primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

on the controllability of linear juggling mechanical systems this paper deals with the controllability of a class of nonsmooth complementarity mechanical systems due to their particular structure they can be decomposed into an object and a robot consequently they are named juggling systems it is shown that the accessibility of the object can be characterized by nonlinear constrained equations or generalized equations examples are presented including a simple model of backlash the main focus of the work is about linear jugglers 2005 elsevier b v all rights reserved

time finite element discretization of hamilton s law of varying action hamilton s law of varying action is used as a variational source for the derivation of finite element discretization procedure in the time domain three different versions of the proposed algorithms are presented and verified for accuracy and stability the first one is the high precision finite time element analogous to the standard finite elements with cutoff frequency the second version is the step by step one time element from which the unconditionally stable with slightly altered accuracy third algorithm is derived the new operator connected with the proposed algorithms bears attractive properties of much greater accuracy than other existing stable methods and easy computer implementation thus the work herein shows that the reservations expressed against the use of finite elements in time domain seem unjustified

levenberg marquardt methods for constrained nonlinear equations with strong local convergence properties we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

modelling of stochastic dynamic systems using hamilton s law of varying action abstract this paper presents a model based on hamilton s law of varying action for stochastic dynamic systems in this model the state variables are approximated as a linear sum of orthogonal polynomials for deterministic systems the coefficients of the polynomials are constant but for stochastic systems the coefficients are random variables the initial conditions are treated as gaussian random variables with specified mean values and covariance matrix and the external forcing functions input processes are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the random input processes are represented in terms of truncated linear sums of orthonormal eigenfunctions with uncorrelated random variables as coefficients this method reduces the infinite dimensional input force vector to one with finite dimensions a method to compute the mean and the variance functions for both the singular and the regular components of the state processes is developed a single degree of freedom spring mass damper systems subjected to random initial conditions and a random forcing function is considered to show the feasibility and effectiveness of the formulation the response is obtained without any reference to the differential equations associated with the dynamics of the system the study shows that the results of this formulation agree well with those obtained using other schemes

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

construction of multisymplectic schemes of any finite order for modified wave equations in this paper we analyze several symplectic discretizations of linear wave equations with periodic boundary conditions and show the discretizations are multisymplectic of arbitrary finite order for wave equations when the periodic boundary condition vanishes

on the solution of the kkt conditions of generalized nash equilibrium problems we consider the solution of generalized nash equilibrium problems by concatenating the kkt optimality conditions of each player s optimization problem into a single kkt like system we then propose two approaches for solving this kkt system the first approach is rather simple and uses a merit function equation based technique for the solution of the kkt system the second approach partially motivated by the shortcomings of the first one is an interior point based method we show that this second approach has strong theoretical properties and in particular that it is possible to establish global convergence under sensible conditions this probably being the first result of its kind in the literature we discuss the results of an extensive numerical testing on four kkt based solution algorithms showing that the new interior point method is efficient and very robust

hamilton s law of varying action part i assumed time modes method abstract without consideration of force equilibrium and differential equations of motion a general derivation of algebraic equations for dynamic systems is presented this is achieved by an assumed time modes approach in conjunction with a direct application of hamilton s law of varying action by assumed time modes it is implied that the dependent variables of the general dynamics problem can be expanded in terms of admissible basis functions in time this approach allows explicita prioriintegration in time of the energy related integrals leading to the general algebraic equations of motion aem in which the constant expansion coefficients constitute the generalized states of motion essential features of the aem are noted and a unified end result is presented identification of system matrices is made in the algebraic form the general motion is considered to be non linear but linear contributions and algebraic reductions to associated linear systems are also shown a simple demonstration is included to show the basic steps to obtain the aem

a note on the existence of the alizadeh haeberly overton direction for semidefinite programming this note establishes a new sufficient condition for the existence and uniqueness of the alizadeh haeberly overton direction for semidefinite programming

analysis of a symplectic difference scheme for a coupled nonlinear schrodinger system in general proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations in this paper a symplectic difference scheme is proposed for an initial boundary value problem of a coupled nonlinear schrodinger system an important lemma and an induction argument are used to prove the unique solvability convergence and stability of numerical solutions an iterative algorithm is also proposed for the symplectic scheme and its convergence is proved numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

polynomial convergence of a new family of primal dual algorithms for semidefinite programming this paper establishes the polynomial convergence of a new class of primal dual interior point path following feasible algorithms for semidefinite programming sdp whose search directions are obtained by applying newton s method to the symmetric central path equation p x p t 1 2 p t s p 1 p x p t 1 2 mu i 0 where p is a nonsingular matrix specifically we show that the short step path following algorithm based on the frobenius norm neighborhood and the semilong step path following algorithm based on the operator 2 norm neighborhood have o sqrt n l and o nl iteration complexity bounds respectively when p i this yields the first polynomially convergent semilong step algorithm based on a pure newton direction restricting the scaling matrix p at each iteration to a certain subset of nonsingular matrices we are able to establish an o n3 2l iteration complexity for the long step path following method the resulting subclass of search directions contains both the nesterov todd direction and the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction

on two interior point mappings for nonlinear semidefinite complementarity problems extending our previous work monteiro and pang 1996 this paper studies properties of two fundamental mappings associated with the family of interior point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices the first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case these trajectories completely fill up the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program unlike the approach based on the theory of maximal monotone maps taken by shida and shindoh 1996 and shida shindoh and kojima 1995 our approach is based on the theory of local homeomorphic maps in nonlinear analysis

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

linear superposition in nonlinear equations even though the kdv and modified kdv equations are nonlinear we show that suitable linear combinations of known periodic solutions involving jacobi elliptic functions yield a large class of additional solutions this procedure works by virtue of some remarkable new identities satisfied by the elliptic functions

solving quasi variational inequalities via their kkt conditions we propose to solve a general quasi variational inequality by using its karush kuhn tucker conditions to this end we use a globally convergent algorithm based on a potential reduction approach we establish global convergence results for many interesting instances of quasi variational inequalities vastly broadening the class of problems that can be solved with theoretical guarantees our numerical testings are very promising and show the practical viability of the approach

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

geometric integration of the paraxial equation abstract evolution of solitary waves in photovoltaic photorefractive crystal satisfy the paraxial equation the paraxial equation is transformed into the symplectic structure of the infinite dimensional hamiltonian system the symplectic structure of the paraxial equation is discretizated by the symplectic method the corresponding symplectic scheme preserves conservation of discrete energy which reflects conservation of energy of the paraxial equation the symplectic scheme is applied to simulate the solitary wave behaviors of the paraxial equation evolution of the solitary waves with the different applied electric field and the different photovoltaic fields are investigated

polynomiality of primal dual algorithms for semidefinite linear complementarity problems based on the kojima shindoh hara family of directions kojima shindoh and hara proposed a family of search directions for the semidefinite linear complementarity problem sdlcp and established polynomial convergence of a feasible short step path following algorithm based on a particular direction of their family the question of whether polynomiality could be established for any direction of their family thus remained an open problem this paper answers this question in the affirmative by establishing the polynomiality of primal dual interior point algorithms for sdlcp based on any direction of the kojima shindoh and hara family of search directions we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of sdlcp

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

shape functions for mixed p version finite elements in the time domain abstract the advantages of mixed formulations are well known in that they simplify the form of the governing equations and can lead to potentially significant savings in computational effort for given accuracy likewise the p version of the finite element method is known to offer savings in computational effort for some problems in this paper shape functions for a mixed p version finite element formulation in the time domain are developed these shape functions produce a high percentage of zeros in certain coefficient matrices the mixed formulation in time used here is based on a weak form of hamilton s extended principle the present formulation allows for accurate and efficient solution of time domain problems the accuracy and power of this formulation are shown for initial value problems such as time marching and for directly obtaining the periodic response of linear systems the method is shown to yield a family of accurate unconditionally stable algorithms for initial value problems based on a simple linear oscillator

general interior point maps and existence of weighted paths for nonlinear semidefinite complementarity problems extending the previous work of monteiro and pang 1998 this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com plementarity problems over the cone of symmetric positive semidefinite matrices instead of focusing our attention on a specific map as was done in the approach of monteiro and pang 1998 this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of monteiro and pang 1998 to this general map each fundamental map leads to a family of weighted continuous trajectories which include the central trajectory as a special case hence for complementarity problems over the cone of symmetric positive semidefinite matrices the notion of weighted central path depends on the fundamental map used to represent the central path

time modes and linear systems abstract a vector space approach for generating the response of linear multi degree of freedom time invariant and time varying dynamic systems using hamilton s law of varying action hlva is presented the boundary initial condition constraints on the temporal basis function expansions of the time dependent variables has been removed while preserving hlva in its original form this provides for the broadest choice of basis functions as a result of this new approach it has been demonstrated that the response of dynamic systems are composed of temporal modes herein denoted as fundamental time modes ftm using these fundamental time modes the general solution for the system response is obtained without reference to initial conditions or forcing functions the unique response of the system is subsequently generated by using the initial conditions and forcing functions to scale the ftm this new approach is demonstrated to provide the exact analytical response as well as provide accurate numerical response solutions to a forced damped spring mass system using admissible temporal basis functions tbf the numerical response solutions using gaussian radial basis functions are compared to those obtained by using power series and third order hermite polynomials the new methodology which will be referred to as the universal method in conjunction with gaussian tbf has permitted use of transition intervals ordinarily referred to as time steps of unprecedented length larger than the period of the motion while still maintaining an accurate response solution

a note on stability of a three stage difference scheme for ordinary differential equations abstract the stability of a three stage difference scheme constructed in 1 is discussed with the construction process given out concisely at first we also present some numerical examples to test our theoretical results

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

numerical analysis of a multi symplectic scheme for a strongly coupled schrodinger system abstract in this paper we analyze a multi symplectic scheme for a strongly coupled nonlinear schrodinger equations we derive a general box scheme which is equivalent to the multi symplectic scheme by reduction method based on the general box scheme we prove that the multi symplectic scheme preserves not only the multi symplectic structure of the equation but also conservation law of mass in general the multi symplectic schemes are not conservative to energy in the nonlinear case so it is difficult to obtain the estimates of numerical solutions in norm hence proofs of convergence and stability are difficult for multi symplectic schemes of nonlinear equations a deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values and second order convergent to the exact solutions in 2 norm a fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

numerical study of the soliton waves of the coupled nonlinear schrodinger system a new six point scheme of the coupled nonlinear schrodinger cnls system is derived from the symplectic scheme to study the collision behaviors of soliton waves we prove that the new six point scheme preserves the square conservation which matches the square conservation of the cnls system numerical experiments show that the new six point scheme has excellent long time numerical behaviors from the numerical experiment results we find that the collisions of the soliton waves in the cnls system are sensitive to the collision velocity and the cross phase modulational coefficient

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

square conservative schemes for a class of evolution equations using lie group methods a new method for constructing square conservative schemes for a class of evolution equations using lie group methods is presented the basic idea is as follows first we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere second we apply lie group methods to the semidiscrete system and then square conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere both exponential and cayley coordinates are used numerical experiments are also reported

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

erratum to levenberg marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

multi stage symplectic schemes of two kinds of hamiltonian systems for wave equations abstract multi stage schemes for wave equation are constructed their stability conditions are discussed

helicopter rotor dynamics by finite element time approximation abstract starting from a variational formulation based on hamilton s principle the paper exploits the finite element technique in the time domain in order to generate by an automated procedure a numerical approximation of the nonlinear periodic equations peculiar to the dynamics of helicopter rotors with this method a unified approach is developed which can be used to solve both the response and stability problem two simple examples are presented that show how the method can be used to study the trimmed response of a fully articulated rigid blade and the mechanical instability analysis of a four bladed rotor

hamilton s law of varying action part ii direct optimal control of linear systems abstract a direct optimal procedure is developed for the control of linear time varying spatially discrete mechanical systems an assumed time modes method in which the dependent variables of the dynamics problem are expanded in terms of admissible basis functions in time is extended to include a similar representation of the control inputs this approach together with direct use of hamilton s law of varying action allows explicita prioriintegration in time of the energy related quantities leading to the algebraic equations of motion thus the conventional first order differential state equations are obviated expansion coefficients for the dependent variables and those for input forces constitute the states and controls respectively a typical performance measure of linear quadratic regulator theory is considered for the optimal control problem and it is transformed into an algebraic objective function by using the assumed time modes expansions the resulting algebraic optimality problem yields closed form solutions for the optimal control gains directly some computational aspects of the proposed methodology are pointed out numerical solutions for single and multi degrees of freedom problems are included

modelling of vibrating systems using time domain finite element method in this paper several dynamic systems are modelled using the time domain finite element method galerkin s weak principle is used to model the general second order mechanical system and is applied to the dynamics of the simple pendulum problems that arise during the approximation of the final momentum are also investigated furthermore additional dynamic analysis methods are suggested for hybrid co ordinate systems that have both slew and flexible modes the proposed methods are based on both extended hamilton s principle and galerkin s weak principle the matrix wave equation is propagated in a space domain satisfying the geometric natural boundary conditions as a result the flexible motion can be obtained and this was compatible with the applied control input a numerical example is used to demonstrate the effectiveness of the proposed modelling methods for the hybrid co ordinate systems

sample average approximation method for a class of stochastic generalized nash equilibrium problems we consider a class of stochastic generalized nash equilibrium problems sgneps under some mild conditions we reformulate the sgnep as a stochastic mixed complementarity problem smcp we apply the well known sample average approximation saa method to solve the smcp we further introduce an interior point method to solve the saa problems comprehensive convergence analysis is given as well in addition we demonstrate the proposed approach on a stochastic generalized nash equilibrium model in river basin pollution game

a predictor corrector interior point algorithm for the semidefinite linear complementarity problem using the alizadeh haeberly overton search direction this paper proposes a globally convergent predictor corrector infeasible interior point algorithm for the monotone semidefinite linear complementarity problem using the alizadeh haeberly overton search direction and shows its quadratic local convergence under the strict complementarity condition

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

construction of symplectic schemes for wave equations via hyperbolic functions sinh x cosh x and tanh x abstract hamiltonian systems are canonical systems on phase space endowed with symplectic structures the dynamical evolutions i e the phase flow of the hamiltonian systems are symplectic transformations which are area preserving the importance of the hamiltonian systems and their special property require the numerical algorithms for them should preserve as much as possible the relevant symplectic properties of the original systems feng kang 1 3 proposed in 1984 a new approach to computing hamiltonian systems from the view point of symplectic geometry he systematically described the general method for constructing symplectic schemes with any order accuracy via generating functions a generalization of the above theory and methods for canonical hamiltonian equations in infinite dimension can be found in 4 using self adjoint schemes we can construct schemes of arbitrary even order 5 these schemes can be applied to wave equation 6 7 and the stability of them can be seen in 7 8 in this paper we will use the hyperbolic functions sinh x cosh x and tanh x to construct symplectic schemes of arbitrary order for wave equations and stabilities of these constructed schemes are also discussed