Applied Mathematics

In this paper we apply the Exp-function method to obtain traveling wave solutions of three nonlinear partial differential equations, namely, generalized sinh-Gordon equation, generalized form of the famous sinh-Gordon equation, and double combined sinh-cosh-Gordon equation. These equations play a very important role in mathematical physics and engineering sciences. The Exp-Function method changes the problem from solving nonlinear partial differential equations to solving a ordinary differential equation. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works.

In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.

Low-rank matrix factorization problems such as non negative matrix factorization (NMF) can be categorized as a clustering or dimension reduction technique. The latter denotes techniques designed to find representations of some high dimensional dataset in a lower dimensional manifold without a significant loss of information. If such a representation exists, the features ought to contain the most relevant features of the dataset. Many linear dimensionality reduction techniques can be formulated as a matrix factorization. In this paper, we combine the conjugate gradient (CG) method with the Barzilai and Borwein (BB) gradient method, and propose a BB scaling CG method for NMF problems. The new method does not require to compute and store matrices associated with Hessian of the objective functions. Moreover, adopting a suitable BB step size along with a proper nonmonotone strategy which comes by the size convex parameter $\eta_k$, results in a new algorithm that can significantly improv...

In this paper, an interior-point algorithm for P∗(κ)-Linear Complementarity Problem (LCP) based on a new parametric trigonometric kernel function is proposed. By applying strictly feasible starting point condition and using some simple analysis tools, we prove that our algorithm has O((1 + 2κ) √ n log n log n ) iteration bound for large-update methods, which coincides with the best known complexity bound. Moreover, numerical results confirm that our new proposed kernel function is doing well in practice in comparison with some existing kernel functions in the literature.

Non-negative matrix factorization (NMF) has become a popular method for representing meaningful data by extracting a non-negative basis feature from an observed non-negative data matrix. Some of the unique features of this method in identifying hidden data put this method amongst the powerful methods in the machine learning area. The NMF is a known non-convex optimization problem and the initial point has a significant effect on finding an efficient local solution. In this paper, we investigate the most popular initialization procedures proposed for NMF so far. We describe each method and present some of their advantages and disadvantages. Finally, some numerical results to illustrate the performance of each algorithm are presented.

In this paper, we propose a new non-monotone conjugate gradient method for solving unconstrained nonlinear optimization problems. We first modify the non-monotone line search method by introducing a new trigonometric function to calculate the non-monotone parameter, which plays an essential role in the algorithm’s efficiency. Then, we apply a convex combination of the Barzilai-Borwein method [Barzilai and Borwein, 1988] for calculating the value of step size in each iteration. Under some suitable assumptions, we prove that the new algorithm has the global convergence property. The efficiency and effectiveness of the proposed method are determined in practice by applying the algorithm to some standard test problems and non-negative matrix factorization problems.

Non-negative matrix factorization (NMF) has become a popular method for representing meaningful data by extracting a non-negative basis feature from an observed non-negative data matrix. Some of the unique features of this method in identifying hidden data put this method amongst the powerful methods in the machine learning area. The NMF is a known non-convex optimization problem and the initial point has a significant effect on finding an efficient local solution. In this paper, we investigate the most popular initialization procedures proposed for NMF so far. We describe each method and present some of their advantages and disadvantages. Finally, some numerical results to illustrate the performance of each algorithm are presented.

In this paper, we propose a new non-monotone conjugate gradient method for solving unconstrained nonlinear optimization problems. We first modify the non-monotone line search method by introducing a new trigonometric function to calculate the non-monotone parameter, which plays an essential role in the algorithm's efficiency. Then, we apply a convex combination of the Barzilai-Borwein method for calculating the value of step size in each iteration. Under some suitable assumptions, we prove that the new algorithm has the global convergence property. The efficiency and effectiveness of the proposed method are determined in practice by applying the algorithm to some standard test problems and non-negative matrix factorization problems.

We give an improved non-monotone line search algorithm for stochastic gradient descent (SGD) for functions that satisfy interpolation conditions. We establish theoretical convergence guarantees for the algorithm for strongly convex, convex and non-convex functions. We conduct a detailed empirical evaluation to validate the theoretical results.

Based on a modified secant equation proposed by Li and Fukushima, we derive a stepsize for the Barzilai–Borwein gradient method. Then, using the newly proposed stepsize and another effective stepsize proposed by Dai et al. in an adaptive scheme that is based on the objective function convexity, we suggest a modified two-point stepsize gradient algorithm. We also show that the limit point of the sequence generated by our algorithm is first-order critical. Finally, our numerical comparisons done on a set of unconstrained optimization test problems from the CUTEr collection are presented. At first, we compare the performance of our algorithm with two other two-point stepsize gradient algorithms proposed by Dai et al. and Raydan. Then, numerical comparisons between the implementations of our algorithm and two conjugate gradient methods proposed by Gilbert and Nocedal, and Hestenes and Stiefel, and also, with the memoryless BFGS algorithm proposed by Liu and Nocedal, are made. Furthermore, to provide a numerical support for our adaptive approach, we compare two other two-point stepsize gradient algorithms, one of which applies the stepsize proposed by Dai et al. and the other applies our newly proposed stepsize, with our algorithm which applies both of these stepsizes together. Numerical results demonstrating the efficiency of our algorithm, in the sense of the performance profile introduced by Dolan and Moré, are reported.