Modification of TLS algorithm for solving ℱ2 linear data fitting problems

It has been proved that the classical TLS algorithm fails to construct a TLS solution of linear data fitting problems AX ≈ B that belong to the class ℱ₂. It will be shown how to modify this algorithm in order to reach a TLS solution. Such solution is not necessarily the minimum 2-norm or Frobenius norm one. A few ideas how to decrease its norm are briefly discussed. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving fully fuzzy linear systems by using implicit Gauss–Cholesky algorithm

This paper analyzes the solution to fully fuzzy linear systems (FFLS). To do so, the implicit Gauss–Cholesky algorithm (IGC) of ABS class is used. Indeed FFLS with different right hand-sides are considered.     

On solving systems of linear algebraic equations by Gibbs’s method

The paper presents an algorithm of approximate solution of a system of linear algebraic equations by the Monte Carlo method superimposed with ideas of simulating Gibbs and Metropolis fields. A solution in the form of a Neumann series is evaluated, the whole vector of solutions is obtained. The dimension of a system may be quite large. Formulas for evaluating the covariance matrix of a single simulation run are given. The method of solution is conceptually linked to the method put forward in a 2009 paper by Ermakov and Rukavishnikova. Examples of 3 × 3 and 100 × 100 systems are considered to compare the accuracy of approximation for the method proposed, for Ermakov and Rukavishnikova’s method and for the classical Monte Carlo method, which consists in consecutive estimation of the components of an unknown vector.     

An algorithm for solving new trust region subproblem with conic model

The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic constraints, while the third one is a nonconvex problem. In this paper, first we analyze the nonconvex problem, and reduce it to two convex problems. Then we discuss some dual properties of these problems and give an algorithm for solving them. At last, we present an algorithm for solving the new trust region subproblem with the conic model and report some numerical examples to illustrate the efficiency of the algorithm.     

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

An efficient reproducing kernel method for solving the Allen–Cahn equation

In this paper, an efficient reproducing kernel method combined with the finite difference method and the Quasi-Newton method is proposed to solve the Allen–Cahn equation. Based on the Legendre polynomials, we construct a new reproducing kernel function with polynomial form. We prove that the semi-scheme can preserve the energy dissipation property unconditionally. Numerical experiments are given to show the efficiency and validity of the proposed scheme.     

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

Retailer��s optimal sourcing strategy by using one major supplier and one emergent supplier

Fast-changing market demand, short product life cycle, and unpredictable events have been enforcing companies in a supply chain to respond to customers’ needs as quickly as possible. While many firms are still seeking a better inventory management within the general business environment, this paper develops an ordering decision support model within a business environment with more unpredictable events. To reduce the uncertainty, a retailer adopts a double sourcing policy with one major supplier and one emergent supplier. Through a two-period dynamic programming model formulation and the simulations based on design of experiments, we analyzed the effects of factors such as penalty cost, backup ratio, purchasing cost ratio, and demand correlation coefficient on retailer’s ordering policy and expected profit. Detailed sensitivity analysis is further conducted based on combinations of penalty cost and backup ratio. Results show that the use of an emergent supplier help increase the retailer’s expected profits. It not only increases the retailer’s upstream sourcing flexibility, but also increases the retailer’s service level with a lower inventory level. As penalty cost or backup ratio increases, the contribution of emergent supplier increases.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

An algorithm for solving convex programs with an additional convex—concave constraint

An implementable decomposition method based on branch-and-bound techniques is proposed for finding a global optimal solution of certain convex programs with an additional convex—concave constrainth(x, y) 0. A nonadaptive simplicial and an adaptive bisection are used for the branching operation, which is performed iny-space only. The bounding operation is based on a relaxation of the convex—concave constrainth(x, y) 0. The algorithm can be applied efficiently for linear programs with an additional affine multiplicative constraint.This work was partially supported by Alexander-von-Humboldt Foundation.     

An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method

This paper provides with a generalization of the work by Cattani (Math. Probl. Eng. (2008) 1–24), who has introduced the connection coefficients of the Shannon wavelets. We apply the Shannon wavelets approximation based on Cattani’s connection coefficients together the collocation points for solving the linear Fredholm integro-differential equations. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results.     

An efficient method for feature selection in linear regression based on an extended Akaike’s information criterion

A method for feature selection in linear regression based on an extension of Akaike’s information criterion is proposed. The use of classical Akaike’s information criterion (AIC) for feature selection assumes the exhaustive search through all the subsets of features, which has unreasonably high computational and time cost. A new information criterion is proposed that is a continuous extension of AIC. As a result, the feature selection problem is reduced to a smooth optimization problem. An efficient procedure for solving this problem is derived. Experiments show that the proposed method enables one to efficiently select features in linear regression. In the experiments, the proposed procedure is compared with the relevance vector machine, which is a feature selection method based on Bayesian approach. It is shown that both procedures yield similar results. The main distinction of the proposed method is that certain regularization coefficients are identical zeros. This makes it possible to avoid the underfitting effect, which is a characteristic feature of the relevance vector machine. A special case (the so-called nondiagonal regularization) is considered in which both methods are identical.     

A potential-function reduction algorithm for solving a linear program directly from an infeasible “warm start”

This paper develops an algorithm for solving a standard-form linear program directly from an infeasible “warm start”, i.e., directly from a given infeasible solution \(\hat x\) that satisfies \(A\hat x = b\) but \(\hat x \ngeqslant 0\) . The algorithm is a potential function reduction algorithm, but the potential function is somewhat different than other interior-point method potential functions, and is given by $$F(x,B) = q\ln (c^T x - B) - \sum\limits_{j = 1}^n {\ln (x_j + h_j (c^T x - B))}$$ where \(q = n + \sqrt n\) is a given constant,h is a given strictly positive shift vector used to shift the nonnegativity constaints, andB is a lower bound on the optimal value of the linear program. The duality gapc ^T x ? B is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints. The algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap (and hence also in the infeasibility) in O(n) iterations. Under more restrictive assumptions regarding the dual feasible region, this algorithm is modified by the addition of a dual barrier term, and will achieve a constant decrease in the duality gap (and in the infeasibility) in \(O(\sqrt n )\) iterations.