a two step matrix splitting iteration for computing pagerank the pagerank algorithm plays an important role in determining the importance of web pages the core of this algorithm involves using the classical power method to compute the pagerank vector which is the principal eigenvector of the matrix representing the web link graph nevertheless it is well known that the power method may perform poorly when the second largest eigenvalue is close to the dominant one in this article we present a new approach that is based on the two step splitting iteration framework the description and convergence of the new algorithm are discussed in detail numerical examples are given to illustrate the performance of this algorithm

refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

a strong dual for conic mixed integer programs mixed integer conic programming is a generalization of mixed integer linear programming in this paper we present an extension of the duality theory for mixed integer linear programming see m guzelsoy and t k ralphs int j oper res taichung 4 2007 pp 118 137 g l nemhauser and l a wolsey integer and combinatorial optimization wiley interscience new york 1988 to the case of mixed integer conic programming in particular we construct a subadditive dual for mixed integer conic programming problems under a simple condition on the primal problem we show that strong duality holds

numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

a reordering for the pagerank problem we describe a reordering particularly suited to the pagerank problem which reduces the computation of the pagerank vector to that of solving a much smaller system and then using forward substitution to get the full solution vector we compare the theoretical rates of convergence of the original pagerank algorithm to that of the new reordered pagerank algorithm showing that the new algorithm can do no worse than the original algorithm we present results of an experimental comparison on five datasets which demonstrate that the reordered pagerank algorithm can provide a speedup of as much as a factor of 6 we also note potential additional benefits that result from the proposed reordering

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

some properties of convex hulls of integer points contained in general convex sets in this paper we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it we first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed this leads to useful results for special classes of convex sets such as pointed cones strictly convex sets and sets containing integer points in their interior we then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron this result generalizes the well known result due to meyer math program 7 223 225 1974 under a simple technical assumption we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron

rational extrapolation for the pagerank vector an important problem in web search is to determine the importance of each page from the mathematical point of view this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1 since this matrix is neither stochastic nor irreducible the power method has convergence problems so the matrix is replaced by a convex combination depending on a parameter c with a rank one matrix its left principal eigenvector now depends on c and it is the pagerank vector we are looking for however when c is close to 1 the problem is ill conditioned and the power method converges slowly so the idea developed in this paper consists in computing the pagerank vector for several values of c and then to extrapolate them by a conveniently chosen rational function at a point near 1 the choice of this extrapolating function is based on the mathematical expression of the pagerank vector as a function of c numerical experiments end the paper

a harmonic restarted arnoldi algorithm for calculating eigenvalues and determining multiplicity abstract a restarted arnoldi algorithm is given that computes eigenvalues and eigenvectors it is related to implicitly restarted arnoldi but has a simpler restarting approach harmonic and regular rayleigh ritz versions are possible for multiple eigenvalues an approach is proposed that first computes eigenvalues with the new harmonic restarted arnoldi algorithm then uses random restarts to determine multiplicity this avoids the need for a block method or for relying on roundoff error to produce the multiple copies

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

the structure of the infinite models in integer programming the infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions in this paper we study the relationships between these two descriptions our results have implications for finite dimensional corner polyhedra one consequence is that nonnegative continuous functions suffice to describe finite dimensional corner polyhedra with rational data we also discover new facts about corner polyhedra with non rational data

linear regression analysis for fuzzy input and output data using the extension principle abstract the method for obtaining the fuzzy least squares estimators with the help of the extension principle in fuzzy sets theory is proposed the membership functions of fuzzy least squares estimators will be constructed according to the usual least squares estimators in order to obtain the membership value of any given value taken from the fuzzy least squares estimator optimization problems have to be solved we also provide the methodology for evaluating the predicted fuzzy output from the given fuzzy input data

unique minimal liftings for simplicial polytopes for a minimal inequality derived from a maximal lattice free simplicial polytope in rn we investigate the region where minimal liftings are uniquely defined and we characterize when this region covers rn we then use this characterization to show that a minimal inequality derived from a maximal lattice free simplex in rn with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points

a gmres power algorithm for computing pagerank problems abstract the pagerank algorithm for determining the importance of web pages has become a central technique in web search we propose a new method to speed up the convergence performance for computing pagerank when the damping factor is close to one called as gmres power which is based on a periodic combination of the power method with the gmres algorithm the description and convergence analysis of the new algorithm are discussed in detail numerical results are reported to confirm the efficiency of the new algorithm

fuzzy linear programming models for npd using a four phase qfd activity process based on the means end chain concept abstract quality function deployment qfd is a customer driven approach in processing new product development npd to maximize customer satisfaction determining the fulfillment levels of the hows including design requirements drs part characteristics pcs process parameters pps and production requirements prs is an important decision problem during the four phase qfd activity process for new product development unlike previous studies which have only focused on determining drs this paper considers the close link between the four phases using the means end chain mec concept to build up a set of fuzzy linear programming models to determine the contribution levels of each how for customer satisfaction in addition to tackle the risk problem in npd processes this paper incorporates risk analysis which is treated as the constraint in the models into the qfd process to deal with the vague nature of product development processes fuzzy approaches are used for both qfd and risk analysis a numerical example is used to demonstrate the applicability of the proposed model

constrained infinite group relaxations of mips recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized in this paper we consider a stronger relaxation of general mixed integer sets by allowing constraints such as bounds on the free integer variables in the continuous group relaxation we generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables

analysis of linear systems with fuzzy parametric uncertainty in this paper the problem of determining the stability margin of a linear system with fuzzy parametric uncertainty is addressed this kind of systems arise when the exact value of a process parameter is not available but an estimation of the value is given by an expert more or less 5 as the cut representation of a fuzzy number is a functional interval interval analysis shows to be a suitable tool to deal with fuzziness based on the argoun s stability test for interval systems a new methodology with a natural graphical interpretation is presented an independent confidence degree for every coefficient is considered allowing a clear understanding on how each coefficient affects stability an analysis on how precise one should be on some coefficients to allow more uncertainty on others can be carried out a comparison with tsypkin polyak plot is also presented

reverse split rank the reverse split rank of an integral polyhedron p p is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is p p already in mathbb r 3 r3 there exist polyhedra with infinite reverse split rank we give a geometric characterization of the integral polyhedra in mathbb r n rn with infinite reverse split rank

equivariant perturbation in gomory and johnson s infinite group problem iii foundations for the k dimensional case with applications to k 2 k 2 we develop foundational tools for classifying the extreme valid functions for the k dimensional infinite group problem in particular we present the general regular solution to cauchy s additive functional equation on restricted lower dimensional convex domains this provides a k dimensional generalization of the so called interval lemma allowing us to deduce affine properties of the function from certain additivity relations next we study the discrete geometry of additivity domains of piecewise linear functions providing a framework for finite tests of minimality and extremality we then give a theory of non extremality certificates in the form of perturbation functions we apply these tools in the context of minimal valid functions for the two dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane under a regularity condition called diagonal constrainedness we show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two dimensional group problem this gives an algorithm for testing the extremality of a given minimal valid function

two dimensional lattice free cuts and asymmetric disjunctions for mixed integer polyhedra in this paper we study the relationship between 2d lattice free cuts the family of cuts obtained by taking two row relaxations of a mixed integer program mip and applying intersection cuts based on maximal lattice free sets in r 2 and various types of disjunctions recently li and richard 2007 studied disjunctive cuts obtained from t branch split disjunctions of mixed integer sets these cu ts generalize split cuts balas 2009 initiated the study of cuts for the two row continuous group relaxation obtained from 2 branch split disjunctions we study these cuts and call them cross cuts for the two row continuous group relaxation and for general mips we also consider cuts obtained from asymmetric 2 branch disjunctions which we call crooked cross cuts for the two row continuous group relaxation we show that unimodular cross cuts the coefficients of the two split inequalities fo rm a unimodular matrix are equivalent to the cuts obtained from maximal lattice free sets other than type 3 triangles we also prove that all 2d lattice free cuts and their s free extensions ar e crooked cross cuts for general mixed integer sets we show that crooked cross cuts can be generated from a structured three row relaxation finally we show that for the corner relaxation of an mip every crooked cross cut is a 2d lattice free cut

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

google pageranking problem the model and the analysis the spectral and jordan structures of the web hyperlink matrix g c cg 1 c ev t have been analyzed when g is the basic stochastic google matrix c is a real parameter such that 0 1y c nv for all v this limit may fail to exist for some v if l is not semisimple as a special case of our results we obtain a complex analog of pagerank for the web hyperlink matrix g c with a complex parameter c we study regularity limits expansions and conditioning of y c and we propose algorithms e g complex extrapolation power method on a modified matrix etc that may provide an efficient way to compute pagerank also with c close or equal to 1 an interpretation of the limit vector nv and a related critical discussion on the model on its adherence to reality and possible ways for its improvement represent the contribution of the paper on modeling issues

maximal lattice free convex sets in linear subspaces we consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice free convex sets in an affine subspace we also show that these sets are polyhedra the latter result extends a theorem of lovasz characterizing maximal lattice free convex sets in rn

a k 1 slope theorem for the k dimensional infinite group relaxation we prove that any minimal valid function for the k dimensional innite group relaxation that is piecewise linear with at most k 1 slopes and does not factor through a linear map with non trivial kernel is extreme this generalizes a theorem of gomory and johnson for k 1 and cornu ejols and molinaro for k 2

monte carlo methods in pagerank computation when one iteration is sufficient pagerank is one of the principle criteria according to which google ranks web pages pagerank can be interpreted as a frequency of visiting a web page by a random surfer and thus it reflects the popularity of a web page google computes the pagerank using the power iteration method which requires about one week of intensive computations in the present work we propose and analyze monte carlo type methods for the pagerank computation there are several advantages of the probabilistic monte carlo methods over the deterministic power iteration method monte carlo methods already provide good estimation of the pagerank for relatively important pages after one iteration monte carlo methods have natural parallel implementation and finally monte carlo methods allow one to perform continuous update of the pagerank as the structure of the web changes

the general inner outer iteration method based on regular splittings for the pagerank problem abstract in this paper combined the regular splittings of the coefficient matrix i p with the inner outer iteration framework 9 a general inner outer gio iteration method is presented for solving the pagerank problem firstly the aor and modified aor maor methods for solving the pagerank problem are constructed and several comparison results are also given next the gio iteration scheme is developed and its overall convergence is analyzed in detail furthermore the preconditioner derived from the gio iteration can be used to accelerate the krylov subspace methods such as gmres method finally some numerical experiments on several pagerank problems are provided to illustrate the efficiency of the proposed algorithm

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

triangular and skew symmetric splitting method for numerical solutions of markov chains in this paper a theorem is presented to indicate that there exists a nonnegative constant e 0 such that the matrix a q email protected is a positive definite matrix where email protected r n x n is an identity matrix and q email protected r n x n is a matrix with positive diagonal and nonpositive off diagonal elements then a class of triangular and skew symmetric splitting tss iteration method is applied to solve the positive definite linear system ax b for obtaining the stationary probability vector of an irreducible markov chain theoretical analyses show that the tss iteration method converges unconditionally to the unique solution of the linear system with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved moreover the inexact triangular and skew symmetric splitting itss iteration method which employs certain krylov subspace methods as the inner iteration processes at each step of the outer tss iteration method is proposed to accelerate the convergence of the tss iteration method numerical experiments are used to illustrate the effectiveness of the tss and itss iteration methods

on computing pagerank via lumping the google matrix computing google s pagerank via lumping the google matrix was recently analyzed in i c f ipsen t m selee pagerank computation with special attention to dangling nodes siam j matrix anal appl 29 2007 1281 1296 it was shown that all of the dangling nodes can be lumped into a single node and the pagerank could be obtained by applying the power method to the reduced matrix furthermore the stochastic reduced matrix had the same nonzero eigenvalues as the full google matrix and the power method applied to the reduced matrix had the same convergence rate as that of the power method applied to the full matrix therefore a large amount of operations could be saved for computing the full pagerank vector in this note we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes called weakly nondangling nodes to another single node and the further reduced matrix is also stochastic with the same nonzero eigenvalues as the google matrix

master corner polyhedron vertices abstract we focus on the vertices of the master corner polyhedron mcp a fundamental object in the theory of integer linear programming we introduce two combinatorial operations that transform vertices to their neighbors this implies that each mcp can be defined by the initial vertices regarding these operations we call them support vertices we prove that the class of support vertices of all mcps over a group is invariant under automorphisms of this group and describe mcp vertex bases among other results we characterize its irreducible points establish relations between a vertex and the nontrivial facets that pass through it and prove that this polyhedron is of diameter 2

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

on an analysis of the strength of mixed integer cutting planes from multiple simplex tableau rows in this paper we derive valid inequalities for general mixed integer linear programs milps by considering several simplex tableau rows simultaneously we explore links between these cutting planes and lattice free convex sets which have a representation as the sum of a polytope and a linear space the codimension of the linear space of such a set is called split dimension we show that in terms of a strength measure introduced by goemans math programming 69 1995 pp 335 349 only lattice free convex sets with full split dimension give rise to a good approximation of the mixed integer hull whereas lattice free convex sets with low split dimension might approximate the mixed integer hull arbitrarily badly in general however we also show that ordinary split cuts approximate the mixed integer hull to within a constant factor when the size of the input data is given

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

an analysis of mixed integer linear sets based on lattice point free convex sets a maximal lattice free polyhedron l has max facet width equal to w if documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document max x in l pi t x min x in l pi t x leq w end document for all facets documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document pi t x leq pi 0 end document of l and documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usep

lifting properties of maximal lattice free polyhedra we study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice free polyhedra we prove a basic invariance property of unique minimal liftings for general maximal lattice free polyhedra this generalizes a previous result by basu et al math oper res 37 2 346 355 2012 for simplicial maximal lattice free polytopes thus completely settling this fundamental question about lifting for maximal lattice free polyhedra we further give a very general iterative construction to get maximal lattice free polyhedra with the unique lifting property in arbitrary dimensions this single construction not only obtains all previously known polyhedra with the unique lifting property but goes further and vastly expands the known list of such polyhedra finally we extend characterizations from basu et al 2012 about lifting with respect to maximal lattice free simplices to more general polytopes these nontrivial generalizations rely on a number of results from discrete geometry including the venkov alexandrov mcmullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces

fom accelerated by an extrapolation method for solving pagerank problems this paper formulates the pagerank problem a x x into a consistent singular linear system i a x 0 and applies the full orthogonalization method fom to solve it this singular system is characterized by index one namely i n d e x i a 1 we analyze the breakdown performance of fom on a general singular linear system and conclude that fom can determine a solution if it converges without any unfortunate breakdowns for our target problem then we propose to use a vector extrapolation method to speed up the convergence performance of fom this extrapolation procedure is based on ritz values which directly stems from the arnoldi extrapolation algorithm wu and wei 2010 eventually numerical experiments are presented to illustrate the effectiveness of our approaches

rating technical attributes in fuzzy qfd by integrating fuzzy weighted average method and fuzzy expected value operator quality function deployment qfd is a planning and problem solving tool that is gaining acceptance for translating customer requirements into the technical attributes of a product deriving the rating order of technical attributes from input variables is a crucial step in applying qfd when the relative weights of customer requirements and the relationship measures between customer requirements and technical attributes are expressed as fuzzy numbers calculating the importance of each technical attribute falls into the category of fuzzy weighted average in which the derived membership function of the fuzzy importance of each technical attribute is not explicitly known thus most ranking methods are not suitable under these circumstances a method is proposed in this paper using fuzzy weighted average method in the fuzzy expected value operator in order to rank technical attributes in fuzzy qfd an example of a flexible manufacturing system design is cited to demonstrate the application of the proposed approach

a new extrapolation method for pagerank computations abstract the pagerank algorithm is widely considered these years because of its great significance in search engine technology and other scientific domains though the power method is the initial measure to settle the pagerank problem it gives poor convergence when the damping factor is sufficiently close to 1 this difficulty encourages researchers to present improved iterative methods for accelerating pagerank computations in this paper a cheap and practical extrapolation approach which is determined by the trace of the google matrix is proposed and it is more attractive when combined with the well known arnoldi type algorithm convergence analysis of our algorithms is given numerical examples illustrate the efficiency of these accelerated schemes

split rank of triangle and quadrilateral inequalities a simple relaxation consisting of two rows of a simplex tableau is a mixed integer set with two equations two free integer variables and nonnegative continuous variables recently andersen et al and cornuejols and margot showed that the facet defining inequalities of this set are either split cuts or intersection cuts obtained from lattice free triangles and quadrilaterals from an example given by cook et al it is known that one particular class of facet defining triangle inequality does not have finite split rank in this paper we show that all other facet defining triangle and quadrilateral inequalities have finite split rank

equivariant perturbation in gomory and johnson s infinite group problem i the one dimensional case we give an algorithm for testing the extremality of minimal valid functions for gomory and johnson s infinite group problem that are piecewise linear possibly discontinuous with rational breakpoints this is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions we also present an extreme function that is a piecewise linear function with some irrational breakpoints whose extremality follows from a new principle

eigenvalues and jordan canonical form of a successively rank one updated complex matrix with applications to google s pagerank problem let a be an nxn complex matrix with eigenvalues l 1 l n counting algebraic multiplicities let x x 1 x k be a rank k matrix such that x 1 x k are right eigenvectors of a corresponding to l 1 l k for 1

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

convergence of polynomial restart krylov methods for eigenvalue computations krylov subspace methods have led to reliable and effective tools for resolving large scale non hermitian eigenvalue problems since practical considerations often limit the dimension of the approximating krylov subspace modern algorithms attempt to identify and condense significant components from the current subspace encode them into a polynomial filter and then restart the krylov process with a suitably refined starting vector in effect polynomial filters dynamically steer low dimensional krylov spaces toward a desired invariant subspace through their action on the starting vector the spectral complexity of nonnormal matrices makes convergence of these methods difficult to analyze and these effects are further complicated by the polynomial filter process the principal object of study in this paper is the angle an approximating krylov subspace forms with a desired invariant subspace convergence analysis is posed in a geometric framework that is robust to eigenvalue ill conditioning yet remains relatively uncluttered the bounds described here suggest that the sensitivity of desired eigenvalues exerts little influence on convergence provided the associated invariant subspace is well conditioned ill conditioning of unwanted eigenvalues plays an essential role this framework also gives insight into the design of effective polynomial filters numerical examples illustrate the subtleties that arise when restarting non hermitian iterations

unique lifting of integer variables in minimal inequalities this paper contributes to the theory of cutting planes for mixed integer linear programs milps minimal valid inequalities are well understood for a relaxation of an milp in tableau form where all the nonbasic variables are continuous they are derived using the gauge function of maximal lattice free convex sets in this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities we characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting this result first appeared in the proceedings of ipco 2010 the answer is a nonconvex region that can be obtained as a finite union of convex polyhedra we then establish a necessary and sufficient condition for the uniqueness of the lifting function

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

cut generating functions and s free sets we consider the separation problem for sets x that are pre images of a given set s by a linear mapping classical examples occur in integer programming as well as in other optimization problems such as complementarity one would like to generate valid inequalities that cut off some point not lying in x without reference to the linear mapping to this aim we introduce a concept cut generating functions cgf and we develop a formal theory for them largely based on convex analysis they are intimately related to s free sets and we study this relation disclosing several definitions for minimal cgf s and maximal s free sets our work unifies and puts into perspective a number of existing works on s free sets in particular we show how cgf s recover the celebrated gomory cuts

mixed integer sets from two rows of two adjacent simplex bases in andersen et al lecture notes in computer science vol 4513 springer berlin pp 1 15 2007 we studied a mixed integer set arising from two rows of a simplex tableau we showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables in this paper we generalize our findings and show that when upper bounds on the non basic variables are also considered further classes of facets arise that cannot be obtained from triangles and quadrilaterals specifically when exactly one upper bound on a non basic variable is introduced stronger inequalities that can be derived from pentagons involving up to six variables also appear