Unconstrained 0–1 nonlinear programming: A nondifferentiable approach

The purpose of this paper is to give new formulations for the unconstrained 0–1 nonlinear problem. The unconstrained 0–1 nonlinear problem is reduced to nonlinear continuous problems where the objective functions are piecewise linear. In the first formulation, the objective function is a difference of two convex functions while the other formulations lead to concave problems. It is shown that the concave problems we obtain have fewer integer local minima than has the classical concave formulation of the 0–1 unconstrained 0–1 nonlinear problem.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Master symmetries for Volterra equation, Belov–Chaltikian and Blaszak–Marciniak lattice equations

It is shown how to derive master symmetries for nonlinear lattice equations systematically using the basic principles but without using either their zero curvature equations or the bi-Hamiltonian structure. This has been illustrated for Volterra equation, two coupled Belov–Chaltikian (BC), and three coupled Blaszak–Marciniak (BM) lattice equations. The existence of a sequence of master symmetries is one of the characteristics of completely integrable nonlinear partial differential and differential–difference equations admitting Hamiltonian structure.     

Binary Transformations and (2 + 1)-Dimensional Integrable Systems

A class of nonlinear nonlocal mappings that generalize the classical Darboux transformation is constructed in explicit form. Using as an example the well-known Davey–Stewartson (DS) nonlinear models and the Kadomtsev–Petviashvili matrix equation (MKP), we demonstrate the efficiency of the application of these mappings in the (2 + 1)-dimensional theory of solitons. We obtain explicit solutions of nonlinear evolution equations in the form of a nonlinear superposition of linear waves.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization

This paper presents a new, ray-oriented method for the global solution of nonscalarized vector optimization problems and a framework for the application of the Karush–Kuhn–Tucker theorem to such problems. Properties of nonlinear multiobjective problems implied by the Karush–Kuhn–Tucker necessary conditions are investigated. The regular case specific to nonscalarized MOPs is singled out when a nonlinear MOP with nonlinearities only in the constraints reduces to a nondegenerate linear system. It is shown that the trajectories of the Lagrange multipliers corresponding to the components of the vector cost function are orthogonal to the corresponding trajectories of the vector deviations in the balance space (to the balance set for Pareto solutions). Illustrative examples are presented.     

Asymptotic methods in the nonlinear theory of viscoelasticity

Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 221–229, March–April, 1974.     

Nonlinear mechanics of polymers

The most general of all possible nonlinear viscoelastic theories is presented. Various partial and, at the same time, fairly broad cases of nonlinear creep and relaxation theories are described, and their physical interpretation is given. Effective methods are proposed for the solution of nonlinear boundary problems, and the regions of existence and uniqueness are elucidated.M. V. Lomonosov State University, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 12–23, January–February, 1972.     

Application of the theory of creep stability to polymer materials

The buckling and loss of stability of a rod made from a nonlinear viscous-elastic polymer material are studied with due allowance for instantaneous nonlinear strains. The case of a hinge-fixed rod is considered for various initial deflections.Scientific-Research Institute of Mechanics of the M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 98–104, January–February, 1972.     

Correspondence principle and the method of approximations for certain nonlinear hereditary media

Problems for two classes of nonlinear hereditary media are reduced by integral transforms to problems of the nonlinear theory of elasticity. An approximation of the elastic solution that makes it possible to solve the formulated problems for nonlinear hereditary materials is demonstrated in the case when the viscoelastic problems reduce to problems of the theory of small elastoplastic strains with active loading.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 66–73, January–February, 1971.     

Nonlinear periodic deformation of thixotropic media

The experimental procedure and the processing of results on the nonlinear periodic deformation of polymeric and dispersed system melts are examined. Experimental data are given on the effect of the amplitude of harmonic deformation on nonlinear stress distortions.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 689–696, July–August, 1972.     

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.     

Estimation of the error of approximation of the analytic solution of problems for nonlinear viscoelastic materials

Estimates are given for the error of approximation of the solution of problems of nonlinear viscoelastic media, for which the nonlinear elastic solution depends analytically on the material characteristics, the body and surface forces, and on the boundary displacements. From the integral estimates particular estimates are derived for viscoelastic materials in cases when the nonlinear elastic solution is majorized by a geometric progression or the sum of products of geometric progressions.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 5, pp. 827–834, September–October, 1971.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Analysis of nonlinear polymer creep

The relaxation properties of polyethylene are analyzed. The nonlinear time-dependent stress-strain relations and the creep and relaxation equations are obtained from the experimental creep data. The analysis is based on an appropriate variant of the nonlinear memory theory with singular functions whose parameters, together with the modulus of elasticity, are determined by the method described in [1].Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 410–414, May–June, 1969.     

The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique

The purpose of this study is to implement Adomian–Pade (Modified Adomian–Pade) technique, which is a combination of Adomian decomposition method (Modified Adomian decomposition method) and Pade approximation, for solving linear and nonlinear systems of Volterra functional equations. The results obtained by using Adomian–Pade (Modified Adomian–Pade) technique, are compared to those obtained by using Adomian decomposition method (Modified Adomian decomposition method) alone. The numerical results, demonstrate that ADM–PADE (MADM–PADE) technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM (MADM).     

Scalar discrete nonlinear multipoint boundary value problems

In this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems. By allowing more general boundary conditions and by imposing less restrictions on the nonlinearities, we obtain results that extend previous work in the area of discrete boundary value problems [Debra L. Etheridge, Jesús Rodriguez, Periodic solutions of nonlinear discrete-time systems, Appl. Anal. 62 (1996) 119–137; Debra L. Etheridge, Jesús Rodriguez, Scalar discrete nonlinear two-point boundary value problems, J. Difference Equ. Appl. 4 (1998) 127–144].