a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

asymptotic mesh independence of newton s method revisited the paper presents a new affine invariant theory on asymptotic mesh independence of newton s method for discretized nonlinear operator equations compared to earlier attempts the new approach is both much simpler and more intuitive from the algorithmic point of view the theory is exemplified at finite element methods for elliptic pde problems

interior point methods in function space a primal dual interior point method for optimal control problems is considered the algorithm is directly applied to the infinite dimensional problem existence and convergence of the central path are analyzed and linear convergence of a short step path following method is established

an eulerian approach to the analysis of krause s consensus models in this paper we analyze a class of multiagent consensus dynamical systems inspired by krause s original model as in krause s model the basic assumption is the so called bounded confidence two agents can influence each other only when their state values are below a given distance threshold r we study the system under an eulerian point of view considering possibly continuous probability distributions of agents and we present original convergence results the limit distribution is always necessarily a convex combination of delta functions at least r far apart from each other in other terms these models are locally aggregating the eulerian perspective provides the natural framework for designing a numerical algorithm by which we obtain several simulations in 1 and 2 dimensions

application of the mesh independence principle to mesh refinement the mesh independence principle mip for a newton iteration applied to an operator equation and its discretization for different stepsizes h states for a fixed starting point the number of iterations of newton s method to obtain a fixed tolerance is eventually independent of h this mip is used to formulate an efficient strategy for the solution of nonlinear equations amounting to the work equivalent of two or three newton iterates for the smallest h examples are given

projective transformations for interior point algorithms and a superlinearly convergent algorithm for the w center problem the purpose of this study is to broaden the scope of projective transformation methods in mathematical programming both in terms of theory and algorithms we start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w center of a polyhedral system which stands for weighted center where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronx we prove basic results regarding contained and containing ellipsoids centered at the w center of the systemx we next shift attention to projective transformations and we exhibit an elementary projective transformation that transforms the polyhedronx to another polyhedronz and that transforms the current interior point to the w center of the transformed polyhedronz we work throughout with a polyhedral system of the most general form namely both inequality and equality costraints

a unified derivation of several error bounds for newton s process abstract this paper gives a unified derivation of the upper and lower bounds of the errors in newton s process obtained by dennis 4 tapia 27 gragg tapia 6 and recently by potra ptak 20 and miel 15 with the use of different techniques our argument is based upon the kantorovich theorem and the kantorovich recurrence relations from the proof it is concluded that the recent upper and lower bounds of miel are finer than those of potra ptak but do not improve the basic error bounds which are directly obtained from the kantorovich theorem further it is shown that the error bounds obtained with the use of the recurrence relations are the same as those obtained by the majorant principle finally results on the newton method in partially ordered space are surveyed a method is also described for estimating the componentwise errors for computed solutions

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

mixed formulation and locking in path following nonlinear analysis abstract the arc length riks strategy has rapidly become a standard tool for path following analysis of nonlinear structures due to its theoretical ability to surpass limit points the aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle locking effect intrinsic to the nonlinear nature of the problem as a consequence its sanitization has to be pursued within a reformulation of the structural model the use of a mixed stress displacement variant of the algorithm in particular appears very promising in this respect the topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in 39 it is shown that with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm

optimization of log x entropy over linear equality constraints in this paper we develop a special purpose iterative algorithm of the row action type for solving the problem of maximizing the log x entropy functional over linear equality constraints the algorithm employs projections onto hyperplanes which we call log x entropy projections a complete proof of convergence is given

nonlinear airfoil limit cycle analysis using continuation method and filtered impulse function in this study an adaptive step size control algorithm incorporated in the continuation method cm is developed and combined with the filtered impulse function method based on aerostructural solve

applications of the modified discrepancy principle to tikhonov regularization of nonlinear ill posed problems in this paper we consider the finite dimensional approximations of tikhonov regularization for nonlinear ill posed problems with approximately given right hand sides we propose an a posteriori parameter choice strategy which is a modified form of morozov s discrepancy principle to choose the regularization parameter under certain assumptions on the nonlinear operator we obtain the convergence and rates of convergence for tikhonov regularized solutions this paper extends the results which were developed by plato and vainikko in 1990 for solving linear ill posed equations to nonlinear problems

a continuation method for strongly monotone variational inequalities we consider the variational inequality problem denoted by vip x f wheref is a strongly monotone function and the convex setx is described by some inequality and possibly equality constraints this problem is solved by a continuation or interior point method which solves a sequence of certain perturbed variational inequality problems these perturbed problems depend on a parameter 0 it is shown that the perturbed problems have a unique solution for all values of 0 and that any sequence generated by the continuation method converges to the unique solution of vip x f under a well known linear independence constraint qualification licq we also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method 1998 the mathematical programming society inc published by elsevier science b v

construction of solution curves for large two dimensional problems of steady state flows of incompressible fluids this work represents a step toward advancing classical methods of bifurcation analysis in conjunction with very large scale scientific computing needed to model realistically processes which involve laminar and steady flows of incompressible fluids a robust method of analysis based on pseudoarclength continuation newton s method and direct solution of linear systems is proposed and applied to the analysis of a representative system of fluid mechanics namely the tilted lid driven cavity accurate solution curves possessing simple singular points were computed for reynolds numbers varying from 0 to 10 000 and for different angles of tilt the results demonstrate that two dimensional models with up to 1 000 000 algebraic equations can be studied feasibly using the methods described here with state of the art vector supercomputers

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

a new model for self organized dynamics and its flocking behavior we introduce a model for self organized dynamics which we argue addresses several drawbacks of the celebrated cucker smale c s model the proposed model does not only take into account the distance between agents but instead the influence between agents is scaled in term of their relative distance consequently our model does not involve any explicit dependence on the number of agents only their geometry in phase space is taken into account the use of relative distances destroys the symmetry property of the original c s model which was the key for the various recent studies of c s flocking behavior to this end we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems which covers systems with non symmetric influence matrices in particular we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with leaders

a mesh independence principle for operator equations and their discretizations the mesh independence principle asserts that when newton s method is applied to a nonlinear equation between some banach spaces as well as to some finite dimensional discretization of that equation then the behavior of the discretized process is asymptotically the same as that for the original iteration and as a consequence the number of steps required by the two processes to converge to within a given tolerance is essentially the same so far this result has been proved only for certain classes of boundary value problems in this paper a proof is presented for a general class of operator equations and discretizations it covers the earlier results and extends them well beyond the cases that have been considered before

an interval step control for continuation methods the authors present a step control for continuation methods that is deterministic in the sense that i it computationally but rigorously verifies that the corrector iteration will converge to a point on the same curve as the previous point i e the predictor corrector iteration will never jump across paths and ii each predictor step is as large as possible subject to verification that the curve is unique with the given interval extension the authors present performance data and comparisons with an approximate step control method pitcon version 6 1 a comparison of plots obtained from both step controls reveals that an approximate step control will behave erratically in situations where the interval step control leads to orderly progression along the curve this is true even if the maximum allowable stepsize for the approximate method is set to be smaller than many of the steps actually taken by the interval algorithm limitations of interval step controls are also discussed

flocking and asymptotic velocity of the cucker smale model with processing delay abstract the processing delay is incorporated into the influence function of the well known cucker smale model for self organized systems with multiple agents both symmetric and non symmetric pairwise influence functions are considered and a lyapunov functional approach is developed to establish the existence of flocking solutions for the proposed delayed cucker smale model an analytic formula is given to calculate the asymptotic flocking velocity in terms of model parameters and the variation of the position during the initial time interval

an automatic load stepping algorithm with error control summary this paper presents an algorithm for controlling the error in non linear finite element analysis which is caused by the use of finite load steps in contrast to most recent schemes the proposed technique is non iterative and treats the governing load deflection relations as a system of ordinary differential equations this permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error the latter is measured by computing the difference between two estimates of the displacement increments for each load step with the initial estimate being found from the first order euler scheme and the improved estimate being found from the second order modified euler scheme if the local truncation error exceeds a specified tolerance then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation local extrapolation is also used to predict the size of the next load step following a successful update in order to control not only the local load path error but also the global load path error the proposed scheme incorporates a correction for the unbalanced forces overall the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step because the solution scheme is non iterative and founded on successful techniques for integrating systems of ordinary differential equations it is particularly robust to illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance detailed results are presented for a variety of elastoplastic boundary value problems

flocking estimates for the cucker smale model with time lag and hierarchical leadership abstract we analyze the cucker smale model under hierarchical leadership in presence of a time delay by using a lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay we also prove a flocking estimate in the case of a free will leader these results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects that frequently appear in real situations

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

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

a method for the parametric center problem with a strictly monotone polynomial time algorithm for linear programming given a system of linear inequalities ax b dt and equalities mx g ht where the right hand sides are parametrically deformed over the scalar t the parametric center problem is to trace the parametric family of approximate solutions x t to the center problems p t where p t is the problem maximize i 1m ln bi dit aix subject to ax b dt and mx g ht we present an algorithm for tracing the parametric family of solutions x t over some given range at the kth iterate of the algorithm the point x tk is an approximate solution to the center problem p tk in an appropriate measure of approximation the value of tk is increased to tk 1 and a newton step is used to generate x tk 1 which is an approximate solution to the center problem p tk 1 the sequence of values of t exhibit at least one of the following geometric rates of change tmax tk 1 tmax tk 1 1 128m tk 1 tmin tk tmin 1 1 128m where tmin tmax is the maximum minimum number t such t

an analysis of a multi level projected steepest descent iteration for nonlinear inverse problems in banach spaces subject to stability constraints we consider nonlinear inverse problems described by operator equations in banach spaces assuming conditional stability of the inverse problem that is assuming that stability holds on a compact convex subset of the domain of the operator we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data we proceed with developing a multi level algorithm based on a nested family of compact convex subsets on which stability holds and the stability constants are ordered growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion after a finite number of steps

predictor and steplength selection in continuation methods for the navier stokes equations abstract continuation methods are in common use for solving the nonlinear equations resulting from discretizations of the navier stokes equations an imoportant ingrediet in these algorithms is the selection of the step size in the continuation parameter for certain values of the reynolds number it is shown that the step size in the reynolds number may be chosen independently of the particular predictor step used within the continuation algorithm estimates for the path derivatives are derived and used to show that the allowable step sizes are substantially greater than would be predicted by theories that apply to general nonlinear mappings

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

a globally convergent lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds we consider the global and local convergence properties of a class of lagrangian barrier methods for solving nonlinear programming problems in such methods simple bound constraints may be treated separately from more general constraints the objective and general constraint functions are combined in a lagrangian barrier function a sequence of such functions are approximately minimized within the domain defined by the simple bounds global convergence of the sequence of generated iterates to a first order stationary point for the original problem is established furthermore possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero this paper is a companion to previous work of ours on augmented lagrangian methods

minimization based formulations of inverse problems and their regularization the conventional way of formulating inverse problems such as identification of a possibly infinite dimensional parameter is via some forward operator which is the concatenation of the observation operator with the parameter to state map for the underlying model recently all at once formulations have been considered as an alternative to this reduced formulation avoiding the use of a parameter to state map which would sometimes lead to overly restrictive conditions here the model and the observation are considered simultaneously as one large system with the state and the parameter as unknowns a still more general formulation of inverse problems containing not only the reduced and all at once formulations but also the well known and highly versatile variational approach also called the kohn vogelius functional approach as special cases is to formulate the inverse problem as a minimization problem instead of an equation for the state and parameter regularization can be incorporated via imp

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

augmented lagrangian sqp methods in hilbert spaces and application to control in the coefficients problems augmented lagrangian methods for optimization problems in hilbert spaces with second order updates of the lagrangian are analyzed the relationship and the differences to sqp methods are pointed out the convergence results depend on the second order sufficient optimality condition applications to parameter estimation problems are given and illustrated by numerical results these illustrate the global behavior of the method

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

a cucker smale model with noise and delay a generalization of the cucker smale model for collective animal behavior is investigated the model is formulated as a system of delayed stochastic differential equations it incorporates two additional processes which are present in animal decision making but are often neglected in modeling i stochasticity imperfections of individual behavior and ii delayed responses of individuals to signals in their environment sufficient conditions for flocking for the generalized cucker smale model are derived by using a suitable lyapunov functional as a by product a new result regarding the asymptotic behavior of delayed geometric brownian motion is obtained in the second part of the paper results of systematic numerical simulations are presented they not only illustrate the analytical results but hint at a somehow surprising behavior of the system namely that the introduction of an intermediate time delay may facilitate flocking

higher order predictors and adaptive steplength control in path following algorithms for tracing the solution path of an underdetermined nonlinear system f y 0 with f r n 1 to r n predictor corrector methods such as the euler newton continuation method have proved to be reliable tools in order to reduce the computational work a more sophisticated pc method is proposed and investigated the method is based on secant length as the local parameter and takes either the second order taylor polynomial or the third order 2 1 pade approximation as the predictor whereas in both cases the directional derivatives containing f and f are efficiently approximated by a few function values it uses a newton type corrector iteration that takes into account the special parametrization and applies an adaptive steplength control strongly related to the higher order predictors numerical tests show that the new algorithm performs very efficiently and essentially reduces the number of evaluations and factorizations of jacobian matrices compared with other pc methods known from lite

non uniqueness of critical points in variational data assimilation abstract in this paper we apply the 4d var data assimilation scheme to the initialization problem for a family of quasilinear evolution equations the resulting variational problem is non convex so it need not have a unique minimizer we comment on the implications of non uniqueness for numerical applications then prove uniqueness results in the following situations 1 the observational times are sufficiently small 2 the prior covariance is sufficiently small we also give an example of a data set where the cost functional has a critical point of arbitrarily large morse index thus demonstrating that the geometry can be highly nonconvex even for a relatively mild nonlinearity

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

the method of resultants for computing real solutions of polynomial systems a new method for determining the real solutions to a set of polynomial equations is presented it is based on the theory of multiresultants the inherently unstable calculation of the determinant is replaced by a stable minimization procedure which is able to take advantage of the sparseness of the resultant matrix two numerical examples illustrate the method the paper contains preliminary work which demonstrates the feasibility of the given approach

unified approach to flutter equations a general formulation of the flutter equation includes gyroscopics viscous damping active controls mach and complex reduced frequency based unsteady aerodynamics quasi linear approximated structural nonlinearities and parameterized matrices solving the flutter equation usually involves a search for neutral stability points the boundary between stability and instability followed by variations in the neutral stability point with various parameters a technique for solving the flutter equation under these conditions must necessarily be capable of solving systems of nonlinear parameterized equations over a range of parameter values continuation methods are specifically designed for solving such problems a continuation method is presented that is capable of solving general formulations of the flutter equation for neutral stability parameter variations optimization model tuning and describing function flutter analyses including an efficient technique for assessing limit cycle stability

asymptotic mesh independence of newton galerkin methods via a refined mysovskii theorem the paper presents a theoretical characterization of the often observed asymptotic mesh independence of newton s method which means that newton s method applied to discretized operator equations behaves essentially the same for all sufficiently fine discretizations unlike previous theoretical approaches the theory presented here does not need any uniform lipschitz assumptions which may be hard to verify in realistic problems or in some cases may not hold at all moreover this approach leads to a significantly simpler presentation the new results are obtained by means of a new refined newton mysovskii theorem in affine invariant formulation this theorem will be of interest in a wider context because it gives both existence and uniqueness of the solution and quadratic convergence for sufficiently good starting points attention is restricted to galerkin approximations even though similar results should hold for finite difference methods but corresponding proofs would certainly be more technical

semilocal convergence for halley s method under weak lipschitz condition the semilocal convergence properties of halley s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak lipschitz condition the method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator an application to a nonlinear hammerstein integral equation of the second kind is provided

following a balanced trajectory from an infeasible point to an optimal linear programming solution with a polynomial time algorithm this paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem a class of trajectories for the problem is defined based on the notion of a balanced solution to the problem given a prespecified positive balancing constant an infeasible solution x is said to be balanced if the optimal value gap is less than or equal to times the infeasibility gap mathematically this can be written as ctx z tx where the linear form tx is the phase i objective function and z is the optimal objective value of the linear program the concept of a balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program each trajectory has the property that all points on or near the trajectory in a suitable metric are balanced the main thrust of the paper is the development of an algorithm that traces a given balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time more specifically the algorithm allows for fixed improvement in the bound on the phase i and the phase ii objectives in on newton steps

a combined homotopy interior point method for convex nonlinear programming abstract in this paper we present a new interior point method combined homotopy interior point method chip method for convex nonlinear programming without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set we prove that for any 0 an solution of the problem can be obtained by the chip method to our knowledge strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well known center path following method therefore the chip method essentially reduces the assumptions of the center path following method and can be applied to more general problems

arclength numerical continuation in free boundary flow abstract using helmholtz s wake model we reduce the study of the free boundary flow past an obstacle consisting of an arc of circle to the investigation of a hammerstein nonlinear integral equation depending on a real parameter the papers dedicated to this problem investigated the case 0 which corresponds to a convex obstacle with respect to the incoming fluid herein we apply for the first time in the literature the arclength continuation method for the case 0 corresponding to a concave arc of a circle for 0 the existence and the uniqueness of the solution was demonstrated but for 0 depending on its value compared to the one of a turning point the integral equation has either no solution or two distinct solutions corresponding to two different obstacles we numerically calculate the free lines the velocity field and the stream lines a diagram of the drag coefficient versus the arc measure for both convex and concave obstacles suggests us to draw some conclusions concerning the optimization of the blades of a vertical axis savonius wind turbine

mesh independence of semismooth newton methods for lavrentiev regularized state constrained nonlinear optimal control problems a class of nonlinear elliptic optimal control problems with mixed control state constraints arising e g in lavrentiev type regularized state constrained optimal control is considered based on its first order necessary optimality conditions a semismooth newton method is proposed and its fast local convergence in function space as well as a mesh independence principle for appropriate discretizations are proved the paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing lavrentiev parameter the latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the lavrentiev parameter

on weakly nonlinear inverse problems in this paper the class of weakly nonlinear inverse problems is introduced these problems are characterized by the property that the second derivative of the parameter to observation mapping can be bounded by the square of the first derivative of that mapping using geometric techniques it is shown that weakly nonlinear inverse problems behave similarly to linear inverse problems in particular their tikhonov regularization leads to a family of quadratically well posed problems examples involving the determination of source terms in semilinear reaction diffusion equations are given

the inverse kinematics problem of spatial 4p3r robot manipulator by the homotopy continuation method with an adjustable auxiliary homotopy function abstract in this paper the concept of an adjustable auxiliary homotopy function for the homotopy continuation method is presented by means of adjusting the auxiliary function we can solve non linear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the newton raphson method and so on

local phase synchronization and clustering for the delayed phase coupled oscillators with plastic coupling abstract in this paper we investigate the local synchronization problem for delayed phase coupled oscillators with plastic coupling also referred as a generalized kuramoto model by linearizing the phase coupled oscillators system at a special solution an approximation phase coupled system with delay effects is deduced moreover we find the multiplicity of semisimple zero eigenvalue for coupling strength coefficients matrix and the rank of vandermonde matrix associated initial values are two sensitive factors the former is closely related to the maximum number of clusters phase of linear coupled system and the latter determines clusters phase synchronization as results both frequency synchronization criteria and phase synchronization criteria are established in this literature also the final synchronization frequency and phase are formulated in terms of coupling strength initial values and time delay

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

symmetry properties for positive solutions of elliptic equations with mixed boundary conditions in this paper we establish symmetry results for positive solutions of semilinear elliptic equations of the type u f u 0 with mixed boundary conditions in bounded domains in particular we prove that any positive solution u of such an equation in a spherical sector r is spherically symmetric if the amplitude of the sector is such that 0 by constructing counterexamples we show that this result is optimal in the sense that it does not hold for sectors be r with amplitude 2 more general symmetry properties are established for positive solutions in domains with axial symmetry these results extend the well known theorems of b gidas w m ni and l nirenberg comm math phys 68 1979 209 243 to sector like domains and mixed boundary conditions

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

a combined phase i phase ii projective algorithm for linear programming we devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution inclusion of such a constraint allows the algorithm to be applied to a possibly infeasible standard form linear program without the addition of any bigm terms or conversion to a primal dual problem

a potential reduction algorithm with user specified phase i phase ii balance for solving a linear program from an infeasible warm start this paper develops a potential reduction algorithm for solving a linear programming problem directly from a warm start initial point that is neither feasible nor optimal the algorithm is an interior point variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement phase i and objective value improvement phase ii the key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense given a prespecified balancing parameter beta 0 the algorithm maintains the following phase i phase ii beta balancing constraint throughout left c t x z right beta xi t x where c t x is the objective function z is the unknown optimal objective value of the linear program and xi t x measures the infeasibility of the current iterate x this balancing constraint can be used to either emphasize rapid attainment of feasibility set beta large

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

theoretical efficiency of a shifted barrier function algorithm for linear programming abstract this paper examines the theoretical efficiency of solving a standard form linear program by solving a sequence of shifted barrier problems of the form minimize x c t x e j 1 n ln x j eh j s t a x b x eh 0 for a given and fixed shift vector h 0 and for a sequence of values of 0 that converges to zero the resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program the advantage of using the shifted barrier approach is that a starting feasible solution is unnecessary and there is no need for a phase i phase ii approach to solving the linear program either directly or through the addition of an artificial variable furthermore the algorithm can be initiated with a warm start i e an initial guess of a primal solution x that need not be feasible the number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible and optimal the number of iterations will also depend on the judicious choice of the shift vector h if an approximate center of the dual feasible region is known in advance then h can be chosen so that the guaranteed fractional decrease in at each iteration is 1 1 6 n which contributes afactor of 6 n to the number of iterations needed to solve the problem

mesh independence for an augmented lagrangian sqp method in hilbert spaces an augmented lagrangian sqp algorithm for optimal control of differential equations in hilbert spaces is analyzed this algorithm has second order convergence rate provided that a second order sufficient optimality condition is satisfied the internal approximation of this method is investigated and convergence results are presented a mesh independence principle for the augmented lagrangian sqp method is proved which asserts that asymptotically the infinite dimensional algorithm and finite dimensional discretizations have the same convergence property more precisely for sufficiently small mesh size there is at most a difference of one iteration step between the number of steps required by the infinite dimensional method and its discretization to converge within a given tolerance varepsilon 0 the theoretical results are demonstrated by two optimal control problems for the burgers equation

parametrization via secant length and application to path following a possible way for parametrizing the solution path of the nonlinear systemh u 0 h n 1 n consists of using the secant length as parameter this idea leads to a quadratic constraint by which the parameter is introduced a newton like method for computing the solution for a given parameter is proposed where the nonlinear system is linearized at each iterate but the quadratic parametrizing equation is exactly satisfied the localq quadratic convergence of the method is proved and some hints for implementing the algorithm are given

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

a parallel and adaptive continuation method for semilinear bifurcation problems in this paper we propose an efficient continuation method for the following of solution branches of parametrized semilinear elliptic problems in order to limit the time required for the computation of a whole branch of accurate solutions we combine a mesh adaptation strategy with parallel programming on a computer network the first computer is devoted to the continuation on a coarse mesh the others share out the correction of every roughly generated solution on a sequence of a posteriori refined meshes a numerical example illustrating the performance of the method is given