superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

improved complexity using higher order correctors for primal dual dikin affine scaling in this paper we show that the primal dual dikin affine scaling algorithm for linear programming of jansen roos and terlaky enhances an asymptotical o sqrt n l complexity by using corrector steps we also show that the result remains valid when the method is applied to positive semi definite linear complementarity problems

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

on the choice of parameters for power series interior point algorithms in linear programming in this paper we study higher order interior point algorithms especially power series algorithms for solving linear programming problems since higher order differentials are not parameter invariant it is important to choose a suitable parameter for a power series algorithm we propose a parameter transformation to obtain a good choice of parameter called ak parameter for general truncated powerseries approximations we give a method to find ak parameter this method is applied to two powerseries interior point algorithms which are built on a primal dual algorithm and a dual algorithm respectively computational results indicate that these higher order power series algorithms accelerate convergence compared to first order algorithms by reducing the number of iterations also they demonstrate the efficiency of thek parameter transformation to amend an unsuitable parameter in power series algorithms

reflectable bases for affine reflection systems abstract the notion of a root base together with its geometry plays a crucial role in the theory of finite and affine lie theory however it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems in this work we consider the notion of a reflectable base for an affine reflection system r a reflectable base for r is a minimal subset of roots such that the non isotropic part of the root system can be recovered by reflecting roots of relative to the hyperplanes determined by we give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types excluding types e 6 7 8 as a by product of our results we show that if the root system under consideration is locally finite then any reflectable base is an integral base

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

logarithmic sumt limits in convex programming the limits of a class of primal and dual solution trajectories associated with the sequential unconstrained minimization technique sumt are investigated for convex programming problems with non unique optima logarithmic barrier terms are assumed for linear programming problems such limits of both primal and dual trajectories are strongly optimal strictly complementary and can be characterized as analytic centers of loosely speaking optimality regions examples are given which show that those results do not hold in general for convex programming problems if the latter are weakly analytic bank et al 3 primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability that class of programming problems contains faithfully convex linear and convex quadratic programming problems as strict subsets in the differential case dual trajectory limits can be characterized similarly albeit under different conditions one of which suffices for strict complementarity

underlying paths in interior point methods for the monotone semidefinite linear complementarity problem an interior point method defines a search direction at each interior point of the feasible region the search directions at all interior points together form a direction field which gives rise to a system of ordinary differential equations odes given an initial point in the interior of the feasible region the unique solution of the ode system is a curve passing through the point with tangents parallel to the search directions along the curve we call such curves off central paths we study off central paths for the monotone semidefinite linear complementarity problem sdlcp we show that each off central path is a well defined analytic curve with parameter ranging over 0 and any accumulation point of the off central path is a solution to sdlcp through a simple example we show that the off central paths are not analytic as a function of sqrt mu and have first derivatives which are unbounded as a function of at 0 in general on the other hand for the same example we can find a subset of off central paths which are analytic at 0 these nice paths are characterized by some algebraic equations

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

karmarkar s linear programming algorithm and newton s method this paper describes a full dimensional version of karmarkar s linear programming algorithm theprojective scaling algorithm which is defined for any linear program in n having a bounded full dimensional polytope of feasible solutions if such a linear program hasm inequality constraints then it is equivalent under an injective affine mappingj n m to karmarkar s original algorithm for a linear program in m havingm n equality constraints andm inequality constraints karmarkar s original algorithm minimizes a potential functiong x and the projective scaling algorithm is equivalent to that version of karmarkar s algorithm whose step size minimizes the potential function in the step direction

optimizing over three dimensional subspaces in an interior point method for linear programming abstract interior point algorithms for solving linear programming problems are considered a three dimensional method is developed that at each iteration solves a subproblem based on minimizing the cost function on low dimensional cross sections of the feasible region the generators for the three dimensional subproblem include the dual affine search direction and two higher order search directions one of the higher order directions is a third order correction to the newton recentering direction and the other is a correction to the dual affine direction that is motivated by the use of rank one updates of the second derivative information numerical results are presented for this method that indicate a nearly 20 reduction in cpu time compared to our best dual affine implementation

analytical properties of the central path at boundary point in linear programming we study the properties of the weighted central paths in linear programming we consider each path as the function of the parameter 0 where the value at d 0 corresponds to the limit point at the boundary of the feasible set we calculate the recursive formulas for the central path derivatives of all orders valid at each 0 we establish the geometric growth of the derivatives and consequently the analyticity of the weighted central path at 0 this paper also provides the analysis of limiting behavior of those projection operators which often appear in the interior point methods the central trajectory is one of the most important concepts in the interior point methods for linear programming the central trajectory or the central path is a distinguished curve that is interior to the feasible set and tends to an optimal point most interior points algorithms follow or are related to the central trajectory although the central path concept is essential for the design and analysis of algorithms relatively few papers study properties of this curve itself for a review on this topic we refer to 3 recently zhao and zhu 16 have studied some analytical properties of the central trajectory in the interior of the feasible set they have established an upper bound for the high order derivatives of the weighted central path and estimated the convergence radius of the corresponding taylor series they have also pointed out the relationship of these properties with the complexity of a path following algorithm where the number of iterations is estimated by a curvature integral see also 15 further guler 3 has analysed some limiting properties of the central trajectory derivatives with respect to the parameter 0 he has established the existence of finite limits for derivatives of all orders as tends to infinity that is the central path stays in the interior and to zero the central path tends to the boundary these results have applications to the construction of polynomial time algorithms which are also locally fast a surprising result of this study is that all derivatives of order 1t end to 0 as 1 which helps to explain why interior point methods for linear programming require so few iterations see 3

minimum seeking properties of analog neural networks with multilinear objective functions in this paper we study the problem of minimizing a multilinear objective function over the discrete set

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

on the continuous trajectories for a potential reduction algorithm for linear programming for a possibly degenerate linear program minx ctx ax b x 0 a is an m n real matrix b rm and c rn whose optimal value is 0 we study the limiting behavior of the trajectories of the family of vector fields i qx i xpax xc q i1ctx1 for all values of q n where x is the diagonal matrix associated with x and pax is the projection operator onto the null space of ax a polynomial algorithm based on the directions i qx has been presented by gonzaga 6 when q n or q n n we show that all trajectories of i qx converge to the unique center of the optimal face of the given linear program when this face consists of a unique vertex it is shown that any trajectory of i qx approaches this vertex along the same direction when the optimal face consists of more than one point we show that there is a threshold value i 0 such that for q i most of the trajectories of i qx converge to the center tangentially to the optimal face and that the direction of approach of a trajectory of i qx depends on the initial condition for q i q i the trajectories of i qx converge to the center along a unique direction along several directions which depend on the initial condition forming a positive angle with the optimal face

containing and shrinking ellipsoids in the path following algorithm we describe a new potential function and a sequence of ellipsoids in the path following algorithm for convex quadratic programming each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors furthermore the volumes of the ellipsoids shrink at the ratio 2 omega sqrt n in comparison to 2 1 in karmarkar s algorithm and 2 1 n in the ellipsoid method we also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming

affine scaling for linear programs with free variables the affine scaling modification of karmarkar s algorithm is extended to solve problems with free variables this extended primal algorithm is used to prove two important results first the geometrically elegant feasibility algorithm proposed by chandru and kochar is the same algorithm as the one obtained by appending a single column of residuals to the constraint matrix second the dual algorithm as first described by adler et al is the same as the extended primal algorithm applied to the dual

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

the classification of locally finite split simple lie algebras one of the great early achievements in lie theory is the classification of the finite dimensional simple complex lie algebras by w killing and e cartan more than 50 years later new aspects were added to this classification by the theory of coxeter groups and the visualization of the classification in terms of dynkin diagrams furthermore serre s description of the simple lie algebras by generators and relations provided a direct way to construct the lie algebras from the cartan matrix corresponding to the choice of a root base i e a system of simple roots

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations we present an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations wherem is the number of constraints andn is the number of variables each operation is performed to a precision of o l bits l is bounded by the number of bits in the input the worst case running time of the algorithm is better than that of karmarkar s algorithm by a factor of sqrt m n

the nonlinear geometry of linear programming iii projective legendre transform coordinates and hilbert geometry this paper studies projective scaling trajectories which are the tra jectories obtained by following the infinitesimal version of karmarkar s linear programming algorithm a nonlinear change of variables projective legendre transform coordinates is introduced to study these trajectories the projective legendre transform mapping has a coordinate free geometric interpretation in terms of the notion of centering by a projective transformation let h be a set of linear programming constraints aj x bj 1 s j s m on rn such that its polytope of feasible solutions ph is bounded and contains 0 in its interior the projective legendre transform mapping i h is given by m a where ph x l b j 1 aj x i x ph x h m ph x x here ph x is the legendre transform coordinate mapping introduced in part ii l h x is a one to one and onto mapping of the interior of the feasible solu tion polytope int ph to the interior of its polar polytope int p the set of projective scaling trajectories with objective function c x co are mappcd under l h to the set of straight line segments in int p passing through the boundary point clco of p as a consequence the projective scaling tra jectories for all objective functions can be interpreted as the complete set of geodesics actually distinguished chords of a projectively invariant metric ge ometry on int ph which is isometric to hilbert geometry on the interior of the polar polytope p

a new vector field method for eigen decomposition of symmetric matrices abstract we study a new approach to eigen decomposition of symmetric matrices called the vector field method in contrast to the methods currently in use the vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere the optimization problems embedded in each iteration of the vector field algorithms admit closed form solutions making the vector field approach promisingly efficient besides establishing the general convergence results about the vector field method and the complexity bounds for computing approximations to eigenvectors we also computationally compare the performance of a family of algorithms called the recursive vector field algorithms with that of the power method the vector field method like the power method offers a significant advantage in that it can exploit sparseness to speed up computation even further the convergence analysis that we present also suggests that one could speed up the vector field algorithms through preprocessing that contracts the spectral radius of the given matrix

long step strategies in interior point primal dual methods in this paper we analyze from a unique point of view the behavior of path following and primal dual potential reduction methods on nonlinear conic problems we demonstrate that most interior point methods with o root n ln 1 epsilon efficiency estimate can be considered as different strategies of minimizing a convex primal dual potential function in an extended primal dual space their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path it is shown that the neighborhood of this path is a region of the fastest decrease of the potential therefore the long step path following methods are in a sense the best potential reduction strategies we present three examples of such long step strategies we prove also an efficiency estimate for a pure primal dual potential reduction method which can be considered as an implementation of a penalty strategy based on a functional proximity measure using the convex primal dual potential we prove efficiency estimates for karmarkar type and dikin type methods as applied to a homogeneous reformulation of the initial primal dual problem

a class of primal affine scaling algorithms abstract we obtain a class of primal affine scaling algorithms which generalize some known algorithms this class depending on a r parameter is constructed through a family of metrics generated by r power r 1 of the diagonal iterate vector matrix we prove the so called weak convergence of the primal class for nondegenerate linearly constrained convex programming we observe the computational performance of the class of primal affine scaling algorithms accomplishing tests with linear programs from the netlib library and with some quadratic programming problems described in the maros and meszaros repository

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

extended affine root supersystems abstract the interaction of a lie algebra l having a weight space decomposition with respect to a nonzero toral subalgebra with its corresponding root system forms a powerful tool in the study of the structure of l this in particular suggests a systematic study of the root system apart from its connection with the lie algebra although there have been a lot of researches in this regard on lie algebra level such an approach has not been considered on lie superalgebra level in this work we introduce and study extended affine root supersystems which are a generalization of both affine reflection systems and locally finite root supersystems extended affine root supersystems appear as the root systems of the super version of extended affine lie algebras and invariant affine reflection algebras including affine lie superalgebras this work provides a framework to study the structure of this kind of lie superalgebras refereed to as extended affine lie superalgebras

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

improving the rate of convergence of interior point methods for linear programming this paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming if x k is the sequence generated by an interior point method the procedure derives an auxiliary sequence bar x k under the suitable assumptions it is shown that the sequence bar x k converges superlinearly faster to the solution than x k application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided

continuous dynamical systems that realize discrete optimization on the hypercube we study the problem of finding a local minimum of a multilinear function e over the discrete set 0 1 n the search is achieved by a gradient like system in 0 1 n with cost function e under mild restrictions on the metric the stable attractors of the gradient like system are shown to produce solutions of the problem even when they are not in the vicinity of the discrete set 0 1 n moreover the gradient like system connects with interior point methods for linear programming and with the analog neural network studied by vidyasagar vid95 in the same context ams subject classifications 34d20 90c51 key words gradient like systems multilinear functions 0 1 programming interior point methods lyapunov stability

limiting behavior of weighted central paths in linear programming we study the limiting behavior of the weighted central paths x s 0 in linear programming at both 0 and we establish the existence of a partition b n of the index set 1 n such thatx i ands j as fori b andj n andx n s b converge to weighted analytic centers of certain polytopes for allk 1 we show that thekth order derivativesx k ands k converge when 0 and consequently the derivatives of each order are bounded in the interval 0 we calculate the limiting derivatives explicitly and establish the surprising result that all higher order derivatives k 2 converge to zero when

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

presentation by conjugation for a1 type extended affine weyl groups abstract there is a well known presentation for finite and affine weyl groups called the presentation by conjugation recently it has been proved that this presentation holds for certain sub classes of extended affine weyl groups the weyl groups of extended affine root systems in particular it is shown that if nullity is 2 an a 1 type extended affine weyl group has the presentation by conjugation we set up a general framework for the study of simply laced extended affine weyl groups as a result we obtain certain necessary and sufficient conditions for an a 1 type extended affine weyl group of arbitrary nullity to have the presentation by conjugation this gives an affirmative answer to a conjecture that there are extended affine weyl groups which are not presented by presentation by conjugation

linear programming and the newton barrier flow in this note we report a simple characteristic of linear programming central trajectories which has a surprising consequence specifically we show that given a bounded polyhedral setp with nonempty interior the logarithmic barrier function with no objective component induces a vector field of negative newton directions which flows from the center ofp along central trajectories to solutions of every possible linear program onp