p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

basis and partition identification for quadratic programming and linear complementarity problems optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions maximally complementary solutions can be characterized by optimal partitions on the other hand the solutions provided by simplex based pivot algorithms are given in terms of complementary bases a basis identification algorithm is an algorithm which generates a complementary basis starting from any complementary solution a partition identification algorithm is an algorithm which generates a maximally complementary solution and its corresponding partition starting from any complementary solution in linear programming such algorithms were respectively proposed by megiddo in 1991 and balinski and tucker in 1969 in this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices the presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

implicit algorithm for finite deformation hypoelastic viscoplasticity in fcc metals an implicit objective stress update algorithm is proposed for a hypoelastic viscoplastic model a thermal dynamic yield function which is derived based on the thermal activation analysis and dislocation interaction mechanisms is used along with the consistency approach and the framework of additive viscoplasticity in deriving the proposed model for fcc metals the corotational formulation approach is utilized in developing the proposed model in the finite deformation field for the case of the newton raphson iteration method a new expression for the consistent algorithmic tangent stiffness matrix of rate dependent metals is derived by direct linearization of the stress update algorithm finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program abaqus numerical implementation for a simple tensile problem is used for validating the material parameters of the ofhc copper under low and high strain rates and temperatures the numerical results of the adiabatic true stress true strain curves compare very well with the experimental data the effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh independent results copyright 2006 john wiley sons ltd

the interpolation theorem for narrow quadrilateral isoparametric finite elements the interpolation theorem for convex quadrilateral isoparametric finite elements is proved in the case when the condition varrho k h k ge varrho 0 0 is not satisfied where h k is the diameter of the element k and varrho k is the radius of an inscribed circle in k the interpolation error is o h k 2 in the l 2 k norm and o h k in the h 1 k norm provided that the interpolated function belongs to h 2 k in the case when the long sides of the quadrilateral k are parallel the constants appearing in the estimates are evaluated

error estimates for the raviart thomas interpolation under the maximum angle condition the classical error analysis for the raviart thomas interpolation on triangular elements requires the so called regularity of the elements or equivalently the minimum angle condition however in the lowest order case optimal order error estimates have been obtained in g acosta and r g duran siam j numer anal 37 2000 pp 18 36 replacing the regularity hypothesis by the maximum angle condition which was known to be sufficient to prove estimates for the standard lagrange interpolation in this paper we prove error estimates on triangular elements for the raviart thomas interpolation of any order under the maximum angle condition also we show how our arguments can be extended to the three dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in g acosta and r g duran siam j numer anal 37 2000 pp 18 36

finite deformation plasticity based on the additive split of the rate of deformation and hyperelasticity abstract finite deformation plasticity formulation based on additive split of rate of deformation and hyperelasticity is presented this approach is valid for finite elastic and plastic strains while rendering the choice and numerical integration of objective stress rates superfluous as the results are automatically objective for small elastic strains our method reduces to the classical hypoelastic corotational formulation provided that the dienes objective stress rate is employed while in the absence of inelastic deformation it coincides with the hyper elastic formulation the validity of the model has been examined on four test problems and the numerical results were found to be in good agreement with either the exact solution or experimental data

analysis of the anisotropic viscoplastic damage response of composite laminates continuum basis and computational algorithms the mathematical structure underlying the rate equations of a recently developed constitutive model for the coupled viscoplastic damage response of anisotropic composites is critically examined in this regard a number of tensor projection operators have been identified and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the euler fully implicit integration method in particular this facilitated i the derivation of explicit expressions of the consistent material tangent stiffnesses that are valid for both three dimensional as well as subspace e g plane stress formulations ii the implications of the symmetry or unsymmetry properties of these tangent operators from a thermodynamic standpoint and iii the development of an effective time step control strategy to ensure accuracy and convergence of the solution in addition the special limiting case of inviscid elastoplasticity is treated the results of several numerical simulations are given to demonstrate the effectiveness of the schemes developed

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

anisotropic error bounds of lagrange interpolation with any order in two and three dimensions abstract in this paper using the newton s formula of lagrange interpolation we present a new proof of the anisotropic error bounds for lagrange interpolation of any order on the triangle rectangle tetrahedron and cube in a unified way

an implicit integration algorithm with a projection method for viscoplastic constitutive equations robinson s viscoplastic model a representative of the so called overstress models is integrated by use of the generalized midpoint rule the solution of the non linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with newton s method consistent tangent moduli are calculated and the quadratic convergence of the global newton equilibrium iteration is shown the time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration various numerical examples are considered to demonstrate the efficiency of the methods

uniform error estimates for certain narrow lagrange finite elements error estimates of dupont and scott are used to derive uniform error estimates for lagrange finite elements in jrn n 2 under the following conditions 1 the elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis and 2 the space of approximating functions ft in each element t must include the space of polynomials of degree 1 n 2 if n is odd this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded for example if n 3 the usual requirement with shape regularity is that each ft contain all first degree polynomials the result of this paper requires that each ft contain all quadratic polynomials and consequently does not apply to linear courant elements in tetrahedrons or trilinear tensor elements in rectangular boxes counterexamples in these two cases are included

an exponential example for terlaky s pivoting rule for the criss cross simplex method recently t terlaky has proposed a new pivoting rule for the criss cross simplex method for linear programming and he proved that his rule is convergent in this note we show that the required number of iterations may be exponential in the number of variables and constraints of the problem

two stress update algorithms for large strains accuracy analysis and numerical implementation two algorithms for the stress update i e time integration of the constitutive equation in large strain solid mechanics are compared from an analytical point of view the order of the truncation error associated to the numerical integration is deduced for each algorithm a priori using standard numerical analysis this accuracy analysis has been performed by means of a convected frame formalism which also allows a unied derivation of both algorithms in spite of their inherent dierences then the two algorithms are adapted from convected frames to a xed cartesian frame and implemented in a small strain nite element code the implementation is validated by means of a set of simple deformation paths simple shear extension extension and compression extension and rotation and two benchmark tests in non linear mechanics the necking of a circular bar and a shell under ring loads in these numerical tests the observed order of convergence is in very good agreement with the theoretical order of convergence thus corroborating the accuracy analysis 1997 john wiley sons ltd

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

implementation of cyclic plasticity models based on a general form of kinematic hardening this paper deals with implementation of cyclic plastic constitutive models in which a general form of strain hardening and dynamic recovery is employed to represent the multilinear as well as non linear evolution of back stress first in order to incorporate such a general form of kinematic hardening in finite element methods successive substitution and its convergence are discussed for implicitly integrating stress moreover a new expression of consistent tangent modulus is derived by introducing a set of fourth rank constitutive parameters into discretized kinematic hardening then the constitutive parameters introduced are specified in three cases of the general form of kinematic hardening the three cases have distinct capabilities of simulating ratcheting and cyclic stress relaxation numerical examples are given to verify the convergence in successive substitution and the new expression of consistent tangent stiffness error maps for implicitly integrating stress under non proportional as well as proportional loading are also given to show that the multilinear case of the general form provides high accuracy even if strain increment is very large copyright 2002 john wiley sons ltd

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

viscoplasticity for instabilities due to strain softening and strain rate softening three viscoplastic approaches are examined in this paper first the overstress viscoplastic models i e the perzyna model and the duvaut lions model are outlined next a consistency viscoplastic approach is presented in the consistency model a rate dependent yield surface is employed while the standard kuhn tucker conditions for loading and unloading remain valid for this reason the yield surface can expand and shrink not only by softening or hardening e ects but also by softening hardening rate e ects a full algorithmic treatment is presented for each of the three models including the derivation of a consistent tangential sti ness matrix based on a limited numerical experience it seems that the consistency model shows a faster global convergence than the overstress approaches for softening problems all three approaches have a regularising e ect in the sense that the initial value problem remains well posed the width of the shear band is determined by the material parameters and if present by the size of an imperfection a relation between the length scales of the three models is given furthermore it is shown that the consistency model can properly simulate the so called s type instabilities which are associated with the occurrence of travelling portevin le chatelier bands 1997 john wiley sons ltd

a generalized elastoplastic plate theory and its algorithmic implementation in continuum mechanics within specific classes of problems one or two dimensional theories are often simpler to apply than the more complete three dimensional one this is for example the case of thin bodies such as plates or shells which may be studied using appropriate two dimensionai theories within this approach the reduction of the dimension is traded for a loss of information relative to the motion in the transverse direction for example in the case of non linear material behaviour classical plasticity plate theories are usually not able to model the effects related to the spreading of plasticity through the cross section in the present paper we discuss a generalized plasticity plate model which can be used to reproduce some of the three dimensional effects in a two dimensional setting we present the continuous and the discrete time model including both isotropic and kinematic hardening mechanisms moreover the form of the tangent matrix consistent with the discrete model is addressed finally some examples cantilever beam clamped circular plate and clamped square plate under monotonic and cyclic loading are studied numerically using a three dimensional classical plasticity theory a classical plasticity plate theory and the proposed plate theory the generalized plasticity plate model matches the three dimensional response with greater accuracy than the classical plasticity plate model

error estimates for 3 d narrow finite elements we obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three dimensional narrow elements first for the lagrange interpolation we prove optimal error estimates both in order and regularity in l p for p 2 for p 2 it is known that this result is not true applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than h 2 these results are valid both for tetrahedral and rectangular elements second for the case of rectangular elements we obtain optimal in order and regularity error estimates for an average interpolation valid for functions in w 1 s p with 1 p and 0 s 1

on the finiteness of the criss cross method abstract in this short paper we prove the finiteness of the criss cross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically this new proof immediately suggests the possibility of relaxing the pivoting selection in the criss cross method without sacrificing the finiteness

a new algorithm for quadratic programming abstract we present a new finite algorithm for quadratic programming our algorithm is based on the solution procedures of linear programming pivoting bland s rule hungarian methods criss cross method however this method does not require the enlargement of the basic tableau as frank wolfe method does it can be considered as a feasible point active set method we solve linear equation systems in oder to reach an active constraint set complementary solutions and we solve a feasibility problem in order to check that optimality can be reached on this active set or to improve the actual solution this algorithm is a straightforward generalization of klafszky s and terlaky s hungarian method it has nearly the same structure as ritter s algorithm which is based on conjugate directions but it does not use conjugate directions

on polynomial approximation in sobolev spaces we obtain new bounds for the error in the polynomial approximation in sobolev spaces in an open bounded set which is star shaped with respect to every point in a set of positive measure b subset omega this estimate follows by applying the hardy littlewood maximal function

new variants of the criss cross method for linearly constrained convex quadratic programming abstract in this paper s zhang s eur j oper res 116 1999 607 new and more flexible criss cross type algorithms with lifo and most often selected variable pivot rules are generalized for linearly constrained convex primal dual quadratic programming problems these criss cross type algorithms are different from the one described in klafszky and terlaky math oper und stat ser optim 24 1992 127 even though the finiteness proof of these new criss cross type algorithms is similar to the original one for the algorithm of klafszky and terlaky in the sense that both these proofs are based on the orthogonality theorem more cases have to be considered due to the flexibility of pivot lifo most often selected variable rules which requires a deeper and more careful analysis when the primal dual problem is a linear programming problem no quadratic terms in the objective function the structure of the corresponding linear complementarity problem is simpler i e the matrix of the problem is skew symmetric for such problem pairs our proof of finiteness simplifies to the proof of illes and meszaros yugoslav j oper res 11 2001 17 and provides a new finiteness proof for s zhang s criss cross type algorithms

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

the linear complimentarity problem sufficient matrices and the criss cross method specially structured linear complementarity problems lcps and their solution by the criss cross method are examined the criss cross method is known to be finite for lcps with positive semidefinite bisymmetric matrices and with p matrices it is also a simple finite algorithm for oriented matroid programming problems recently cottle pang and venkateswaran identified the class of column row sufficient matrices they showed that sufficient matrices are a common generalization of p and psd matrices cottle also showed that the principal pivoting method with a clever modification can be applied to row sufficient lcps in this paper the finiteness of the criss cross method for sufficient lcps is proved further it is shown that a matrix is sufficient if and only if the criss cross method processes all the lcps defined by this matrix and all the lcps defined by the transpose of this matrix and any parameter vector

the s monotone index selection rules for pivot algorithms of linear programming abstract in this paper we introduce the concept of s monotone index selection rule for linear programming problems we show that several known anti cycling pivot rules like the minimal index last in first out and the most often selected variable pivot rules are s monotone index selection rules furthermore we show a possible way to define new s monotone pivot rules we prove that several known algorithms like the primal dual simplex mbu simplex algorithms and criss cross algorithm with s monotone pivot rules are finite methods we implemented primal simplex and primal mbu simplex algorithms in matlab using three s monotone index selection rules the minimal index the last in first out lifo and the most often selected variable mosv index selection rule numerical results demonstrate the viability of the above listed s monotone index selection rules in the framework of pivot algorithms

the finite criss cross method for hyperbolic programming in this paper the finite criss cross method is generalized to solve hyperbolic fractional linear programming problems just as in the case of linear or quadratic programming the criss cross method can be initialized with any not necessarily feasible basic solution it is known that if the feasible region of the problem is unbounded then some of the known algorithms fail to solve the problem our criss cross algorithm does not have such drawback finiteness of the procedure is proved under the usual mild assumptions some small numerical examples illustrate the main features of the algorithm and show that our method generates different iterates than other earlier published methods

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

convergence of the newton raphson algorithm in elastic plastic incremental analysis a spatially continuous time discrete formulation of the loading of an elastic perfectly plastic body governed by a von mises yield condition is presented it is assumed that incremental changes in strain occur along minimum work paths which is equivalent to a backward difference implicit integration algorithm or the radial return method

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

the role of pivoting in proving some fundamental theorems of linear algebra abstract we give a constructive proof for some classical theorems of linear algebra steinitz rank of matrices rouche kronecker capelli farkas wey minkowski the construction is based on pivoting and its most important part is the pivot selection rule which is based on bland s least index resolution

new variants of finite criss cross pivot algorithms for linear programming abstract in this paper we generalize the so called first in last out pivot rule and the most often selected variable pivot rule for the simplex method as proposed in s zhang operations research letters 10 1991 189 192 to the criss cross pivot setting where neither the primal nor the dual feasibility is preserved the finiteness of the new criss cross pivot variants is proven

on extremal behaviors of murty s least index method in this small note we observe some extremal behaviors of murty s least index method for solving linear complementarity problems in particular we show that the expected number of steps for solving murty s exponential example with a random permutation of variable indices is exactly equal ton wheren is the size of the input square matrix

a note on the edmonds fukuda pivoting rule for simplex algorithms abstract the pivot selection rule of edmonds fukuda for simplex algorithms was described by jack edmonds in 1982 at the xith international symposium on mathematical programming also the essential characteristics of the rule namely that it prevents cycling in simplex algorithms was proved by edmonds contrary to other pivot selection rules however it is not obvious that the edmonds fukuda rule maintains feasibility of the solution during the solution process this was stated without proof by edmonds and no proof has hitherto appeared a proof is given in the present note also edmonds proof of the anti cycling property is stated finally the relation to the recursive rule ii method of bland is discussed