimizing sequences of non-trivial variational integrals containing quasiminimizers of an inhomogeneous p-Dirichlet integral. Employing the concept of variational capacity, we show that the gradients of these quasiminimizers possess global higher integrability

A Class of Diagonally Preconditioned Limited Memory Quasi-Newton Methods for Large-Scale Unconstrained Optimization

The focus of this thesis is to diagonally precondition on the limited memory quasi-Newton method for large scale unconstrained optimization problem. Particularly, the centre of discussion is on diagonally preconditioned limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method. L-BFGS method has been widely used in large scale unconstrained optimization due to its effectiveness. However, a major drawback of the L-BFGS method is that it can be very slow on certain type of problems. Scaling and preconditioning have been used to boost the performance of the L-BFGS method. In this study, a class of diagonally preconditioned L-BFGS method will be proposed. Contrary to the standard L-BFGS method where its initial inverse Hessian approximation is the identity matrix, a class of diagonal preconditioners has been derived based upon the weak-quasi-Newton relation with an additional parameter. Choosing different parameters leads the research to some well-known diagonal updating formulae which enable the R-linear convergent for the L-BFGS method. Numerical experiments were performed on a set of large scale unconstrained minimization problem to examine the impact of each choice of parameter. The computational results suggest that the proposed diagonally preconditioned L-BFGS methods outperform the standard L-BFGS method without any preconditioning. Finally, we discuss on the impact of the diagonal preconditioners on the L-BFGS method as compared to the standard L-BFGS method in terms of the number of iterations, the number of function/gradient evaluations and the CPU time in second

Diagonal quasi-Newton updating formula using log-determinant norm

Quasi-Newton method has been widely used in solving unconstrained optimization problems. The popularity of this method is due to the fact that only the gradient of the objective function is required at each iterate. Since second derivatives (Hessian) are not required, quasi-Newton method is sometimes more efficient than the Newton method, especially when the computation of Hessian is expensive. On the other hand, standard quasi-Newton methods required full matrix storage that approximates the (inverse) Hessian. Hence, they may not be suitable to handle problems of large-scale. In this paper, we develop quasi-Newton updating formula diagonally using log-determinant norm such that it satisfies the weaker secant equation. The Lagrange multiplier is approximated using the Newton-Raphson method that is associated with weaker secant relation. An executable code is developed to test the efficiency of the proposed method with some standard conjugate-gradient methods. Numerical results show that the proposed method performs better than the conjugate gradient method

Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization

In this paper, we aim to propose some spectral gradient methods via variational technique under log-determinant norm. The spectral parameters satisfy the modified weak secant relations that inspired by the multistep approximation for solving large scale unconstrained optimization. An executable code is developed to test the efficiency of the proposed method with spectral gradient method using standard weak secant relation as constraint. Numerical results are presented which suggest a better performance has been achieved

A class of diagonal preconditioners for limited memory BFGS method.

A major weakness of the limited memory BFGS (LBFGS) method is that it may converge very slowly on ill-conditioned problems when the identity matrix is used for initialization. Very often, the LBFGS method can adopt a preconditioner on the identity matrix to speed up the convergence. For this purpose, we propose a class of diagonal preconditioners to boost the performance of the LBFGS method. In this context, we find that it is appropriate to use a diagonal preconditioner, in the form of a diagonal matrix plus a positive multiple of the identity matrix, so as to fit information of local Hessian as well as to induce positive definiteness for the diagonal preconditioner at a whole. The property of hereditary positive definiteness is maintained by a careful choice of the positive scalar on the scaled identity matrix while the local curvature information is carried implicitly on the other diagonal matrix through the variational techniques, commonly employed in the derivation of quasi-Newton updates. Several preconditioning formulae are then derived and tested on a large set of standard test problems to access the impact of different choices of such preconditioners on the minimization performance

Discovering factors of graph polynomials

One of the most common approaches in studying any polynomial is by looking at its factors. Over the years, different graph polynomials have been defined for both undirected and directed graphs, including the Tutte polynomial, chromatic polynomial, greedoid polynomial and cover polynomial. We consider two graph polynomials, one for undirected graphs and one for directed graphs. We first give an overview of these two polynomials. We then discuss the factors of these polynomials as well as the information that are encapsulated by these factors

Rasch model assessment of algebraic word problem among year 8 Malaysian students

Word problems continue to be a challenge for students today. All students must meet the prerequisites for problem solving and reasoning skills, which are important components of the critical thinking component of 21st century skills. This study is being conducted to assess students’ strategies for solving word problems with numbers, consecutives, and ages. The Rasch model is used to analyze the item difficulty level of word problems and students’ strategies for solving ten-word problems at various levels of item difficulty in a similar trait. Then, Pearson correlation analysis is used to investigate the item difficulty level in relation to linguistic, algebraic, and arithmetic factors of word problems before evaluating students’ performance on solving these word problems using various strategies. Rasch model found these algebraic word-problem questions are slightly harder for year 8 Malaysian students in relative to an international standard. Meanwhile, the item difficulty of word problems is driven by linguistic and algebra factors where students can score accurately if the word problems contained explicit information. However, the students encountered difficulties while losing their solution strategy when the questions contained implicit data that demanded critical thinking ability

Diagonal quasi-Newton updating formula via variational principle under the log-determinant measure

Quasi-Newton method has been widely used in solving unconstrained optimization problems. The popularity of this method is due to the fact that only the gradient of the objective function is required at each iterate. Since second derivatives (Hessian) are not required, quasi-Newton method is sometimes more efficient than the newton method, especially when the computation of hessian is expensive. On the other hand, standard quasi-Newton methods required full matrix storage that approximates the (inverse) Hessian. Hence, they may not be suitable to handle problems of large-scale. In this paper, we develop quasi-Newton updating formula diagonally using log-determinant norm such that it satisfies the weaker secant equation. The Lagrangian dual of the variational problem is solved to obtain some approximations for the Lagrange multiplier that is associated with the weak secant equation. An executable code is developed to test the efficiency of the proposed method with some standard conjugate-gradient methods. Numerical results show that the proposed method performs better than the conjugate gradient method

A new exponentiated beta burr type X distribution : model, theory, and applications

In recent years, many attempts have been carried out to develop the Burr type X distribution, which is widely used in fitting lifetime data. These extended Burr type X distributions can model the hazard function in decreasing, increasing and bathtub shapes, except for unimodal. Hence, this paper aims to introduce a new continuous distribution, namely exponentiated beta Burr type X distribution, which provides greater flexibility in order to overcome the deficiency of the existing extended Burr type X distributions. We first present its density and cumulative function expressions. It is then followed by the mathematical properties of this new distribution, which include its limit behaviour, quantile function, moment, moment generating function, and order statistics. We use maximum likelihood approach to estimate the parameters and their performance is assessed via a simulation study with varying parameter values and sample sizes. Lastly, we use two real data sets to illustrate the performance and flexibility of the proposed distribution. The results show that the proposed distribution gives better fits in modelling lifetime data compared to its sub-models and some extended Burr type X distributions. Besides, it is very competitive and can be used as an alternative model to some nonnested models. In summary, the proposed distribution is very flexible and able to model various shaped hazard functions, including the increasing, decreasing, bathtub, and unimodal