worst case analysis of demand point aggregation for the euclidean p median problem abstract solving large scale p median problems is usually time consuming people often aggregate the demand points in a large scale p median problem to reduce its problem size and make it easier to solve most traditional research on demand point aggregation is either experimental or assuming uniformly distributed demand points in analytical studies in this paper we study demand point aggregation for planar p median problem when demand points are arbitrarily distributed efficient demand aggregation approaches are proposed with the corresponding attainable worst case aggregation error bounds measured we demonstrate that these demand aggregation approaches introduce smaller worst case aggregation error bounds than that of the honeycomb heuristic papadimitriou c h 1981 worst case and probabilistic analysis of a geometric location problem siam journal on computing 10 542 557 when demand points are arbitrarily distributed we also conduct numerical experiments to show their effectiveness

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

an ode based trust region method for unconstrained optimization problems in this paper a new trust region algorithm is proposed for solving unconstrained optimization problems this method can be regarded as a combination of trust region technique fixed step length and ode based methods a feature of this proposed method is that at each iteration only a system of linear equations is solved to obtain a trial step another is that when a trial step is not accepted the method generates an iterative point whose step length is defined by a formula under some standard assumptions it is proven that the algorithm is globally convergent and locally superlinear convergent preliminary numerical results are reported

on the global convergence of the bfgs method for nonconvex unconstrained optimization problems this paper is concerned with the open problem of whether the bfgs method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems we propose a cautious bfgs update and prove that the method with either a wolfe type or an armijo type line search converges globally if the function to be minimized has lipschitz continuous gradients

on the performance of a new symmetric rank one method with restart for solving unconstrained optimization problems quasi newton qn methods are generally held to be the most efficient minimization methods for solving unconstrained optimization problems among the qn methods symmetric rank one sr1 is one of the very competitive formulas in the present paper we propose a new sr1 method the new technique attempts to improve the quality of the sr1 hessian by employing the scaling of the identity in a certain sense however since at some iterations these updates might be singular indefinite or undefined this paper proposes an updates criterion based on the eigenvalues of the sr1 update to measure this quality hence the new method is employed only to improve the approximation of the sr1 hessian it is shown that the numerical results support the theoretical considerations for the usefulness of this criterion and show that the proposed method improves the performance of the sr1 update substantially

scaled memoryless symmetric rank one method for large scale optimization this paper concerns the memoryless quasi newton method that is precisely the quasi newton method for which the approximation to the inverse of hessian at each step is updated from the identity matrix hence its search direction can be computed without the storage of matrices in this paper a scaled memoryless symmetric rank one sr1 method for solving large scale unconstrained optimization problems is developed the basic idea is to incorporate the sr1 update within the framework of the memoryless quasi newton method however it is well known that the sr1 update may not preserve positive definiteness even when updated from a positive definite matrix therefore we propose the memoryless sr1 method which is updated from a positive scaled of the identity where the scaling factor is derived in such a way that positive definiteness of the updating matrices are preserved and at the same time improves the condition of the scaled memoryless sr1 update under very mild conditions it is shown that for strictly convex objective functions the method is globally convergent with a linear rate of convergence numerical results show that the optimally scaled memoryless sr1 method is very encouraging

a nonmonotone trust region method for unconstrained optimization in this paper we propose a nonmonotone trust region method for unconstrained optimization our method can be regarded as a combination of nonmonotone technique fixed steplength and trust region method when a trial step is not accepted the method does not resolve the subproblem but generates a iterative point whose steplength is defined by a formula we only allow increase in function value when trial steps are not accepted in close succession of iterations under mild conditions we prove that the algorithm is global convergence and superlinear convergence primary numerical results are reported

analysis of centroid aggregation for the euclidean distance p median problem demand data aggregation results in loss of information and thereby induces errors in the locational decision being made both in the facility location configurations optimality error and in the computed value of the objective function cost error the aggregation effect is quite problem specific depending on the aggregation scheme used and on the demand pattern in this paper we perform a theoretical analysis for the centroid aggregation effect on the euclidean distance p median location problem we study the worst case and average case errors and in the multi facility location model source c error is closely examined the results of the paper are illustrated via numerical examples and some empirical findings of previous work are interpreted using our analytical results

finite dominating sets for network location problems a research theme involving location on networks since its inception has been the identification of a finite dominating set fds or a finite set of points to which an optimal solution must belong we attempt to unify and generalize results of this sort we survey the literature and then prove some theorems that subsume most previous results and that are at the same time more general than previous results the paper is aimed primarily at investigators who wish to know whether an fds exists for a specific problem

optimal conditioning and convergence in rank one quasi newton updates we show that the optimally conditioned update proposed by spedicato and greenstadt for solving nonlinear equations is obtainable in closed form however unlike its rank two counterpart designed for minimization conditioning in the rank one case is not necessarily a good measure of closeness we propose updates that compromise the two properties local and superlinear convergence is proved for a class of updates

classical particles with internal structure ii second order internal spaces a systematic study of classical relativistic particles with internal structure initiated in a previous paper is continued and a study of second order internal spaces sos is presented within the framework of the lagrangian form of constrained dynamics such internal spaces q are those for which a phase space treatment must necessarily use the cotangent bundle t q the large variety of possible sos s ten discrete cases and two one parameter families is separated into those capable of a manifestly covariant description and those for which special methods based on the transitive action of sl 2 c on a coset space are needed the concept of the isotopy representation plays an important role in this context seven of the possible discrete sos s are shown to be describable in a manifestly covariant way two discrete and one one parameter family of sos s are shown to be unphysical in the sense that no lagrangians can be written in which the internal and the space time position variables are nontrivially coupled and the remaining single discrete and one one parameter family are shown to be physical though not describable in a manifestly covariant way general phase space methods uniformly applicable to all sos s are developed and as an illustrative example a lagrangian model for an sos in which the internal space is spanned by two orthonormal spacelike unit vectors is presented

new quasi newton methods for unconstrained optimization problems abstract many methods for solving minimization problems are variants of newton method which requires the specification of the hessian matrix of second derivatives quasi newton methods are intended for the situation where the hessian is expensive or difficult to calculate quasi newton methods use only first derivatives to build an approximate hessian over a number of iterations this approximation is updated each iteration by a matrix of low rank in unconstrained minimization the original quasi newton equation is bk 1sk yk where yk is the difference of the gradients at the last two iterates in this paper we first propose a new quasi newton equation b k 1 s k y k in which y k is decided by the sum of yk and aksk where ak is some matrix then we give two choices of ak which carry some second order information from the hessian of the objective function the three corresponding bfgs type algorithms are proved to possess global convergence property the superlinear convergence of the one algorithm is proved extensive numerical experiments have been conducted which show that the proposed algorithms are very encouraging

a framework for demand point and solution space aggregation analysis for location models abstract many location problems can be formulated as minimizing some location objective function subject to upper bounds on other location constraint functions when such functions are subadditive and nondecreasing in the distances a common occurrence worst case demand point aggregation error bounds are known we show how to solve a relaxation and a restriction of the aggregated problem in such a way as to obtain lower and upper bounds on the optimal value of the original problem we consider some applications to covering and related problems

an adaptive multiphase approach for large unconditional and conditional p median problems abstract a multiphase approach that incorporates demand points aggregation variable neighbourhood search vns and an exact method is proposed for the solution of large scale unconditional and conditional p median problems the method consists of four phases in the first phase several aggregated problems are solved with a local search with shaking procedure to generate promising facility sites which are then used to solve a reduced problem in phase 2 using vns or an exact method the new solution is then fed into an iterative learning process which tackles the aggregated problem phase 3 phase 4 is a post optimisation phase applied to the original disaggregated problem for the p median problem the method is tested on three types of datasets which consist of up to 89 600 demand points the first two datasets are the birch and the tsp datasets whereas the third is our newly geometrically constructed dataset that has guaranteed optimal solutions the computational experiments show that the proposed approach produces very competitive results the proposed approach is also adapted to cater for the conditional p median problem with interesting results

aggregation error bounds for a class of location models many location models involve distances and demand points in their objective function in urban contexts there can be millions of demand points this leads to demand point aggregation which produces error we identify a general model structure that includes most such location models and present a means of obtaining error bounds for all models with this structure the error bounds suggest how to do the demand point aggregation so as to keep the error small

matroid theory and chern simons it is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum yang mills physics but also for m theory our discussion is focused in an action consisting purely of the chern simons term but in principle the main ideas can be applied beyond such an action in our treatment the theorem due to thistlethwaite which gives a relationship between the tutte polynomial for graphs and jones polynomial for alternating knots and links plays a central role before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the fano matroid and d 11 supergravity our approach also seems to be related to loop solutions of quantum gravity based in an ashtekar formalism

worst case and probabilistic analysis of a geometric location problem we consider the problem of choosing k medians among n points on the euclidean plane such that the sum of the distances from each of the n points to its closest median is minimized we show that this problem is np complete we also present two heuristics that produce arbitrarily good solutions with probability going to 1 one is a partition heuristic and works when k grows linearly or almost so with n the other is the honeycomb heuristic and is applicable to rates of growth of k of the form k sim n varepsilon 0 varepsilon 1

thomas rotation and the mixed state geometric phase it is shown that uhlmann s parallel transport of purifications along a path of mixed states represented by 2 2 density matrices is just the path ordered product of thomas rotations these rotations are invariant under hyperbolic translations inside the bloch sphere that can be regarded as the poincare ball model of hyperbolic geometry a general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the bures fidelities is derived the formula gives back the solid angle result well known from studies of the pure state geometric phase it is also shown that this mixed state anholonomy can be reinterpreted as the pure state non abelian anholonomy of entangled states living in a suitable restriction of the quaternionic hopf bundle in this picture uhlmann s parallel transport is just the pancharatnam transport of quaternionic spinors

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

a theoretical and experimental study of the symmetric rank one update this paper first discusses computational experience using the sri update in conventional line search and trust region algorithms for unconstrained optimization the experiments show that the sri is very competitive with the widely used bfgs method they also indicate two interesting features the final hessian approximations produced by the sri method are not generally appreciably better than those produced by the bfgs and the sequences of steps produced by the sri do not usually seem to have the uniform linear independence property that is assumed in recent convergence analysis this paper presents a new analysis that shows that the sri method with a line search is n 1 step q superlinearly convergent without the assumption of linearly independent iterates this analysis assumes that the hessian approximations are positive definite and bounded asymptotically which from computational experience are reasonable assumptions

a mixed integer programming formulation and solution for traffic analysis zone delineation considering zone amount decision abstract due to the spatial aggregation effect the amount of zones aoz has a direct influence on the optimality of traffic analysis zone delineation tazd problems unlike existing studies this paper addresses a tazd problem in which both the aoz and zone partitions are optimized for the tazd problem we propose a mixed integer programming mip model with the optimization objective of minimizing the sum of geographical errors within each zone to solve the tazd model the following works have been done first the tazd model is strengthened by our proposed ir influence region construction algorithm second based on the strengthened model we analyse the lower bound of the optimal aoz from the theoretical perspective and then the drmi domain reduction method i is proposed to specify a tighter lower bound also to reduce the amount of decision variables we put forward the drmii domain reduction method ii which enumerates all feasible aoz values third to settle a non solution situation we propose the ahcbha aggregation hierarchical clustering based heuristic algorithm in order to rebuild the solution space fourth we then design two methods to solve the tazd model the former method combines the ahcbha with the drmi and the latter method combines the ahcbha with the drmii finally the performance of the proposed tazd and the developed methods is explored using a numerical example and a real world case the obtained results constitute good solutions

on the stable equilibrium points of gradient systems this paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f in particular it is shown that under the assumption that f is real analytic local minimality is necessary and sufficient for stability under the weaker assumption that f is indefinitely continuously differentiable local minimality is neither necessary nor sufficient for stability

classical particles with internal structure general formalism and application to first order internal spaces group theoretic methods are used to systematically classify all possible internal structures for an elementary classical relativistic particle in terms of coset spaces of sl 2 c with respect to its continuous subgroups the allowed internal spaces q are separated into first and second order ones depending on whether a canonical description can be given using q itself or if it needs the cotangent bundle t q three of the former are found one corresponding to the use of a majorana spinor as the internal variable the other two related to orbits in the lie algebra of so 3 1 under the adjoint action for the latter two a lagrangian description of an elementary object with the corresponding internal space is set up and the dynamics studied

soldered bundle background for the de sitter top we prove that the mathematical framework for the de sitter top system is the de sitter fiber bundle in this context the concept of soldering associated with a fiber bundle plays a central role we comment on the possibility that our formalism may be of particular interest in different contexts including macdowell mansouri theory two time physics and oriented matroid theory

sizing and least change secant methods oren and luenberger introduced in 1974 a strategy for replacing hessian approximations by their scalar multiples and then performing quasi newton updates generally least change secant updates such as the bfgs or dfp updates oren and luenberger management sci 20 1974 pp 845 862 in this paper the function omega a left frac operatorname trace a n operatorname det a 1 n right is shown to be a measure of change with a direct connection to the oren luenberger strategy this measure is interesting because it is related to the ell 2 condition number but it takes all the eigenvalues of a into account rather than just the extremes if the class of possible updates is restricted to the broyden class i e scalar premultiples are not allowed then the optimal update depends on the dimension of the problem it may or may not be in the convex class but it becomes the bfgs update as the dimension increases this seems to be yet another explanation for why

multi step quasi newton methods for optimization abstract quasi newton methods update at each iteration the existing hessian approximation or its inverse by means of data deriving from the step just completed we show how multi step methods employing in addition data from previous iterations may be constructed by means of interpolating polynomials leading to a generalization of the secant or quasi newton equation the issue of positive definiteness in the hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original single step methods the results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach by comparison with single step methods particularly as the dimension of the problem increases

improved hessian approximation with modified secant equations for symmetric rank one method symmetric rank one sr1 is one of the competitive formulas among the quasi newton qn methods in this paper we propose some modified sr1 updates based on the modified secant equations which use both gradient and function information furthermore to avoid the loss of positive definiteness and zero denominators of the new sr1 updates we apply a restart procedure to this update three new algorithms are given to improve the hessian approximation with modified secant equations for the sr1 method numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard sr1 and bfgs updates is clearly observed

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

wigner rotations bargmann invariants and geometric phases the concept of the wigner rotation familiar from the composition law of pure lorentz transformations is described in the general setting of lie group coset spaces and the properties of coset representatives examples of abelian and non abelian wigner rotations are given the lorentz group wigner rotation occurring in the coset space sl 2 r so 2 simeq so 2 1 so 2 is shown to be an analytic continuation of a wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations belonging to the coset space su 2 u 1 simeq so 3 so 2 the possibility of interpreting these two wigner rotations as geometric phases is shown in detail essential background material on geometric phases bargmann invariants and null phase curves all of which are needed for this purpose is provided

euclidean hub and spoke networks the hub and spoke distribution paradigm has been a fundamental principle in geographic network design for more than 40 years one of the primary advantages that such networks possess is their ability to exploit economies of scale in transportation by aggregating network flows through common sources in this paper we consider the problem of designing an optimal hub and spoke network in continuous euclidean space the spokes of the network are distributed uniformly over a service region and our objective is to determine the optimal number of hub nodes and their locations we consider seven different backbone network topologies for connecting the hub nodes namely the steiner and minimum spanning trees a travelling salesman tour a star network a capacitated vehicle routing tour a complete bipartite graph and a complete graph we also perform an additional analysis on a multilevel network in which network flows move through multiple levels of transshipment before reaching the service region we desc

quasi newton methods based on ordinary differential equation approach for unconstrained nonlinear optimization in this paper we propose a hybrid ode based quasi newton qn method for unconstrained optimization problems which combines the idea of low order implicit runge kutta rk techniques for gradient systems with the qn type updates of the jacobian matrix such as the symmetric rank one sr1 update the main idea of this approach is to associate a qn matrix to approximate numerically the jacobian matrix in the gradient system fundamentally this is a gradient system based on the first order optimality conditions of the optimization problem to further extend the methods for solving large scale problems a feature incorporated to the proposed methods is that a limited memory setting for matrix vector multiplications is used thus avoiding the computational and storage issues when computing jacobian information under suitable assumptions global convergence of the proposed method is proved practical insights on the effectiveness of these approaches on a set of test functions are given by a numerical comparison with that of the limited memory bfgs algorithm l bfgs and conjugate gradient algorithm cg

searching for a connection between matroid theory and string theory we make a number of observations about matter ghost string phase which may eventually lead to a formal connection between matroid theory and string theory in particular in order to take advantage of the already established connection between matroid theory and chern simons theory we propose a generalization of string theory in terms of some kind of kahler metric we show that this generalization is closely related to the kahler chern simons action due to nair and schiff in addition we discuss matroid string connection via matroid bundles and a schild type action and we add new information about the relationship between matroid theory d 11 supergravity and chern simons formalism

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

a modified bfgs method and its global convergence in nonconvex minimization in this paper we propose a modification of the bfgs method for unconstrained optimization a remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function under certain conditions we also establish superlinear convergence of the method

berry phases and unitary transformations berry phases for spin are defined for any closed loop made by a vector changing direction in three dimensional space a sequence of rotations moves the vector along the loop each rotation is around the axis perpendicular to the moving vector the berry phases come from the eigenvalues of the unitary operator for the product of these rotations the angle of the product rotation is shown to be the solid angle enclosed by the loop the proof uses the ordinary language of quantum mechanics the product is calculated from the commutation relations for spin a general framework is set up to define berry phases for other transformations and states like those for rotations and spin the integral formula is derived alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics an example is used to show how one hamiltonian may be simpler than others adiabatic evolution is obtained in the limit as a coupling constant goes to zero so the adiabatic changes are m