convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

interior point methods an old and new approach to nonlinear programming in this paper we discuss the main concepts of structural optimization a field of nonlinear programming which was formed by the intensive development of modern interior point schemes

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

general paradigm for distilling classical key from quantum states in this paper we develop a formalism for distilling a classical key from a quantum state in a systematic way expanding on our previous work on a secure key from bound entanglement horodecki et al 2005 more detailed proofs discussion and examples are provided of the main results namely we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory with the only change being that instead of distilling einstein podolsky rosen epr pairs the parties distill private states the form of these general private states are given and we show that there are a number of useful ways of expressing them some of the private states can be approximated by certain states which are bound entangled thus distillable entanglement is not a requirement for a private key we find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel pqc we also find a general class of states which have negative partial transpose are npt but which appear to be bound entangled the relative entropy distance is shown to be an upper bound on the rate of a key this allows us to compute the exact value of a distillable key for a certain class of private states

an o n 3 l adaptive path following algorithm for a linear complementarity problem in this paper we propose an o n3l algorithm which is a modification of the path following algorithm 8 for a linear complementarity problem the path following algorithm has to take a short step size in each iteration in order to bound the number of overall arithmetic operations by o n3l in practical computation we can determine the step size adaptively mizuno yoshise and kikuchi 11 reported that such an adaptive algorithm required about o l iterations for some test problems here we show that we can use a rank one update technique in the adaptive algorithm so that the number of overall arithmetic operations is theoretically bounded by o n3l

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

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

generalized q functions the modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of q or husimi functions a parameter lambda specifies orderings in a mapping from the operator psi sigma to the corresponding phase space wavefunction where sigma is a given fiducial vector the choice lambda 0 specifies the weyl mapping and the q function so obtained is the usual one when sigma is the vacuum state more generally any choice of of lambda in the range 1 1 corresponds to orderings varying between standard and anti standard for all such orderings the generalized q functions are non negative by construction they are shown to be proportional to expectation of the system state rho with respect to a generalized displaced squeezed state which depends on lambda and position p q in phase space thus when a system has been prepared in the state rho a generalized q function is proportional to the probability of finding it in the generalized squeezed state any such q function can also be written as the smoothing of the wigner function for the system state rho by convolution with the wigner function for the generalized squeezed state

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

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

long steps in an o n 3 l algorithm for linear programming we consider partial updating in ye s affine potential reduction algorithm for linear programming we show that using a goldstein armijo rule to safeguard a linesearch of the potential function during primal steps is sufficient to control the number of updates we also generalize the dual step construction to apply with partial updating the result is the first o n3l algorithm for linear programming whose steps are not constrained by the need to remain approximately centered the fact that the algorithm has a rigorous primal only initialization actually reduces the complexity to less than o m1 5n1 5l

complexity of circumscribed and inscribed ellipsoid methods for solving equilibrium economical models an exchange production model and the arrow debreu model with linear utility functions are considered these models are reduced to special convex programming problems these problems can be solved by the inscribed and circumscribed ellipsoid methods it is proved that the inscribed ellipsoid method s complexity in this case is less than the complexity of the circumscribed ellipsoid method

entanglement of the antisymmetric state we analyse the entanglement of the antisymmetric state in dimension d d and present two main results first we show that the amount of secrecy that can be extracted from the state is low more precisely the distillable key is bounded by o 1 d second we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state entanglement cost is larger than a constant independent of d the second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant finally we note that the regularised relative entropy of entanglement is asymptotically continuous in the state elementary and advanced facts from the representation theory of the unitary group including the concept of plethysm play a central role in the proofs of the main results

quadratic convergence in a primal dual method we show that the mizuno todd ye o nl iteration predictor corrector primal dual interior point algorithm for linear programming is quadratically convergent our proof does not assume that the problems be nondegenerate we do not assume that the iterate generated by the algorithm be convergent an assumption common to all previous asymptotic convergence analysis

on the complexity of nonnegative matrix scaling an n n nonnegative matrix a is said to be doubly stochastic scalable if there exist two positive diagonal matrices x and y such that xay is doubly stochastic we derive an upper bound on the norms of the scaling factors x and y and give a polynomial time complexity bound on the problem of computing the scaling factors to a prescribed accuracy

on the complexity of general matrix scaling and entropy minimization via the ras algorithm given an n m nonnegative matrix a a ij and positive integral vectors r e n and c e m having a common one norm h the r c scaling problem is to obtain positive diagonal matrices x and y if they exist such that xay has row and column sums equal to r and c respectively the entropy minimization problem corresponding to a is to find an n m matrix z z ij having the same zero pattern as a the sum of whose entries is a given number h its row and column sums are within given integral vectors of lower and upper bounds and such that the entropy function consisting of the sum of the terms z ij ln z ij a ij is minimized when the lower and upper bounds coincide matrix scaling and entropy minimization are closely related in this paper we present several complexity bounds for the e approximate r c scaling problem polynomial in n m h 1 e and ln v v where v and v are the largest and the smallest positive entries of a respectively these bounds although not polynomial in ln 1 1 e not only provide alternative complexities for the polynomial time algorithms but could result in better overall complexities in particular our theoretical analysis supports the practicality of the well known ras algorithm in our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities but also a complexity for square nonnegative matrices having positive permanent in particular our results extend nontrivially many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by kalantari and khachiyan to the case of general r c scaling finally we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals and the sum of all entries is also prescribed balinski and demange described an ras type algorithm for its continuous version but did not analyze its complexity we show that this algorithm produces an 1 e approximate solution within complexity polynomial in n m h v v and 1 1 e

on the scaling of multidimensional matrices abstract elementary proofs are given for theorems of bapat and raghavan on the scaling of nonnegative multidimensional matrices theorems of sinkhorn and of brualdi parter and schneider are derived as corollaries for positive two dimensional matrices hilbert s projective metric and a theorem of g birkhoff are used to prove that sinkhorn s original iterative procedure converges geometrically the ratio of convergence is estimated from the given data

wigner measures in noncommutative quantum mechanics we study the properties of quasi distributions or wigner measures in the context of noncommutative quantum mechanics in particular we obtain necessary and sufficient conditions for a phase space function to be a noncommutative wigner measure for a gaussian to be a noncommutative wigner measure and derive certain properties of the marginal distributions which are not shared by ordinary wigner measures moreover we derive the robertson schrodinger uncertainty principle finally we show explicitly how the set of noncommutative wigner measures relates to the sets of liouville and commutative wigner measures

polynomial affine algorithms for linear programming the method of steepest descent with scaling affine scaling applied to the potential functionq logc x i 1n logxi solves the linear programming problem in polynomial time forq n ifq n then the algorithm terminates in no more than o n2l iterations if q n sqrt n withq o n then it takes no more than o nl iterations a modified algorithm using rank 1 updates for matrix inversions achieves respectively o n4l and o n3 5l arithmetic computions

diagonal matrix scaling and linear programming a positive semidefinite symmetric matrix either has a nontrivial nonnegative zero or can be scaled by a positive diagonal matrix into a doubly quasi stochastic matrix this paper describes a simple path following newton algorithm of the complexity o sqrt n l iterations to either scale an n times n matrix or give a nontrivial nonnegative zero the latter problem is well known to be equivalent to linear programming

husimi transform of an operator product it is shown that the series derived by mizrahi giving the husimi transform or covariant symbol of an operator product is absolutely convergent for a large class of operators in particular the generalized liouville equation describing the time evolution of the husimi function is absolutely convergent for a large class of hamiltonians in contrast the series derived by groenewold giving the weyl transform of an operator product is often only asymptotic or even undefined the result is used to derive an alternative way of expressing expectation values in terms of the husimi function the advantage of this formula is that it applies in many of the cases where the anti husimi transform or contravariant symbol is so highly singular that it fails to exist as a tempered distribution

generalized scalings satisfying linear equations abstract we unify and generalize a broad class of problems referred in the literature as scaling problems by extending the applicability of a formulation suggested recently by bapat and raghavan specifically let a r n b r m and c r m n be given where a is strictly positive a c scaling of the vector a is defined to be a vector a r n with a i a i m k 1 u c kj k for some strictly positive vector u r m the problem of finding a c scaling of the vector a which satisfies the linear system cx b will be called a generalized scaling problem in this paper it is shown that previously studied matrix scaling problems e g finding scalings with prespecified row sums and column sums or finding scalings with row sums equaling the corresponding column sums or finding scalings of multidimensional matrices with prespecified margins are special instances of generalized scaling problems generalized scaling problems are reduced to convex optimization problems and the reduction is used to characterize solutions to develop necessary and sufficient conditions for their existence to establish uniqueness results and to characterize approximate solutions

general multiprocessor task scheduling approximate solutions in linear time we study the problem of scheduling n independent tasks on a set of m parallel processors where the execution time of a task is a function of the subset of processors assigned to the task for any fixed m we propose a fully polynomial approximation scheme that for any fixed epsilon 0 finds a preemptive schedule of length at most 1 epsilon times the optimum in o n time we also discuss the nonpreemptive variant of the problem and present for any fixed m a polynomial approximation scheme that computes an approximate solution of any fixed accuracy in linear time in terms of the running time this linear complexity bound gives a substantial improvement of the best previously known polynomial bound j chen and a miranda siam j comput 31 2001 pp 1 17

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

barrier functions in interior point methods we show that the universal barrier function of a convex cone introduced by nesterov and nemirovskii is the logarithm of the characteristic function of the cone this interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone this provides a simple method for calculating the universal barrier for homogeneous cones we identify some known barriers as the universal barrier scaled by an appropriate constant we also calculate some new universal barrier functions our results connect the field of interior point methods to several branches of mathematics such as lie groups jordan algebras siegel domains differential geometry complex analysis of several variables etc

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

speeding up karmarkar s algorithm for multicommodity flows we show how to speed up karmarkar s linear programming algorithm for the case of multicommodity flows the special structure of the constraint matrix is exploited to obtain an algorithm for the multicommodity flow problem which requires o s3 5v2 5el arithmetic operations each operation being performed to a precision of o l bits herev is the number of vertices ande is the number of edges in the given network s is the number of commodities andl is bounded by the number of bits in the input we obtain a speed up of the order of e0 5 v0 5 e2 5 v2 5s2 over karmarkar s modified algorithm which is substantial for dense networks the techniques in the paper can also be used to speed up any interior point algorithm for any linear programming problem whose constraint matrix is structurally similar to the one in the multicommodity flow problem

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

a procedure of chvatal for testing feasibility in linear programming and matrix scaling the problems of testing the feasibility of a system of linear inequalities or strict inequalities are well known to be the most fundamental problems in the theory and practice of linear programming from gordan s theorem it follows that ax b is feasible if and only if the homogeneous problem aty 0 bty s 0 0 0 y s 0 0 is infeasible we prove a stronger result if ax b is feasible then there is a feasible point satisfying x atw for some w 0 moreover there exists a feasible x atw satisfying aatw b w 1 where is a positive scalar and w 1 1 w1 1 wn t the existence of w and its computation is motivated by a procedure suggested by chvatal for solving linear programming as homogeneous problems as well as results on diagonal matrix scaling of positive semidefinite matrices not only these reveal the significance of the homogeneous problem but also practical and theoretical relevance of khachiyan and kalantari s diagonal matrix scaling algorithm in computing an interior point of a linear system of inequalities or in solving linear programming itself over the reals or the rationals

network coding a computational perspective in this work we study the computational perspective of network coding focusing on two issues first we address the computational complexity of finding a network code for acyclic multicast networks second we address the issue of reducing the amount of computation performed by network nodes in particular we consider the problem of finding a network code with the minimum possible number of encoding nodes i e nodes that generate new packets by performing algebraic operations on packets received over incoming links we present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph g v e in time 0 e kh v k2h2 h4k3 k h where k is the number of destinations and h is the number of packets our algorithm improves the best known running time for network code construction in addition our algorithm guarantees that the number of encoding nodes in the obtained network code is upper bounded by 0 h3k2 next we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes we prove that in the majority of settings this problem is np hard however we show that if h o 1 k o 1 and the underlying communication graph is acyclic then there exists an algorithm that solves this problem in polynomial time

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

semidefinite programming and matrix scaling over the semidefinite cone abstract let e be the hilbert space of real symmetric matrices with block diagonal form diag a m where a is n n and m is an l l diagonal matrix with the inner product x y trace xy we assume n l 1 i e allow n 0 or l 0 given x e we write x 0 x 0 if it is positive semidefinite positive definite let q e e be a symmetric positive semidefinite linear operator and min x 0 5 trace xqx x 1 x 0 the problem of testing if 0 is a significant problem called homogeneous programming on the one hand the feasibility problem in semidefinite programming sdp can be formulated as a homogeneous programming problem on the other hand it is related to the generalization of the classic problem of matrix scaling let 0 1 be a given accuracy u qe e e the identity matrix in e and n n l we describe a path following algorithm that in case 0 in o n ln n u newton iterations produces d 0 d 1 such that d if 0 in o n ln n u ln ln 1 newton iterations the algorithm produces d 0 such that dqde e where d is the operator that maps w e to d1 2wd1 2 moreover we use the algorithm to prove 0 if and only if there exists d 0 such that dqde e if and only if there exists d 0 such that qd 0 thus via this duality the matrix scaling problem is a natural dual to the feasibility problem in sdp this duality also implies that in blum et al bull ams 21 1989 1 real number model of computation the decision problem of testing the solvability of matrix scaling is both in np and co np although the above complexities can be deduced from our path following algorithm for general self concordant homogeneous programming and for matrix scaling obtained in scaling dualities and self concordant homogeneous programming in finite dimensional spaces technical report lcsr tr 359 department of computer science rutgers university new brunswick nj 1999 for the problems considered here the present analysis is quite elementary short and complete this simplicity is mainly due to a new inequality derived in this paper that relates the norm of scaled quantities at two successive newton iterations and implies a quadratic rate of convergence the present algorithm is not only a simple path following algorithm for testing the solvability of feasibility problem in sdp but is also capable of testing solvability of the matrix scaling problem when n 0 the algorithm reduces to the diagonal matrix scaling linear programming algorithm of khachiyan and kalantari siam j optim 4 1992 668 as in the case of lp the algorithm of this paper can be used to solve the general sdp problem

mixed states with positive wigner functions the wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent according to a result of hudson the characterization of mixed states with a positive wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters i e functions with given wigner spectrum in the sense of narcowich if a state is a convex combination of coherent states it has the property that the wigner spectrum contains the unit interval which in turn implies that the wigner function is positive it is shown by explicit examples that the converses of both implications are false the examples are taken from a low dimensional section of the state space in which all wigner spectra can be computed in this set counterexamples to a conjecture by narcowich concerning the wigner spectrum of products are also found as well as a state whose wigner spectrum is a convergent sequence of discrete points

on the diagonal scaling of euclidean distance matrices to doubly stochastic matrices we consider the problem of scaling a nondegenerate predistance matrix a to a doubly stochastic matrix b if a is nondegenerate then there exists a unique positive diagonal matrix c such that b cac we further demonstrate that if a is a euclidean distance matrix then b is a spherical euclidean distance matrix finally we investigate how scaling a nondegenerate euclidean distance matrix a to a doubly stochastic matrix transforms the points that generate a we find that this transformation is equivalent to an inverse stereographic projection

on the existence of supervisory policies that enforce liveness in discrete event dynamic systems modeled by controlled petri nets we consider discrete state plants represented by controlled petri nets ctlpns where a subset of transitions can be prevented from firing by a supervisor a transition in a ctlpn can fire at a marking if there are sufficient tokens in its input places and it is permitted to fire by the supervisor a ctlpn is live if it is possible to fire any transition from every marking that is reachable under supervision in this paper we derive a necessary and sufficient condition for the existence of a supervisory policy that enforces liveness in ctlpns we show this condition cannot be tested for an arbitrary ctlpn however for bounded ctlpns or ctlpns where each transition is individually controllable we show the existence of a supervisory policy which enforces that liveness is decidable we also show the existence of a supervisory policy that enforces liveness is necessary and sufficient for the existence of a minimally restrictive supervisory policy

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

o n3 noniterative heuristic algorithm for linear programs with error free implementation we present an o n 3 noniterative heuristic algorithm and describe an implementation of p adic arithmetic to solve the linear programming problems error free an optimality test as well as numerical examples depict the scope of such an algorithm in the few cases where it did not lead to the optimal solution it did provide a nonnegative solution of the system ax b which could be used as the basic feasible solution for any one of the widely used simplex methods say the revised simplex algorithm without the need to include artificial variables in ax b for consistency check thus the algorithm could be at least a pre processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix a and consequent additional space computational complexities and errors

scaling symmetric positive definite matrices to prescribed row sums we give a constructive proof of a theorem of marshall and olkin that any real symmetric positive definite matrix can be symmetrically scaled by a positive diagonal matrix to have arbitrary positive row sums we give a slight extension of the result showing that given a sign pattern there is a unique diagonal scaling with that sign pattern and we give upper and lower bounds on the entries of the scaling matrix

asymptotic relative entropy of entanglement we present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for werner states of arbitrary dimension we then demonstrate its validity using methods from convex optimization this is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

locating tree shaped facilities using the ordered median objective in this paper we consider the location of a tree shaped facility s on a tree network using the ordered median function of the weighted distances to represent the total transportation cost objective this function unifies and generalizes the most common criteria used in location modeling e g median and center if there are n demand points at the nodes of the tree t v e this function is characterized by a sequence of reals 1 n satisfying 1 2 n 0 for a given subtree s let x s x1 xn be the set of weighted distances of the n demand points from s the value of the ordered median objective at s is obtained as follows sort the n elements in x s in nonincreasing order and then compute the scalar product of the sorted list with the sequence two models are discussed in the tactical model there is an explicit bound l on the length of the subtree and the goal is to select a subtree of size l which minimizes the above transportation cost function in the strategic model the length of the subtree is variable and the objective is to minimize the sum of the transportation cost and the length of the subtree we consider both discrete and continuous versions of the tactical and the strategic models we note that the discrete tactical problem is np hard and we solve the continuous tactical problem in polynomial time using a linear programming lp approach we also prove submodularity properties for the strategic problem these properties allow us to solve the discrete strategic version in strongly polynomial time moreover the continuous version is also solved via lp for the special case of the k centrum objective we obtain improved algorithmic results using a dynamic programming dp algorithm and discretization and nestedness results

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

on complexity of matrix scaling abstract line sun scaling problem for a nonnegative matrix a is to find positive definite diagonal matrices y z which result in prescribed row and column sums of the scaled matrix yaz the matrix balancing problem for a nonnegative square matrix a is to find a positive definite diagonal matrix x such that the row sums in the scaled matrix xax are equal to the corresponding column sums we demonstrate that versions of both these problems same as those of other scaling problems for non negative multiindex arrays can be reduced to a specific geometric programming problem for the latter problem we develop a polynomial time algorithm thus deriving polynomial time solvability of a number of generic scaling problems for nonnegative multiindex arrays our results extend those previously known for the problems of matrix balancing 3 and of double stochastic scaling of a square nonnegative matrix 2

a scaling technique for finding the weighted analytical center of a polytope let a bounded full dimensional polytope be defined by the systemax b wherea is anm n matrix letai denote theith row of the matrixa and define theweighted analytic center of the polytope to be the point that minimizes the strictly convex barrier function i 1mwi ln aitx bi the proper selection of weightswi can make any desired point in the interior of the polytope become the weighted analytic center as a result the weighted analytic center has applications in both linear and general convex programming for simplicity we assume that the weights are positive integers if some of thewi s are much larger than others then newton s method for minimizing the resulting barrier function is very unstable and can be very slow previous methods for finding the weighted analytic center relied upon a rather direct application of newton s method potentially resulting in very slow global convergence we present a method for finding the weighted analytic center that is based on the scaling technique of edmonds and karp and is an enhancement of newton s method the scaling algorithm runs in o sqrt m log w iterations wherem is the number of constraints defining the polytope andw is the largest weight given on any constraint each iteration involves taking a step in the newton direction and its complexity is dominated by the time needed to solve a system of linear equations

a comparative study of algorithms for matrix balancing the problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics urban planning statistics demography and stochastic modeling these problems are called matrix balancing problems we describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing matrix scaling and nonlinear optimization the algorithms we study are the ras algorithm a diagonal similarity scaling algorithm and a truncated newton algorithm for network optimization we present results from computational experiments with large scale problems based on producing consistent estimates of social accounting matrices for developing countries

algorithms for proportional matrices in reals and integers letr be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh 0 closely related axiomatic approaches have been developed to ascribe meanings to the statements the real matrixf r and the integer matrixa r are proportional to a given matrixp 0

converse bounds for quantum and private communication over holevo werner channels werner states have a host of interesting properties which often serve to illuminate the unusual properties of quantum information starting from these states one may define a family of quantum channels known as the holevo werner channels which themselves afford several unusual properties in this paper we use the teleportation covariance of these channels to upper bound their two way assisted quantum and secret key capacities this bound may be expressed in terms of relative entropy distances such as the relative entropy of entanglement and also in terms of the squashed entanglement most interestingly we show that the relative entropy bounds are strictly sub additive for a sub class of the holevo werner channels so that their regularisation provides a tighter performance these information theoretic results are first found for point to point communication and then extended to repeater chains and quantum networks under different types of routing strategies

karmarkar s algorithm with improved steps we define a function of the interior of the feasible region of karmarkar s canonical linear program with the property that its supremum is bounded above by one if and only if the optimal value of the linear program is zero we use this function to obtain data dependent steps for karmarkar s algorithm the corresponding reduction in the potential function improves upon the data independent reduction derived by padberg we analyze the general step and consider its potential practical advantages

quantum bochner s theorem for phase spaces built on projective representations bochner s theorem gives the necessary and sufficient conditions on a function such that its fourier transform corresponds to a true probability density function in the wigner phase space picture quantum bochner s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid and possibly mixed quantum state and such that its fourier transform is a true probability density we extend this theorem to discrete phase space representations which possess enough symmetry more precisely we show that discrete phase space representations that are built on projective unitary representations of abelian groups with a slight restriction on admissible two cocycles enable a quantum bochner s theorem

time evolution of the wigner function in this paper we give a partial answer to the problem when does an initially non negative wigner function remain non negative under the effect of the time evolution we show that for pure states this is the case for linear systems only to prove this we use the fact that the wigner function is non negative if and only if the wavefunction is gaussian we also prove that the green s solution of the evolution equation of the wigner function which in the framework of probability theory corresponds to the conditional probability density takes on negative values we utilize a theorem about moments borrowed from pawula we conclude that the wigner phase space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

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

the nonlinear geometry of linear programming 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

on the complexity of quantified linear systems in this paper we explore the computational complexity of the conjunctive fragment of the first order theory of linear arithmetic quantified propositional formulas of linear inequalities with k 1 quantifier alternations are log space complete in s k p or p k p depending on the initial quantifier we show that when we restrict ourselves to quantified conjunctions of linear inequalities i e quantified linear systems the complexity classes collapse to polynomial time in other words the presence of universal quantifiers does not alter the complexity of the linear programming problem which is known to be in p our result reinforces the importance of sentence formats from the perspective of computational complexity

wigner functions on non standard symplectic vector spaces we consider the weyl quantization on a flat non standard symplectic vector space we focus mainly on the properties of the wigner functions defined therein in particular we show that the sets of wigner functions on distinct symplectic spaces are different but have non empty intersections this extends previous results to arbitrary dimension and arbitrary constant symplectic structure as a by product we introduce and prove several concepts and results on non standard symplectic spaces which generalize those on the standard symplectic space namely the symplectic spectrum williamson s theorem and narcowich wigner spectra we also show how wigner functions on non standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

tight uniform continuity bounds for quantum entropies conditional entropy relative entropy distance and energy constraints we present a bouquet of continuity bounds for quantum entropies falling broadly into two classes first a tight analysis of the alicki fannes continuity bounds for the conditional von neumann entropy reaching almost the best possible form that depends only on the system dimension and the trace distance of the states almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators as applications we give new proofs with tighter bounds of the asymptotic continuity of the relative entropy of entanglement er and its regularization e r infty as well as of the entanglement of formation ef using a novel quantum coupling of density operators which may be of independent interest we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation aka entanglement cost e c e f infty second we derive analogous continuity bounds for the von neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint most importantly systems of multiple quantum harmonic oscillators while without an energy bound the entropy is discontinuous it is well known to be continuous on states of bounded energy however a quantitative statement to that effect seems not to have been known here under some regularity assumptions on the hamiltonian we find that quite intuitively the gibbs entropy at the given energy roughly takes the role of the hilbert space dimension in the finite dimensional fannes inequality

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

polynomial time computation of strong and n present value optimal policies in markov decision chains this paper studies the problem of finding a stationary strong present value optimal and more generally an n present value optimal policy for an infinite horizon stationary finite state action substochastic markov decision chain a policy is strong present value optimal if it has maximum present value for all small positive interest rates a policy is n present value optimal if its present value falls below the maximum present value for all small positive by o n 1 the importance of stationary n present value optimal policies arises from the following known facts the set of such policies diminishes with n and for n s is precisely the set of stationary strong present value optimal policies for 0 n s stationary n present value optimal policies are nearly strong present value optimal and are of independent interest the best algorithms for finding stationary strong present value optimal policies find stationary n present value policies for n 1 s in that order this paper decomposes th

a combinatorial interior point method for network flow problems for solving minimum cost flow problems we develop a combinatorial interior point method based on a variant of the algorithm of karmarkar described in gonzaga 3 4 gonzaga proposes search directions generated by projecting certain directions onto the nullspace ofa by the special combinatorial structure of networks any projection onto the nullspace ofa can be interpreted as a flow in the incremental graph ofg in particular to evaluate the new search direction it is sufficient to choose a negative circuit subject to costs on the arcs depending on the current solution that approach results in an o mn2l algorithm wherem denotes the number of vertices n denotes the number of arcs andl denotes the total length of the input data

solving linear programming problems exactly a linear programming problem in an inequality form having a bounded solution is solved error free using an algorithm that sorts the inequalities removes the redundant ones and uses the p adic arithmetic c elsevier science inc 1997

when is the wigner function of multidimensional systems nonnegative it is shown that for systems with an arbitrary number of degrees of freedom a necessary and sufficient condition for the wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form this result generalizes the one obtained by hudson rep math phys 6 249 1974 for one dimensional systems

ellipsoids that contain all the solutions of a positive semi definite linear complementarity problem this paper deals with the lcp linear complementarity problem with a positive semi definite matrix assuming that a strictly positive feasible solution of the lcp is available we propose ellipsoids each of which contains all the solutions of the lcp we use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the lcp we can apply the lower bound to test whether a given variable is positive over the solution set of the lcp that is if the lower bound is positive we know that the variable is positive over the solution set of the lcp hence by the complementarity condition its complement is zero in this case we can eliminate the variable and its complement from the lcp we also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path following algorithm proposed by the authors for the lcp if the lcp has a unique non degenerate solution the lower bound and the upper bound for the solution computed at each iteration of the path following algorithm both converge to the solution of the lcp

the generalized weyl correspondence and time dependent stochastic processes the definition and some general properties of the generalized weyl correspondence between stochastic processes and operator valued real functions on a hilbert space plus a trace class operator are given the relation between the derivatives of an operator valued function and the derivative of the corresponding stochastic process are studied when the operator valued function is the position or momentum in the heisenberg picture a condition for the positivity of the joint distribution functions of the corresponding process is given provided that the evolution hamiltonian be quadratic in the position and momentum finally the case of an arbitrary hamiltonian evolution for the position operator is studied and the two dimensional density functions of the process is related to the wigner function associated to some state and a necessary condition for the positivity of the densities is given

on linear programming and matrix scaling over the algebraic numbers abstract adler and beling considered the linear programming problem over the real algebraic numbers they obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial time solvability using the ellipsoid method and under some models of computation based on a better notion of size than that used by adler and beling we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint matrix dimensions for these canonical problems as well as for the matrix scaling problem shown to be a more general problem than linear programming we obtain the necessary bounds demonstrate simple rounding schemes justify the applicability of two polynomial time interior point algorithms under some models of computation describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior point algorithms under an input model and give an alternative method to the traditional duality based approach for the conversion of a general linear programming problem into a feasibility problem

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

o nρl iteration and o n3l operation potential reduction algorithms for linear programming abstract in this paper we propose two potential reduction algorithms which we call algorithm 1 and algorithm 2 for linear programming algorithm 1 has parameters in the potential function and which determines a step size suppose that n 1 and 0 2 n 0 5 for 0 5 1 then algorithm 1 requires at most o n l iterations and o n 3 l arithmetic operations in total if we take n and 0 2 algorithm 1 is similar to ye s primal form algorithm algorithm 2 has a parameter n 1 for 0 5 1 it requires at most o n l iterations and o n 3 l arithmetic operations in total if n algorithm 2 is similar to the long step algorithm of anstreicher and bosch

wigner quantisation of arrival time and oscillator phase by means of the wigner distribution function an expression is derived for the quasiprobability that a free particle in one dimension arrives at the origin during a given time interval and also for the quasiprobability that the phase of a one dimensional harmonic oscillator lies in a given segment of the circle it is shown that the best bounds on these quasiprobabilities p are 0 1559 or p or 1 0077 independent of the size of the interval it is shown that on disjoint unions of infinitely many intervals the quasiprobabilities may add up to any positive or negative real value it is briefly indicated how negative probabilities can be avoided altogether in an alternative approach to the quantisation of arrival time and oscillator phase

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

random walks on polytopes and an affine interior point method for linear programming let k be a polytope in rn defined by m linear inequalities we give a new markov chain algorithm to draw a nearly uniform sample from k the underlying markov chain is the first to have a mixing time that is strongly polynomial when started from a central point we use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

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

a potential reduction variant of renegar s short step path following method for linear programming abstract we propose a new polynomial potential reduction method for linear programming which can also be seen as a large step path following method we do an approximate linesearch along the newton direction with respect to renegar s strictly convex potential function if the iterate is far away from the central trajectory if the iterate lies close to the trajectory we update the lower bound for the optimal value dependent on this updating scheme the iteration bound can be proved to be o n l or o nl our method differs from the recently published potential reduction methods in the choice of the potential function and the search direction

a theorem of the alternative for multihomogeneous functions and its relationship to diagonal scaling of matrices abstract we prove that given a multihomogeneous function satisfying some initial conditions either it has a certain nonnegative zero over a given subspace or an associated logarithmic barrier function has a constrained stationary point under convexity precisely one of these conditions is satisfied the main ingredients of the proof are the derivation of significant properties of the constrained stationary points of the logarithmic barrier function and their relationship to corresponding points of an associated karmarkar potential function corollaries of the theorem include a duality for karmarkar s canonical linear program which happens to be a stronger version of gordan s theorem a more general duality than an existing one concerning the diagonal scaling of symmetric matrices also shown to be a stronger version of gordan s theorem and a diagonal scalability result for a class of multihomogeneous polynomials which is more general than a previously known result on the scalability of positive multidimensional matrices we also give an algorithmic proof of the theorem of the alternative through a projective algorithm which is a generalization of a modified karmarkar linear programming algorithm and in the context of nonnegative matrix scaling becomes a variant of the well known ras algorithm