fuzzy goal programming complementary slackness conditions and computational schemes abstract goal programming gp is an important category in linear programming and multiobjective versions which minimize the deviation among value of each objective function and its goal with satisfying other constraints of problem in this paper we deal with fuzzy goal programming fgp which is a generalized problem in vague nature where parameters relations and goals are given as fuzzy quantities with respect to imprecise data recently inuiguchi et al m inuiguchi j ramik t tanio m vlach satisficing solutions and duality in interval and fuzzy linear programming fuzzy sets and systems 135 2003 151 177 have proved some important dual theorems on fgp which are very close to the same crisp versions in continuation to their works based on fuzzy extensions of relations we get complementary slackness conditions for fgp which have an essential role in many optimization techniques moreover we obtain the most optimistic and pessimistic satisficing solutions employing these conditions and present a convex combination of them which covers the satisficing solutions for a fixed level of certainty our practical schemes implement an efficient interior point algorithm as sub procedure so they are polynomial time algorithms

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

design of on line algorithms using hitting times random walks are well known for playing a crucial role in the design of randomized off line as well as on line algorithms in this work we prove some basic identities for ergodic markov chains e g an interesting characterization of reversibility in markov chains is obtained in terms of first passage times besides providing new insight into random walks on weighted graphs we show how these identities give us a way of designing competitive randomized on line algorithms for certain well known problems

efficiency test of pseudorandom number generators using random walks markov chain monte carlo methods and computer simulations usually use long sequences of random numbers generated by deterministic rules so called pseudorandom number generators their efficiency depends on the convergence rate to the stationary distribution and the quality of random numbers used for simulations various methods have been employed to measure the convergence rate to the stationary distribution but the effect of random numbers has not been much discussed we present how to test the efficiency of pseudorandom number generators using random walks

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems multicommodity network flow models arise in a wide variety of contexts typical among which is the dimensioning of telecommunication networks in this paper we present various approaches based on specialization of the simplex algorithm and interior point methods to solve nonoriented multicommodity flowproblems algorithms are tested with data from the france telecom paris district transmission network first we focus on a specialization for the node arc formulation of the problem a primal simplex and dual affine scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared numerical results are provided for problems up to about 800 000 constraints and 6 000 000 variables however much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem a dantzig wolfe decomposition method is designed and compared with a specialized implementation of the analytic center cutting plane method accpm partitioning techniques are used to exploit the structure of the master programs involved in those methods

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

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation

markov chain algorithms for planar lattice structures consider the following markov chain whose states are all domino tilings of a 2n 2n chessboard starting from some arbitrary tiling pick a 2 2 window uniformly at random if the four squares appearing in this window are covered by two parallel dominoes rotate the dominoes 90 rm o in place repeat many times this process is used in practice to generate a random tiling and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics analogous markov chains are used to randomly generate other structures on various two dimensional lattices this paper presents techniques which prove for the first time that in many interesting cases a small number of random moves suffice to obtain a uniform distribution

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

improved bounds for sampling colorings we consider the problem of sampling uniformly at random from the set of proper k colorings of a graph with maximum degree our main result is the design of a simple markov chain that converges in o nk log n time to the desired distribution when k 116

on the number of iterations of karmarkar s algorithm for linear programming karmarkar s algorithm for linear programming was published in 1984 and it is highly important to both theory and practice on the practical side some of its variants have been found to be far more efficient than the simplex method on a wide range of very large calculations while its polynomial time properties are fundamental to research on complexity these properties depend on the fact that each iteration reduces a potential function by an amount that is bounded away from zero the bound being independent of all the coefficients that occur it follows that under mild conditions on the initial vector of variables the number of iterations that are needed to achieve a prescribed accuracy in the final value of the linear objective function is at most a multiple ofn wheren is the number of inequality constraints by considering a simple example that allowsn to be arbitrarily large we deduce analytically that the magnitude of this complexity bound is correct specifically we prove that the solution of the example by karmarkar s original algorithm can require aboutn 20 iterations further we find that the algorithm makes changes to the variables that are closely related to the steps of the simplex method

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

some thoughts about random walks on figure eight abstract in this paper we discuss the issues of recurrence and transience of random walks in multiply connected spaces and in particular in the presence of knots such problems are of interest in polymer physics since almost all sufficiently long linear polymer chains are quasi knotted at least at theta solvent quality conditions and hence the transience recurrence provides mathematically rigorous definition of the entanglement concept these problems are also of biological interest since they clarify the role of topology in problems involving molecular recognition obtained results extend and generalize the results of earlier published rapid communication kholodenko phys rev e 58 1998 r5213 they also provide some answers to problems which were just mentioned in our earlier published report kholodenko phys reports 298 1998 251

interior point algorithms for linear programming with inequality constraints interior methods for linear programming were designed mainly for problems formulated with equality constraints and non negative variables the formulation with inequality constraints has shown to be very convenient for practical implementations and the translation of methods designed for one formulation into the other is not trivial this paper relates the geometric features of both representations shows how to transport data and procedures between them and shows how cones and conical projections can be associated with inequality constraints

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

combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods abstract in this work the optimal adjustment algorithm for p coordinates which arose from a generalization of the optimal pair adjustment algorithm is used to accelerate the convergence of interior point methods using a hybrid iterative approach for solving the linear systems of the interior point method its main advantages are simplicity and fast initial convergence at each interior point iteration the preconditioned conjugate gradient method is used in order to solve the normal equation system the controlled cholesky factorization is adopted as the preconditioner in the first outer iterations and the splitting preconditioner is adopted in the final outer iterations the optimal adjustment algorithm is applied in the preconditioner transition in order to improve both speed and robustness numerical experiments on a set of linear programming problems showed that this approach reduces the total number of interior point iterations and running time for some classes of problems furthermore some problems were solved only when the optimal adjustment algorithm for p coordinates was used in the change of preconditioners

limit analysis with the dual affine scaling algorithm the collapse state of a rigid plastic material with the linearized mises yield condition is computed we use an infeasible point variant of the dual affine scaling algorithm for linear programming which is extremely efficient for this large sparse and ill conditioned problem for a classical test problem we obtain better results than previously both for the limit load and for the collapse fields thus setting a new standard for computations in limit analysis

special functions as solutions to discrete painleve equations we present results on special solutions of discrete painleve equations these solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second order discrete equations these linear equations define the discrete analogues of special functions

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

the pinney equation and its discretization the pinney equation is part of the original ermakov system which has been the subject of intensive study recently here we show that it may be related to a two dimensional conformal riccati equation leading to a new method for its linearization a discrete analogue of the pinney equation is constructed using the above connection with the conformal group an alternative discretization is obtained by using a discrete schwarz derivative both of these nonlinear difference equations are linearizable

decomposed block cholesky factorization in the karmarkar algorithm solving a class of super large lp problems abstract with the introduction of the karmarkar algorithm hope for optimally solving super large lp problems often encountered in the communication industry had risen high however a general code of the original karmarkar algorithm takes a long computation time to solve such problems employing proper factorization schemes substantially reduces the computation time of the karmarkar algorithm for solving super large linear programming lp problems in this paper a modified version of the karmarkar algorithm is described and the complexities involved in applying the algorithm to a class of super large lp problems are identified a factorization scheme decomposed block cholesky factorization is proposed in order to reduce the complexities of a matrix inversion operation that occurs within every iteration of the karmarkar algorithm the factorization scheme capitalizes on the generally well behaved block structure of the constraint matrix of the super large lp problems the computation time required for solving a class of super large lp problems using this scheme is estimated to be reduced by a factor of two orders of magnitude the proposed factorization scheme is even more promising in solving the super large lp problems on parallel processors using the karmarkar algorithm

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

interior point methods for nonconvex nonlinear programming orderings and higher order methods the paper extends prior work by the authors on loqo an interior point algorithm for nonconvex nonlinear programming the specific topics covered include primal versus dual orderings and higher order methods which attempt to use each factorization of the hessian matrix more than once to improve computational efficiency results show that unlike linear and convex quadratic programming higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming but that a variant of mehrotra s predictor corrector algorithm can definitely improve performance

rational solutions of the discrete time toda lattice and the alternate discrete painleve ii equation the yablonskii vorob ev polynomials yn t which are defined by a second order bilinear differential difference equation provide rational solutions of the toda lattice they are also polynomial tau functions for the rational solutions of the second painleve equation pii here we define two variable polynomials yn t h on a lattice with spacing h by considering rational solutions of the discrete time toda lattice as introduced by suris these polynomials are shown to have many properties that are analogous to those of the yablonskii vorob ev polynomials to which they reduce when h 0 they also provide rational solutions for a particular discretization of pii namely the so called alternate discrete pii and this connection leads to an expression in terms of the umemura polynomials for the third painleve equation piii it is shown that the backlund transformation for the alternate discrete painleve equation is a symplectic map and the shift in time is also symplectic finally we present a lax pair for the alternate discrete pii which recovers jimbo and miwa s lax pair for pii in the continuum limit h 0

large scale convex optimization via saddle point computation this article proposes large scale convex optimization problems to be solved via sad dle points of the standard lagrangian a recent approach for saddle point computation is specialized by way of a specific perturbation technique and unique scaling method to convex optimization problems with differentiable objective and constraint funct ions in each iteration the update directions for primal and dual variables are determined by gradients of the lagrangian these gradients are evaluated at perturbed points that are generated from current points via auxiliary mappings the resulting algorithm suits massively parallel computing though in this article we consider only a serial implementation we test a version of our code embedded within gams on 16 nonlinear problems which are mainly large these models arise from multistage optimization of economic systems for larger problems with adequate precision requirements our implementation appears faster than minos

perfect reconstruction filter banks having linear phase fir filters with equiripple response the design of equiripple linear phase analysis and synthesis fir filters of two channel perfect reconstruction pr filter banks is formulated as the minimization of a weighted peak error under both linear inequality arising from the desired responses of the analysis filters and nonlinear equality pr constraints the effectiveness of a proposed method to solve the design problem a modified dual affine scaling variant of karmarkar s 1989 algorithm and an approximation scheme is illustrated through several design examples

special relations for continuous and discrete painleve equations we investigate a relation between solutions of the painleve ii equation corresponding to values of the parameter of pii which cannot be connected through a schlesinger transformation which was first derived by gambier here we present the discrete analogue of this relation relating the discrete pii to the alternate d pii the latter turns out to be a consequence of a quadratic relation existing between two different families of solutions of pv a q discrete analogue of the latter relation is also presented

a short cut potential reduction algorithm for linear programming as most interior point algorithms iterate they repeatedly perform costly matrix operations such as projections on the entire constraint matrix for large scale linear programming problems such operations consume the great majority of the computation time required however for problems where the number of variables far exceeds the number of constraints operations over the entire constraint matrix are unnecessary we will examine and extend decomposition techniques which greatly reduce the amount of work required by such interior point methods as the dual affine scaling and the dual potential reduction algorithms in an effort to judge the practical viability of the decompositioning we compare the performance of the dual potential reduction algorithm with and without decompositioning over a set of randomly generated transportation problems accompanying a theoretical justification of these techniques we focus on the implementation details and computational results of one such technique

on finding a vertex solution using interior point methods abstract an approach is proposed to generate a vertex solution while using a primal dual interior point method to solve linear programs a controlled random perturbation is made to the cost vector a method to identify the active constraints at the vertex to which the solutions are converging is given this basic method is further refined to save computational effort the proposed approach is tested by using a variation of the primal dual interior point method our method is developed by taking a predictor corrector approach in practice this method takes considerably fewer iterations to solve linear programs than methods described by choi monma and shanno lustig marsten and shanno and domich boggs donaldson and witzgall computational results on problems from the netlib test set are reported to test our approach for finding vertex solutions these results show that one perturbation is enough to force the solutions to converge to a vertex the results indicate that the proposed approach is insensitive to the number of degenerate variables the results also indicate that the effort required to generate a vertex solution is comparable to that required to solve the problem using an interior point method

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

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

tatonnement procedures for linearly constrained convex optimization the emphasis in this article is to exploit the fact that precision requirements for solutions of most economic models in practice are moderate only a simple approach is introduced for solving linearly constrained partial equilibrium models based on an iterative scheme similar to the simplex method it allows large scale models to be solved within a practical tolerance efficiently even in a micro computer environment extensions to linearly constrained convex optimization problems are presented finally a set of computational tests on 68 linear programs from the netlib library is reported comparison of our approach with the simplex method using minos 5 1 and with karmarkar s algorithm is reported for moderate precision requirements these preliminary results are highly encouraging

random walks arising in random number generation random number generators often work by recursively computing x n 1 equiv ax n b mod p various schemes exist for combining these random number generators in one scheme a and b are themselves chosen each time from another generator assuming that this second source is truly random we investigate how long it takes for x n to become random for example if a 1 and b 0 1 or 1 each with probability frac 1 3 then cp 2 steps are required to achieve randomness on the other hand if a 2 and b 0 1 or 1 each with probability frac 1 3 then c log p log log p steps always suffice to guarantee randomness and for infinitely many p are necessary as well although in fact for almost all odd p 1 02 log 2 p steps are enough

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

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

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

random walks on a finite graph with congestion points we consider a random walk on a finite graph with congestion points which can be decomposed into finite groups we derive the explicit formulas of the probability generating function of the first hitting time using group representations

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

minimax design of two channel nonuniform division filterbanks using iir allpass filters the design of two channel linear phase nonuniform division filter ndf banks constructed by infinite impulse response iir digital allpass filters dafs in the sense of l sub spl infin error criteria is considered first the theory of two channel ndf bank structures using two iir dafs is developed then the design problem is appropriately formulated to result in a simple optimization problem utilizing a variant of karmarkar s algorithm we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the iir dafs the resulting two channel ndf banks can possess approximately linear phase response without magnitude distortion the effectiveness of the proposed technique is achieved by forming an appropriate chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations several simulation examples are presented for illustration and comparison

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

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

comparative analysis of affine scaling algorithms based on simplifying assumptions we analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions we show that under a strong probabilistic assumption regarding the distribution of the data in an iteration the decrease in the primal potential function will be omega rho sqrt log n with high probability compared to the guaranteed 1 2n is a parameter in the potential function andn is the number of variables under the same assumption we further show that the objective reduction rate of dikin s affine scaling algorithm is 1 1 sqrt log n with high probability compared to no guaranteed convergence rate

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

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

continued fraction solutions to the riccati equation and integrable lattice systems continued fraction solutions to the matrix riccati equation are discussed which are constructed by using the concept of form invariance it is demonstrated that this technique is related to the akns method of deriving integrable nonlinear lattice systems this gives an explanation why continued fraction solutions related to the toda lattice were obtained in a previous work continued fractions corresponding to kac van moerbeke discrete nonlinear schrodinger and discrete modified kdv lattice equations are constructed a method for linearising the kac van moerbeke lattice equations is rederived and particular solutions are generated the authors approach demonstrates the crucial role played by the boundary condition at the finite end of the lattice for the existence of this method these results are extended to the other two lattice systems above in the semi infinite case and corresponding particular solutions generated in terms of bessel functions

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

a remark on the hankel determinant formula for solutions of the toda equation we consider the hankel determinant formula of the functions of the toda equation we present a relationship between the determinant formula and the auxiliary linear problem which is characterized by a compact formula for the functions in the framework of the kp theory similar phenomena that have been observed for the painleve ii and iv equations are recovered the case of finite lattice is also discussed

dual versus primal dual interior point methods for linear and conic programming we observe a curious property of dual versus primal dual path following interior point methods when applied to unbounded linear or conic programming problems in dual form while primal dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility unboundedness detectors

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

a primal dual interior point method for robust optimal control of linear discrete time systems this paper describes how to efficiently solve a robust optimal control problem using recently developed primal dual interior point methods one potential application is model predictive control the optimization problem considered consists of a worst case quadratic performance criterion over a finite set of linear discrete time models subject to inequality constraints on the states and control signals the scheme has been prototyped in matlab to give a rough idea of the efficiencies obtained it is possible to solve problems with more than 10 000 primal variables and 40 000 constraints on a workstation the key to the efficient implementation is an iterative solver in conjunction with a riccati recursion invertible pre conditioner for computing the search directions

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

hopfield neural networks in large scale linear optimization problems abstract hopfield neural networks and affine scaling interior point methods are combined in a hybrid approach for solving linear optimization problems the hopfield networks perform the early stages of the optimization procedures providing enhanced feasible starting points for both primal and dual affine scaling interior point methods thus facilitating the steps towards optimality the hybrid approach is applied to a set of real world linear programming problems the results show the potential of the integrated approach indicating that the combination of neural networks and affine scaling interior point methods can be a good alternative to obtain solutions for large scale optimization problems

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

ultradiscretization without positivity we present a new ultradiscretization approach which can be applied to discrete systems the solutions of which are not positive definite this was made possible thanks to an ansatz involving the hyperbolic sine function we apply this new procedure to simple mappings for the linear and homographic mappings we obtain ultradiscrete forms and explicitly construct their solutions two discrete painleve ii equations are also analysed and ultradiscretized we show how to construct the ultradiscrete analogues of their rational and airy type solutions