variational q calculus abstract we propose q versions of some basic concepts of continuous variational calculus such as the euler lagrange equation and its applications to the isoperimetric lagrange and optimal control problems the maximum principle and also to the hamilton systems and commutation equations

a non monotone regularization newton method for the second order cone complementarity problem based on the smoothing newton method and the tikhonov regularization method we construct a regularization newton method for the second order cone complementarity problem the method uses a non monotone line search scheme which contains the usual monotone line search as a special case by using the theory of euclidean jordan algebras we prove that the proposed method is globally and locally quadratically convergent under suitable assumptions some numerical results are reported which indicate the effectiveness of the method

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

global convergence of a class of non interior point algorithms using chen harker kanzow smale functions for nonlinear complementarity problems we propose a class of non interior point algorithms for solving the complementarity problems cp find a nonnegative pair x y 2n satisfying y f x and xiyi 0 for every i 1 2 n where f is a continuous mapping from n to n the algorithms are based on the chen harker kanzow smale smoothing functions for the cp and have the following features a it traces a trajectory in 3n which consists of solutions of a family of systems of equations with a parameter b it can be started from an arbitrary not necessarily positive point in 2n in contrast to most of interior point methods and c its global convergence is ensured for a class of problems including not strongly monotone complementarity problems having a feasible interior point to construct the algorithms we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition and propose a class of algorithms involving suitable neighborhoods of the trajectory we also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

kneser s theorem in q calculus while difference equations deal with discrete calculus and differential equations with continuous calculus so called q difference equations are considered when studying q calculus in this paper we obtain certain oscillation criteria for second order q difference equations among them a q calculus version of the famous kneser theorem

a neural network based on the generalized fischer burmeister function for nonlinear complementarity problems in this paper we consider a neural network model for solving the nonlinear complementarity problem ncp the neural network is derived from an equivalent unconstrained minimization reformulation of the ncp which is based on the generalized fischer burmeister function f p a b email protected a b p a b we establish the existence and the convergence of the trajectory of the neural network and study its lyapunov stability asymptotic stability as well as exponential stability it was found that a larger p leads to a better convergence rate of the trajectory numerical simulations verify the obtained theoretical results

variational calculus on q nonuniform lattices abstract we introduce the variational calculus on q nonuniform lattices in particular we discuss the basic concepts such as the euler lagrange equation and its applications to the isoperimetric the lagrange and the optimal control problems on q nonuniform lattices

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

special nonuniform lattice snul orthogonal polynomials on discrete dense sets of points difference calculus compatible with polynomials i e such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree can only be made on special lattices well known in contemporary q calculus orthogonal polynomials satisfying difference relations on such lattices are presented in particular lattices which are dense on intervals q 1 are considered

finite difference method of characteristics scheme for the equations of gasdynamics a finite difference method for the characteristic equations of gasdynamics is developed the method is based on a directed discretization of the characteristic compatibility conditions the choice of wave propagation directions is arbitrary and there is no need to fit interpolating functions to the initial data the proposed method is not conservative and therefore must be used with a shock fitting procedure in regions wherever the flow is discontinuous numerical results are presented for sample two dimensional problems umerical schemes based on the method of character istics have been applied successfully to a large number of problems in multidimensio nal gasdynamics schemes of this type are attractive because they are typically robust in no small part because of their direct relationship to the physics of nonlinear wave propagation the review paper by roe 1 out lines a number of characteristi cs based methods including algorithms for both the conservative and nonconservative forms of the governing equations in this paper we present a nonconservative discretization for the characteristic compati bility conditions it is important to keep in mind that al gorithms of this type must in general incorporate a capability to track discontinuities in the flowfield explicitly to ensure that the rankine hugoniot jump conditions are satisfied in these regions a number of conservative and nonconservative schemes for multidimensional flow are constructed under the assumption that the multidimensional problem can be split into quasi one dimensional pieces in which waves travel normal to the mesh lines this is true of approximate riemann solvers 2 3 flux vector splitting schemes 4 5 and also of the nonconservative scm scheme6 how this assumption affects the quality of the numerical results is not entirely clear especially if the compar ison is made keeping the total computational effort fixed nevertheless there is evidence that numerical methods that account for the multidimensional nature of the problem can produce high resolution solutions one family of methods of this type has evolved from colella s conservative unsplit godunov method 7 multidimensional characteristics informa tion is propagated to the faces of the computational cell to compute the numerical fluxes required in the solution of the integral form of the conservation equations perhaps even more sophisticated is roe s model 8 for which unfortunately little data is available several recent papers report research and opinions on this subject 8 9 10 many nonconservative truly multidimensional schemes based on the method of characteristi cs for multidimensional flow have evolved from the method of butler 11 according to this method a number of characteristi c compatibility condi tions are combined to yield a set of equations in which the spatial derivatives of the dependent variables are evaluated along the projections onto space of the stream and wave type characteristic rays the advantage in this approach is clear the upwind directions are defined by the characteristic rays

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

orthogonal polynomials on systems of non uniform lattices from compatibility conditions abstract we deduce difference equations in the matrix form for laguerre hahn orthogonal polynomials on systems of non uniform lattices the so called compatibility conditions involving the transfer matrices as a consequence we obtain closed form expressions for the recurrence relation coefficients of the laguerre hahn polynomials of class zero

a nonmonotone line search technique and its application to unconstrained optimization a new nonmonotone line search algorithm is proposed and analyzed in our scheme we require that an average of the successive function values decreases while the traditional nonmonotone approach of grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 requires that a maximum of recent function values decreases we prove global convergence for nonconvex smooth functions and r linear convergence for strongly convex functions for the l bfgs method and the unconstrained optimization problems in the cute library the new nonmonotone line search algorithm used fewer function and gradient evaluations on average than either the monotone or the traditional nonmonotone scheme

a regularization semismooth newton method based on the generalized fischer burmeister function for p0 ncps we consider a regularization method for nonlinear complementarity problems with f being a p 0 function which replaces the original problem with a sequence of the regularized complementarity problems in this paper this sequence of regularized complementarity problems are solved approximately by applying the generalized newton method for an equivalent augmented system of equations constructed by the generalized fischer burmeister fb ncp functions f p with p 1 we test the performance of the regularization semismooth newton method based on the family of ncp functions through solving all test problems from mcplib numerical experiments indicate that the method associated with a smaller p for example email protected 1 1 2 usually has better numerical performance and the generalized fb functions f p with email protected 1 1 2 can be used as the substitutions for the fb function f 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs