the worst casr step in karmarkar s algorithm in this note we consider the worst case performance in a single step of karmarkar s projective algorithm for linear programming in the transformed problem which arises on each iteration we show that the critical ratio r r can be improved asymptotically by a factor of two we also show that in the original problem where performance is characterized by reduction in the potential function the worst case reduction can be improved to approximately 0 72 moreover we demonstrate that both of these bounds are tight so that no further improvement based on the analysis of a single step is possible

duhamel s principle for temporally inhomogeneous evolution equations in banach space on precise les conditions sous lesquelles le principe de duhamel est valable pour des equations d evolution de type hyperbolique dans un espace de banach

discrete breathers in a mass in mass chain with hertzian local resonators we report on the existence of discrete breathers in a one dimensional mass in mass chain with linear intersite coupling and nonlinear precompressed hertzian local resonators which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates after predicting theoretically the existence of discrete breathers in the continuum and anticontinuum limits of intersite coupling we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain where the displacement profiles of the breathers are localized around one lattice site we then analyze the frequency amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode vanishing amplitude solution of the system near the upper band gap edge finally we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes in energy conserving and dissipative scenarios including within settings that may be relevant to future experimental studies

a first order interior point method for linearly constrained smooth optimization we propose a first order interior point method for linearly constrained smooth optimization that unifies and extends first order affine scaling method and replicator dynamics method for standard quadratic programming global convergence and in the case of quadratic program sub linear convergence rate and iterate convergence results are derived numerical experience on simplex constrained problems with 1000 variables is reported

chaotic behavior of the affine scaling algorithm for linear programming the affine scaling algorithm for linear programming involves a step size parameter t that must be chosen in the interval 0 1 it is known that the algorithm converges to an optimal solution for values of t le 2 3 in this paper we examine the behavior of the algorithm for values of t 2 3 we show that for certain values of t in this range the algorithm can exhibit chaotic behavior

shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

on the performance of karmarkar s algorithm over a sequence of iterations karmarkar s projective algorithm for linear programming is considered with real arithmetic and exact linesearch of the potential function it is shown that for every n geqq 3 there is a linear program with n variables such that the algorithm obtains a potential reduction of about 1 3 on each iteration for the same problems the algorithm requires theta ln n varepsilon iterations to reduce the objective gap to a factor varepsilon of its initial value it is thus proved that in the worst case the convergence of karmarkar s algorithm with exact linesearch cannot be independent of n and moreover potential reduction may be a poor indicator of algorithm performance

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

on affine scaling algorithms for nonconvex quadratic programming we investigate the use of interior algorithms especially the affine scaling algorithm to solve nonconvex indefinite or negative definite quadratic programming qp problems although the nonconvex qp with a polytope constraint is a hard problem we show that the problem with an ellipsoidal constraint is easy when the hard qp is solved by successively solving the easy qp the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions

superlinear convergence of the affine scaling algorithm in this paper we show that a variant of the long step affine scaling algorithm with variable stepsizes is two step superlinearly convergent when applied to general linear programming lp problems superlinear convergence of the sequence of dual estimates is also established for homogeneous lp problems having the origin as the unique optimal solution we also show that 2 3 is a sharp upper bound on the fixed stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates this result gives a plausible but not accurate theoretical explanation for why 2 3 is a sharp upper bound on the fixed stepsize that guarantees the convergence of the dual estimates

an iterative method for the calculation of narrow solitary excitations on atomic chains abstract we discuss the quasicontinuum approximation of collins and give a general condition for its validity for the solution of the difference differential equations of the full discrete model we develop an iteration procedure in fourier space the method is tested for solitary excitations with widths of the order of the lattice constant by a comparison with both exact results and computer simulations an additional result of the new mwthod is a simple derivation of the collins approximation which allows several generalizations

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

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

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

approximate cone factorizations and lifts of polytopes in this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix this provides a robust generalization of the famous result of yannakakis that polyhedral lifts of a polytope are controlled by exact nonnegative factorizations of its slack matrix our approximations behave well under polarity and have efficient representations using second order cones we establish a direct relationship between the quality of the factorization and the quality of the approximations and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron

global convergence of a long step affine scaling algorithm for degenerate linear programming problems in this paper we present new global convergence results on a long step affine scaling algorithm obtained by means of the local karmarkar potential functions this development was triggered by dikin s interesting result on the convergence of the dual estimates associated with a long step afflne scaling algorithm for homogeneous lp problems with unique optimal solutions without requiring any assumption on degeneracy we show that moving a fixed proportion lambda up to two thirds of the way to the boundary at each iteration ensures convergence of the iterates to an interior point of the optimal face as well as the dual estimates to the analytic center of the dual optimal face where the asymptotic reduction rate of the value of the objective function is 1 lambda we also give an example showing that this result is tight to obtain convergence of the dual estimates to the analytic center of the dual optimal face

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

constant potential primal dual algorithms a framework we start with a study of the primal dual affine scaling algorithms for linear programs using ideas from kojima et al mizuno and nagasawa and new potential functions we establish a framework for primal dual algorithms that keep a potential function value fixed we show that if the potential function used in the algorithm is compatible with a corresponding neighborhood of the central path then the convergence proofs simplify greatly our algorithms have the property that all the iterates can be kept in a neighborhood of the central path without using any centering in the search directions

an o n l iteration bound primal dual cone affine scaling algorithm for linear programming in this paper we introduce a primal dual affine scaling method the method uses a search direction obtained by minimizing the duality gap over a linearly transformed conic section this direction neither coincides with known primal dual affine scaling directions jansen et al 1993 monteiro et al 1990 nor does it fit in the generic primal dual method kojima et al 1989 the new method requires o sqrt n l main iterations it is shown that the iterates follow the primal dual central path in a neighbourhood larger than the conventional mathcal n 2 neighbourhood the proximity to the primal dual central path is measured by trigonometric functions

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

convergence and boundary behavior of the projective scaling trajectories for linear programming we analyze the convergence and boundary behavior of the continuous trajectories of the vector field induced by the projective scaling algorithm as applied to possibly degenerate linear programming problems in karmarkar s standard form we show that a projective scaling trajectory tends to an optimal solution which in general depends on the starting point when the optimal solution is unique we show that all projective scaling trajectories approach the optimal solution through the same asymptotic direction our analysis is based on the affine scaling trajectories for the homogeneous standard form linear program that arises from karmarkar s standard form linear program by removing the unique nonhomogeneous constraint

combined interior point method and semismooth newton method for frictionless contact problems in the present paper a solution scheme is proposed for frictionless contact problems of linear elastic bodies which are discretized using the finite element method with lower order elements an approach combining the interior point method and the semismooth newton method is proposed in this method an initial active set for the semismooth newton method is obtained from the approximate optimal solution by the interior point method the simplest node to node contact model is considered in the present paper that is pairs of matching nodes exist on the contact surfaces however the discussions can be easily extended to a node to segment or segment to segment contact model in order to evaluate the proposed method a number of illustrative examples of the frictionless contact problem are shown the proposed combined method is compared with the interior point method and the semismooth newton method two numerical examples that are difficult to solve using the semismooth newton method are solved effectively using the proposed combined method it is shown that the proposed method converges within far fewer iterations than the semismooth newton methods or the interior point method

a primal projective interior point method for linear programming we present a new projective interior point method for linear programming with unknown optimal value this algorithm requires only that an interior feasible point be provided it generates a strictly decreasing sequence of objective values and within polynomial time either determines an optimal solution or proves that the problem is unbounded we also analyze the asymptotic convergence rate of our method and discuss its relationship to other polynomial time projective interior point methods and the affine scaling method

an extension of karmarkar projective algorithm for convex quadratic programming we present an extension of karmarkar s linear programming algorithm for solving a more general group of optimization problems convex quadratic programs this extension is based on the iterated application of the objective augmentation and the projective transformation followed by optimization over an inscribing ellipsoid centered at the current solution it creates a sequence of interior feasible points that converge to the optimal feasible solution in o ln iterations each iteration can be computed in o ln3 arithmetic operations wheren is the number of variables andl is the number of bits in the input in this paper we emphasize its convergence property practical efficiency and relation to the ellipsoid method

affine scaling algorithm fails for semidefinite programming in this paper we introduce an affine scaling algorithm for semidefinite programming sdp and give an example of a semidefinite program such that the affine scaling algorithm converges to a non optimal point both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and they are non degenerate everywhere

a strong bound on the integral of the central path curvature and its relationship with the iteration complexity of primal dual path following lp algorithms the main goals of this paper are to i relate two iteration complexity bounds derived for the mizuno todd ye predictor corrector mty p c algorithm for linear programming lp and ii study the geometrical structure of the lp central path the first iteration complexity bound for the mty p c algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path the second iteration complexity bound derived recently by the authors using the notion of crossover events introduced by vavasis and ye is expressed in terms of a scale invariant condition number associated with m n constraint matrix of the lp in this paper we establish a relationship between these bounds by showing that the first one can be majorized by the second one we also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by vavasis and ye in view of the behavior of their layered least squares path following lp method that the central path consists of mathcal o n 2 long but straight continuous parts while the remaining curved part is relatively short

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

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

on the improvement per iteration in karmarkar s algorithm for linear programming we investigate the decrease in potential at an iteration of karmarkar s projective method for linear programming for a fixed step length parameter so that we must have 0 1 the best possible guarantee n inn dimensional space is essentially ln 2 0 69 and to achieve this we must take about 1 indeed we show the precise result that n equals ln 1 ln 1 n 1 forn sufficiently large if we choose an optimal step length at each iteration then this guarantee increases only to about 0 72 we also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem

global convergence of the affine scaling algorithm for convex quadratic programming in this paper we give a global convergence proof of the second order affine scaling algorithm for convex quadratic programming problems where the new iterate is the point which minimizes the objective function over the intersection of the feasible region with the ellipsoid centered at the current point and whose radius is a fixed fraction beta in 0 1 of the radius of the largest scaled ellipsoid inscribed in the nonnegative orthant the analysis is based on the local karmarkar potential function introduced by tsuchiya for any beta in 0 1 and without assuming any nondegeneracy assumption on the problem it is shown that the sequences of primal iterates and dual estimates converge to optimal solutions of the quadratic program and its dual respectively

scattering theory for temporally inhomogeneous hamilton equations in hilbert space in the scattering theory for the equation y p t y 0 in banach space x it is known that for a perturbation which is summable in time the scattering operator is everywhere defined in the space of the generalized solutions and onto we prove here that a stronger notion of scattering operator defined in the space of the true solutions is everywhere defined and onto provided that x is reflexive and that p satisfies the following property if p denotes the average operator p t tt 1p s ds then the cauchy problem for the equation y p t y 0 is uniformly well posed on the real line this result is applied to the hamilton equation in hilbert space jx h t x generalizing those of the author arch rational mech anal 86 1984 147 180 for the abstract wave equation u a t u 0

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

an interior affine scaling cubic regularization algorithm for derivative free optimization subject to bound constraints abstract in this paper we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function we first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method at each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique the global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions preliminary numerical results are reported to show the effectiveness of the proposed algorithm

a convergence proof for an affine scaling algorithm for convex quadratic programming without nondegeneracy assumptions this paper presents a theoretical result on convergence of a primal affine scaling method for convex quadratic programs it is shown that as long as the stepsize is less than a threshold value which depends on the input data only ye and tse s interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions in addition its local convergence rate is at least linear and the dual iterates have an ergodically convergent property

solving nonlinear multicommodity flow problems by the analytic center cutting plane method the paper deals with nonlinear multicommodity flow problems with convex costs a decomposition method is proposed to solve them the approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network it is thus formulated as a shortest path problem and solved with dijkstra s d heap algorithm an implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations computational results show the efficiency of this approach on well known nondifferentiable problems and also large scale randomly generated problems up to 1000 arcs and 5000 commodities

a polynomial primal dual dikin type algorithm for linear programming in this paper we present a new primal dual affine scaling method for linear programming the method yields a strictly complementary optimal solution pair and also allows a polynomial time convergence proof the search direction is obtained by using the original idea of dikin namely by minimizing the objective function which is the duality gap in the primal dual case over some suitable ellipsoid this gives rise to completely new primal dual affine scaling directions having no obvious relation with the search directions proposed in the literature so far the new directions guarantee a significant decrease in the duality gap in each iteration and at the same time they drive the iterates to the central path in the analysis of our algorithm we use a barrier function which is the natural primal dual generalization of karmarkar s potential function the iteration bound is onl which is a factor ol better than the iteration bound of an earlier primal dual affine scaling method monteiro adler and resende monteiro r d c i adler m g c resende 1990 a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension math oper res 15 191 214

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

a polynomial predictor corrector trust region algorithm for linear programming in this paper we present a scaling invariant interior point predictor corrector type algorithm for linear programming lp whose iteration complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the lp constraint matrix at the predictor stage the algorithm either takes the step along the standard affine scaling as direction or a new trust region type direction whose construction depends on a scaling invariant bipartition of the variables determined by the as direction this contrasts with the layered least squares direction introduced in s vavasis and y ye math program 74 1996 pp 79 120 whose construction depends on multiple layered partitions of the variables that are not scaling invariant moreover it is shown that the overall arithmetic complexity of the algorithm weakly depends on the right hand side and the cost of the lp in view of the work involved in the computation of the trust region steps

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

global convergence of the affine scaling methods for degenerate linear programming problems in this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy the behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence of the affine scaling methods

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

global exponential stability and global attractivity of impulsive hopfield neural networks with time delays in this paper by means of constructing the extended impulsive delayed halanay inequality and by lyapunov functional methods we analyze the global exponential stability and global attractivity of impulsive hopfield neural networks with time delays some new sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive hopfield neural networks with time delays are obtained those conditions are more feasible than that given in the earlier references to some extent some numerical examples are also discussed in this work to illustrate the advantage of the results we obtained

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

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 strategy of global convergence for the affine scaling algorithm for convex semidefinite programming the affine scaling algorithm is one of the earliest interior point methods developed for linear programming this algorithm is simple and elegant in terms of its geometric interpretation but it is notoriously difficult to prove its convergence it often requires additional restrictive conditions such as nondegeneracy specific initial solutions and or small step length to guarantee its global convergence this situation is made worse when it comes to applying the affine scaling idea to the solution of semidefinite optimization problems or more general convex optimization problems in math program 83 1 3 393 406 1998 muramatsu presented an example of linear semidefinite programming for which the affine scaling algorithm with either short or long step converges to a non optimal point this paper aims at developing a strategy that guarantees the global convergence for the affine scaling algorithm in the context of linearly constrained convex semidefinite optimization in a least restrictive manner we propose a new rule of step size which is similar to the armijo rule and prove that such an affine scaling algorithm is globally convergent in the sense that each accumulation point of the sequence generated by the algorithm is an optimal solution as long as the optimal solution set is nonempty and bounded the algorithm is least restrictive in the sense that it allows the problem to be degenerate and it may start from any interior feasible point

convergence analysis of the projective scaling algorithm based on a long step homogeneous affine scaling algorithm in this paper we give a new convergence analysis of a projective scaling algorithm we consider a long step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem this algorithm takes a fixed fraction 2 3 of the way towards the boundary of the nonnegative orthant at each iteration the iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection which generates the same search direction as the original projective scaling algorithm at each iterate the recent convergence results for the long step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm specifically we will show i polynomiality of the algorithm with complexities of o nl and o n2l iterations for 2 3 and 2 3 respectively ii global covnergence of the algorithm when the optimal face is unbounded iii convergence of the primal iterates to a relative interior point of the optimal face iv convergence of the dual estimates to the analytic center of the dual optimal face and v convergence of the reduction rate of the objective function value to 1

matching free trees maximal cliques and monotone game dynamics motivated by our recent work on rooted tree matching in this paper we provide a solution to the problem of matching two free i e unrooted trees by constructing an association graph whose maximal cliques are in one to one correspondence with maximal common subtrees we then solve the problem using simple payoff monotonic dynamics from evolutionary game theory we illustrate the power of the approach by matching articulated and deformed shapes described by shape axis trees experiments on hundreds of larger uniformly random trees are also presented the results are impressive despite the inherent inability of these simple dynamics to escape from local optima they always returned a globally optimal solution

a new iteration complexity bound for the mty predictor corrector algorithm in this paper we present a new iteration complexity bound for the mizuno todd ye predictor corrector mty p c primal dual interior point algorithm for linear programming the analysis of the paper is based on the important notion of crossover events introduced by vavasis and ye for a standard form linear program min ctx ax b x ge 0 with decision variable x in re n e show that the mty p c algorithm started from a well centered interior feasible solution with duality gap n mu 0 finds an interior feasible solution with duality gap less than n eta in cal o t mu 0 eta n 3 5 log bar chi a iterations where t t equiv min n 2 log log t log t for all t 0 and bar chi a is a scaling invariant condition number associated with the matrix a more specifically bar chi a is the infimum of all the conditions numbers bar chi ad where d varies over the set of positive diagonal matrices under the setting of the turing machine model our analysis yields an cal o n 3 5 l a min n 2 log l l iteration complexity bound for the mty p c algorithm to find a primal dual optimal solution where la and l are the input sizes of the matrix a and the data a b c respectively this contrasts well with the classical iteration complexity bound for the mty p c algorithm which depends linearly on l instead of log l

regularity versus degeneracy in dynamics games and optimization a unified approach to different aspects in this paper links are established between optimality conditions for quadratic optimization problems qualitative properties in the nonlinear selection replicator dynamics and central solution concepts of evolutionary game theory with particular emphasis on several regularity conditions that are desirable in any of the three fields mentioned above as strictness conditions for locally optimal solutions as hyperbolicity conditions for fixed points and as quasi strictness conditions for game equilibria

a variant of the vavasis ye layered step interior point algorithm for linear programming in this paper we present a variant of vavasis and ye s layered step path following primal dual interior point algorithm for linear programming our algorithm is a predictor corrector type algorithm which uses from time to time the layered least squares lls direction in place of the affine scaling as direction it has the same iteration complexity bound of vavasis and ye s algorithm namely cal o n 3 5 log bar chi a n where n is the number of nonnegative variables and bar chi a is a certain condition number associated with the constraint matrix a vavasis and ye s algorithm requires explicit knowledge of bar chi a which is very hard to compute or even estimate in order to compute the layers for the lls direction in contrast our algorithm uses the as direction at the current iterate to determine the layers for the lls direction and hence does not require the knowledge of bar chi a a variant with similar properties and with the same complexity has been developed by megiddo mizuno and tsuchiya math programming 82 1998 pp 339 355 however their algorithm needs to compute n lls directions on every iteration while ours computes at most one lls direction on any given iteration

traveling waves and their tails in locally resonant granular systems in the present study we revisit the theme of wave propagation in locally resonant granular crystal systems also referred to as mass in mass systems we use three distinct approaches to identify relevant traveling waves the first consists of a direct solution of the traveling wave problem the second one consists of the solution of the fourier tranformed variant of the problem or more precisely of its convolution reformulation upon an inverse fourier transform in real space finally our third approach will restrict considerations to a finite domain utilizing the notion of fourier series for important technical reasons namely the avoidance of resonances which will be discussed in detail all three approaches can be utilized in either the displacement or the strain formulation typical resulting computations in finite domains result in the solitary waves bearing symmetric non vanishing tails at both ends of the computational domain importantly however a countably infinite set of anti resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise