On Pontryagin’s first direct method in pursuit problems

We describe an example of a three-dimensional linear differential game with convex compact sets of control. In this example, the integrand in Pontryagin’s first direct method is discontinuous on a set of positive measure.     

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.     

Application of Lyapunov’s direct method to the study of the stability of solutions to systems of impulsive differential equations

Classical theorems on the stability of the solutions of impulsive differential equations are further developed.     

On the efficiency of two variants of Kurchatov’s method for solving nonlinear systems

We consider Kurchatov’smethod and construct two variants of this method for solving systems of nonlinear equations and deduce their local R-orders of convergence in a direct symbolic computation. We also propose a generalization to several variables of the efficiency used in the scalar case and analyse the efficiencies of the three methods when they are used to solve systems of nonlinear equations.     

On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method

Affine generalized Nash equilibrium problems (AGNEPs) represent a class of non-cooperative games in which players solve convex quadratic programs with a set of (linear) constraints that couple the players’ variables. The generalized Nash equilibria (GNE) associated with such games are given by solutions to a linear complementarity problem (LCP). This paper treats a large subclass of AGNEPs wherein the coupled constraints are shared by, i.e., common to, the players. Specifically, we present several avenues for computing structurally different GNE based on varying consistency requirements on the Lagrange multipliers associated with the shared constraints. Traditionally, variational equilibria (VE) have been amongst the more well-studied GNE and are characterized by a requirement that the shared constraint multipliers be identical across players. We present and analyze a modification to Lemke’s method that allows us to compute GNE that are not necessarily VE. If successful, the modified method computes a partial variational equilibrium characterized by the property that some shared constraints are imposed to have common multipliers across the players while other are not so imposed. Trajectories arising from regularizing the LCP formulations of AGNEPs are shown to converge to a particular type of GNE more general than Rosen’s normalized equilibrium that in turn includes a variational equilibrium as a special case. A third avenue for constructing alternate GNE arises from employing a novel constraint reformulation and parameterization technique. The associated parametric solution method is capable of identifying continuous manifolds of equilibria. Numerical results suggest that the modified Lemke’s method is more robust than the standard version of the method and entails only a modest increase in computational effort on the problems tested. Finally, we show that the conditions for applying the modified Lemke’s scheme are readily satisfied in a breadth of application problems drawn from communication networks, environmental pollution games, and power markets.     

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplaces equation in R² is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poissons equation, which the derivatives of particular solutions are estimated under the L² norm. The convergent order of solving the Dirichlet problem of Poissons equation by the method of particular solution and method of fundamental solution is derived. Dedicated to Charles A. Micchelli with esteem on the occasion of his 60th birthdayAMS subject classification 35J05, 31A99     

The Lyapunov–Movchan method in problems of the stability of flows and deformation processes

The extension of Lyapunov's method to continuous mechanical systems are discussed. An annotated bibliography of papers is given in which, based on the Lyapunov–Movchan method, with the construction of corresponding functionals, a direct analysis is carried out of the stability of motion (deformation) of continuous mechanical systems. The material is divided into sections, devoted to the following: (a) the extension of the mathematical apparatus as a whole to continuous and dynamic systems, (b) the stability of elastic, elastoplastic and viscoelastic deformable solids, (c) stability in aeroelasticity and hydroelasticity theory, (d) the linearized theory of hydrodynamic stability, and (e) the stability with reference to perturbations of material functions in the theory of constitutive relations.     

On the numerical stability of Floater-Hormann’s rational interpolant

In this note we extend the definition of the first barycentric formula for Lagrange interpolation to Floater-Hormann interpolants and present an algorithm to evaluate it which is backward stable on the entire real line. We also discuss in detail the numerical stability of the second barycentric formula for Floater-Hormann interpolants.     

On the stability of Hosszú’s functional equation

Two stability results are proved. The first one states that Hosszú’s functional equation $$f(x+y-xy)+f(xy)=f(x)-f(y)=0\ \ \ \ \ (x,y \in \rm R)$$ is stable. The second is a local stability theorem for additive functions in a Banach space setting.     

Lyapunov’s direct method for linear systems of functional-differential equations in Sobolev space

We establish a criterion for exponential stability in the H¹-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov’s direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.     

High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.     

Sinc-Galerkin method for numerical solution of the Bratu’s problems

Study of the performance of the Galerkin method using sinc basis functions for solving Bratu’s problem is presented. Error analysis of the presented method is given. The method is applied to two test examples. By considering the maximum absolute errors in the solutions at the sinc grid points are tabulated in tables for different choices of step size. We conclude that the Sinc-Galerkin method converges to the exact solution rapidly, with order, $O(\exp{(-c \sqrt{n}}))$ accuracy, where c is independent of n.     

Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials

In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving the twelfth-order boundary-value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Accelerating the cubic regularization of Newton’s method on convex problems

In this paper we propose an accelerated version of the cubic regularization of Newton’s method (Nesterov and Polyak, in Math Program 108(1): 177–205, 2006). The original version, used for minimizing a convex function with Lipschitz-continuous Hessian, guarantees a global rate of convergence of order \(O\big({1 \over k^2}\big)\), where k is the iteration counter. Our modified version converges for the same problem class with order \(O\big({1 \over k^3}\big)\), keeping the complexity of each iteration unchanged. We study the complexity of both schemes on different classes of convex problems. In particular, we argue that for the second-order schemes, the class of non-degenerate problems is different from the standard class.     

On the sufficient descent property of the Shanno’s conjugate gradient method

Satisfying in the sufficient descent condition is a strength of a conjugate gradient method. Here, it is shown that under the Wolfe line search conditions the search directions generated by the memoryless BFGS conjugate gradient algorithm proposed by Shanno satisfy the sufficient descent condition for uniformly convex functions.     

On the complexity of extending the convergence region for Traub’s method

The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results.     

On the enumerative nature of Gomory’s dual cutting plane method

For 30 years after their invention half a century ago, cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use. When they finally proved their practical usefulness in the late eighties, that happened in the framework of branch and bound procedures, as an auxiliary tool meant to reduce the number of enumerated nodes. To this day, pure cutting plane methods alone have poor convergence properties and are typically not used in practice. Our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form. In this paper, the second in a sequence, we address some important issues arising when designing a computationally sound pure cutting plane method. We analyze the dual cutting plane procedure proposed by Gomory in 1958, which is the first (and most famous) convergent cutting plane method for integer linear programming. We focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version (as documented in our previous paper on the subject), and we propose new versions of Gomory’s cutting plane procedure with an improved performance. In particular, the new versions are based on enumerative schemes that treat the objective function implicitly, and redefine the lexicographic order on the fly to mimic a sound branching strategy. Preliminary computational results are reported.