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

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

revetements du demi plan de drinfeld et correspondance de langlands p adique nous decrivons le complexe de de rham des revetements du demi plan de drinfeld pour gl 2 q p cette description conjecturee par breuil et strauch fournit une realisation geometrique de la correspondance de langlands locale p adique pour certaines representations de de rham de dimension de gal q p

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

a novel augmented lagrangian based formulation for upper bound limit analysis summary this paper describes a novel upper bound formulation of limit analysis this formulation is an innovative variant of an existing two field mixed formulation based on the augmented lagrangian method also developed by the authors a natural approach is used to describe the deformation of each finite element furthermore and in contrast to the previous formulation two independent field approximations are now both used to define the velocity field defined globally and at element level it is shown that this feature allows a governing system of uncoupled linear equations to be obtained some numerical examples in plane strain conditions are presented in order to illustrate the current model performance in conclusion the potential and advantages of this new approach are discussed copyright 2011 john wiley sons ltd

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

computation of limit loads for computing the collapse state of a perfect plastic continuum we suggest a mixed finite element approximation to the infinite dimensional mathematical programming problem of limit analysis the convergence of the limit load to the exact solution is discussed as well as solution methods for the discrete problem finally the method is applied to a classical problem in plane strain

splitting dense columns in sparse linear systems abstract we consider systems of equations of the form aa t x b where a is a sparse matrix having a small number of columns which are much denser than the other columns these dense columns in a cause aat to be very or even completely dense which greatly limits the effectiveness of sparse matrix techniques for directly solving the above system of equations in the literature on interior point methods for linear programming the usual technique for dealing with this problem is to split a into a sparse part s and a dense part d a s d and to solve systems involving aat in terms of the solution of systems involving sst using either conjugate gradient techniques or the sherman morrison woodbury formula this approach has the difficulty that sst is often rank deficient even when aat has full rank in this paper we propose an alternative method which avoids the rank deficiency problem and at the same time allows for the effective use of sparse matrix techniques the resulting algorithm is both efficient and robust

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

modular representations of pgl2 and automorphic forms for shimura curves in this paper we study certain infinite dimensional representations of the general linear group gl2 over a local field k in vector spaces over k these representations are closely related to the theory of analytic functions on the p adic upper half plane over k they were originally introduced by morita mor who studied them intensively in the case when the characteristic of k is zero morita s work has subsequently been greatly extended by schneider and stuhler ss and schneider s to representations of gl k and higher p adic symmetric spaces in this paper we re consider some of morita s representations of gl2 from the point of view of integral and modular representation theory it seems quite natural to study p adic representations by reducing them modp and we supply a technique for doing this in particular suppose that ydenotes the bruhat tits tree of pglz k we consider the representations of pgl2 k on the harmonic functions on w of e v e n weight that is the harmonic functions on the edges of j taking values in even symmetric powers of the standard representation of gl2 in this paper we find an invariant integral lattice in these representations and we describe the modular representations of pgl2 k obtained by reducing this lattice modp the modular representations obtained from morita s representations are quite simple and natural objects although they seem a bit peculiar at first suppose that f is the residue field of k that v is a vector space over f and that p gl2 f end v is a finite dimensional modular representation notice that p may be viewed in a natural way as a modular representation of gl r where r is the valuation ring of k we show that a typical infinite dimensional representation of morita s when reduced mod p has a filtration whose factors are isomorphic to i cl2 r wj t n l2 t note that there are only finitely many irreducible finite dimensional modular representations of glz f the appearance of these finitely many

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

an upper bound limit analysis of mindlin plates using cs dsg3 method and second order cone programming the paper presents a numerical procedure for kinematic limit analysis of mindlin plate governed by von mises criterion the cell based smoothed discrete shear gap method cs dsg3 is combined with a second order cone optimization programming socp for determining the upper bound limit load of the mindlin plates the limit analysis problem of mindlin plates is formulated by minimizing the dissipation power subjected to a set of constraints of boundary conditions and unitary external work this minimization problem is then transformed into an explicit form suitable for the solution using the socp the numerical results of some benchmark problems show that the proposal procedure can provide the reliable upper bound collapse multipliers for the mindlin plates cs dsg3 is combined with socp for determining upper bound limit load of mindlin plates limit analysis problem is transformed into an explicit form suitable for solution using socp and cs dsg3 numerical results show that cs dsg3 is free of shear locking and can provide reliable upper bound collapse multipliers for mindlin plates

computation of collapse states with von mises type yield condition a new computational approach to the problem of limit analysis with quadratic yield condition is developed and tested the problem is solved with the exact convex yield condition and the general case of unbounded yield set corresponding to unrestricted hydrostatic pressure is treated the solution method simultaneously approximates the collapse fields for stresses and flow the approach consists of two separate steps first the continuous problem is discretized by the finite element method into a finite dimensional optimization problem which turns out to be of a very specific form then this problem is solved by a suitable optimization method this way it is not necessary to be an expert in optimization in order to solve collapse problems and we can take advantage of the rapid development in optimization methods for large sparse problems without changing the discretization program the efficiency and accuracy of the method for large problems is demonstrated by solving a classical problem in the plane strain model approximately 90000 finite element nodes with 3 stress components and 2 velocity components at each node in two space dimensions this may be overkill but it shows that we are able to solve problems in three space dimensions

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

mesh adaptive computation of upper and lower bounds in limit analysis summary an efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented with a formulation that explicitly considers the exact convex yield condition the approach consists of two main steps first the continuous problem under the form of the static principle of limit analysis is discretized twice one per bound using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound the second step consists of solving the resulting discrete non linear optimization problems these are reformulated as second order cone programs which allows for the use of primal dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model to benefit from the fact that collapse mechanisms are typically highly localized a novel method for adaptive meshing is introduced the method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions the efficiency of the methodology is illustrated with applications in plane stress and plane strain problems copyright q 2008 john wiley sons ltd

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

a combined homotopy interior point method for general nonlinear programming problems abstract a combined homotopy interior point method for solving general nonlinear programming is proposed the algorithm generated by this method to kuhn tucker points of the general nonlinear programming problems is proved to be globally convergent under the normal cone condition about the constraints probably without the convexity

a cell based smoothed finite element method for kinematic limit analysis this paper presents a new numerical procedure for kinematic limit analysis problems which incorporates the cell based smoothed finite element method with second order cone programming the application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort the non smooth optimization problem is formulated as a problem of minimizing a sum of euclidean norms ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm plane stress and plane strain problems governed by the von mises criterion are considered but extensions to problems with other yield criteria having a similar conic quadratic form or 3d problems can be envisaged

a random matrix model for elliptic curve l functions of finite conductor we propose a random matrix model for families of elliptic curve l functions of finite conductor a repulsion of the critical zeros of these l functions away from the centre of the critical strip was observed numerically by miller 2006 exp math 15 257 79 such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros for large conductors this distribution is expected to approach the one level density of eigenvalues of orthogonal matrices after appropriate rescaling our purpose here is to provide a random matrix model for miller s surprising discovery we consider the family of even quadratic twists of a given elliptic curve the main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra we call this sub ensemble of so 2n the excised orthogonal ensemble the sieving off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of l functions implied by the formulae of waldspurger and kohnen zagier the cut off scale appropriate to modelling elliptic curve l functions is exponentially small relative to the matrix size n the one level density of the excised ensemble can be expressed in terms of that of the well known jacobi ensemble enabling the former to be explicitly calculated it exhibits an exponentially small on the scale of the mean spacing hard gap determined by the cut off value followed by soft repulsion on a much larger scale neither of these features is present in the one level density of so 2n when n we recover the limiting orthogonal behaviour our results agree qualitatively with miller s discrepancy choosing the cut off appropriately gives a model in good quantitative agreement with the number theoretical data

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

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

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

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

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

a bilevel programming method for pipe network optimization this paper is concerned with the optimal dimensioning problem in which the layout of a pipe network is given but several available diameters can be selected for each pipe the purpose is to minimize total cost while satisfying certain restrictions a new method is proposed the main features of which are to treat the problem as a bilevel programming problem then to simplify the lower level problem by conjugate duality theory and to handle the upper level problem by its piecewise linear and convex nature some numerical testing results are also presented

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

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

an extended variant of karmarkar s interior point algorithm abstract in this paper extended algorithm of karmarkar s interior point algorithm is considered theoretical aspects of the new algorithm are discussed we show that the new iterative algorithm converges faster than former karmarkar s algorithms successful convergence for problems of different sizes is obtained numerical results show that when we use a new parameter in the classical karmarkar s algorithm number of iterations is less than the number of iterations for two mentioned methods

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

computation of the collapse state in limit analysis using the lp primal affine scaling algorithm abstract the object of this work is twofold the first goal is to demonstrate that for the duality problem of limit analysis with linearized yield condition the lp primal affine scaling algorithm shows properties which are significantly different from those of the simplex method and that as a consequence of this better results can be obtained the second goal is to compute the collapse state for a classical and hard problem in plane strain tension of a rectangular bar with symmetric thin cuts both the collapse multiplier and collapse fields are computed to a better accuracy and detail than previously

formulating two stage stochastic programs for interior point methods this paper describes an approach for modeling two stage stochastic programs that yields a form suitable for interior point algorithms a staircase constraint structure is created by replacing first stage variables with sparse split variables in conjunction with side constraints dense columns are thereby eliminated the resulting model is larger than traditional stochastic programs but computational savings are substantial over a tenfold improvement for the problems tested a series of experiments with stochastic networks drawn from financial planning demonstrates the attained efficiencies comparisons with minos and the dual block angular stochastic programming model are provided as benchmarks the split variable approach is applicable to general two stage stochastic programs and other dual block angular models

a fundamental class of stress elements in lower bound limit analysis there is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite element method once the stress field has been discretized the yield criterion and the

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

computing limit loads by minimizing a sum of norms this paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition such as for example the von mises condition after discretization with the finite element method using divergence free elements for the plastic flow the kinematic formulation reduces to the problem of minimizing a sum of euclidean vector norms subject to a single linear constraint this is a nonsmooth minimization problem since many of the norms in the sum may vanish at the optimal point recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form the approach is applied to test problems in limit analysis in two different plane models plane strain and plates in the first case more than 80 of the terms in the objective function are zero in the optimal solution causing extreme ill conditioning in the second case all terms are nonzero in both cases the method works very well and problems are solved which are larger by at least an order of magnitude than previously reported the relative accuracy for the solution of the discrete problems measured by duality gap and feasibility is typically of the order 10 8

theoretical convergence of large step primal dual interior point algorithms for linear programming this paper proposes two sets of rules rule g and rule p for controlling step lengths in a generic primal dual interior point method for solving the linear programming problem in standard form and its dual theoretically rule g ensures the global convergence while rule p which is a special case of rule g ensures the o nl iteration polynomial time computational complexity both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions they rely neither on neighborhoods of the central trajectory nor on potential function these rules allow large steps without performing any line search rule g is especially flexible enough for implementation in practically efficient primal dual interior point algorithms

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 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

low lying zeros of l functions with orthogonal symmetry we investigate the moments of a smooth counting function of the zeros near the central point of l functions of weight k cuspidal newforms of prime level n we split by the sign of the functional equations and show that for test functions whose fourier transform is supported in 1 n 1 n as n the first n centered moments are gaussian by extending the support to 1 n 1 1 n 1 we see non gaussian behavior in particular the odd centered moments are nonzero for such test functions if we do not split by sign we obtain gaussian behavior for support in 2 n 2 n if 2k n the nth centered moments agree with random matrix theory in this extended range providing additional support for the katz sarnak conjectures the proof requires calculating multidimensional integrals of the nondiagonal terms in the bessel kloosterman expansion of the petersson formula we convert these multidimensional integrals to one dimensional integrals already considered in the work of iwaniec luo and sarnak ils and derive a new and more tractable expression for the nth centered moments for such test functions this new formula facilitates comparisons between number theory and random matrix theory for test functions supported in 1 n 1 1 n 1 by simplifying the combinatorial arguments as an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point

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

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

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

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

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

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

computing l functions with large conductor an algorithm is given to efficiently compute l functions with large conductor in a restricted range of the critical strip examples are included for about 24000 dihedral galois representations with conductor near 10 7 the data shows good agreement with a symplectic random matrix model

an interior point algorithm for elastoplasticity the problem of small deformation rate independent elastoplasticity is treated using convex programming theory and algorithms a finite step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the lagrangian associated with a convex mathematical program next an algorithm based on the classical primal dual interior point method is developed several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem the resulting method is compared to state of the art elastoplastic procedures for which both similarities and differences are found finally a number of examples are solved demonstrating the capabilities of the algorithm when applied to standard perfect plasticity hardening multisurface plasticity and problems involving softening copyright 2006 john wiley sons ltd

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

on affine scaling and semi infinite programming we consider an extension of the affine scaling algorithm for linear programming problems with free variables to problems having infinitely many constraints and explore the relationship between this algorithm and the finite affine scaling method applied to a discretization of the problem

a specialized interior point algorithm for multicommodity network flows despite the efficiency shown by interior point methods in large scale linear programming they usually perform poorly when applied to multicommodity flow problems the new specialized interior point algorithm presented here overcomes this drawback this specialization uses both a preconditioned conjugate gradient solver and a sparse cholesky factorization to solve a linear system of equations at each iteration of the algorithm the ad hoc preconditioner developed by exploiting the structure of the problem is instrumental in ensuring the efficiency of the method an implementation of the algorithm is compared to state of the art packages for multicommodity flows the computational experiments were carried out using an extensive set of test problems with sizes of up to 700 000 variables and 150 000 constraints the results show the effectiveness of the algorithm

quadratic velocity linear stress interpolations in limit analysis summary we compare a family of mixed interpolations on triangles with straight edges as applied to limit analysis the aim of this paper is to prove theoretically that the approximate collapse factors obtained with these finite elements always comply with certain inequalities that exist among them two of these interpolations are used in limit analysis for the first time in this article the inequalities in the proposition are also demonstrated via numerical applications to this end the most frequently used benchmark problems in limit analysis are revisited and discussed with respect to the results computed with these finite elements copyright 2014 john wiley sons ltd

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

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

a 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

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

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

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

fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems abstract branch and cut methods for integer programming problems solve a sequence of linear programming problems traditionally these linear programming relaxations have been solved using the simplex method the reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one if the solution to a relaxation is fractional additional constraints can be generated which cut off the solution to the relaxation but donot cut off any feasible integer points gomory cutting planes and other classes of cutting planes are generated from the final tableau in this paper we consider using an interior point method to solve the linear programming relaxations we show that it is still possible to generate gomory cuts and other cuts without having to recreate a tableau and we also show how variables can be fixed without using the optimal reduced costs the procedures we develop do not require that the current relaxation be solved to optimality this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation

on the collapse solution in limit analysis the existence and regularity of collapse solutions in limit analysis of a plastic continuum is examined collapse fields for stresses and velocity exist as a saddle point for the duality between the static and kinematic formulations the velocity field is defined as a pair u u ut where u is of bounded deformation in while ut is the velocity of the surface a generalized green s formula for the collapse fields is proved under certain regularity conditions

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

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

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

lower bound limit analysis of cohesive frictional materials using second order cone programming the formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints here we present a method for obtaining strict lower bound solutions using second order cone programming socp for which efficient primaldual interior point algorithms have recently been developed following a review of previous work we provide a brief introduction to socp and describe how lower bound limit analysis can be formulated in this way some methods for exploiting the data structure of the problem are also described including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage the benefits of employing socp are then illustrated with numerical examples through the use of an effective algorithm software very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine the numerical examples concern plane strain conditions and the mohr coulomb criterion however we show that socp can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form notable examples being the drucker prager criterion in 2d or 3d and nielsen s criterion for plates

on dual convergence of the generalized proximal point method with bregman distances the use of generalized distances e g bregman distances instead of the euclidean one in the proximal point method for convex optimization allows for elimination of the inequality constraints from the subproblems in this paper we consider the proximal point method with bregman distances applied to linearly constrained convex optimization problems and study the behavior of the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem under rather general assumptions which cover most bregman distances of interest we obtain an ergodic convergence result namely that a sequence of weighted averages of the dual sequence converges to a specific point of the dual optimal set as an intermediate result we prove under the same assumptions that the dual central path generated by a large class of barriers including the generalized bregman distances converges to the same point

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems