the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

linear time algorithm to cover and hit a set of line segments optimally by two axis parallel squares abstract this paper discusses the problem of covering and hitting a set of line segments l in r 2 by a pair of axis parallel congruent squares of minimum size we also discuss the restricted version of covering where each line segment in l is to be covered completely by at least one square the proposed algorithms assume that the input segments are given in a read only array for each of these problems i e covering hitting and restricted covering problems our proposed algorithm reports the optimum result by executing only two passes of reading the input array sequentially all these algorithms need only o 1 extra space the solution of these problems also give a 2 approximation for covering and hitting those line segments l by two congruent disks of minimum radius with same computational complexity

a column generation approach to radiation therapy treatment planning using aperture modulation this paper considers the problem of radiation therapy treatment planning for cancer patients during radiation therapy beams of radiation pass through a patient this radiation kills both cancerous and normal cells so the radiation therapy must be carefully planned to deliver a clinically prescribed dose to certain targets while sparing nearby organs and tissues currently a technique called intensity modulated radiation therapy imrt is considered to be the most effective radiation therapy for many forms of cancer in imrt the patient is irradiated from several different directions from each direction one or more irregularly shaped radiation beams of uniform intensity are used to deliver the treatment this paper deals with the problem of designing a treatment plan for imrt that determines an optimal set of such shapes called apertures and their corresponding intensities this is in contrast with established two stage approaches where in the first phase each radiation beam is viewed as consisting of a set of individual beamlets each with its own intensity a second phase is then needed to approximate and decompose the optimal intensity profile into a set of apertures with corresponding intensities the problem is formulated as a large scale convex programming problem and a column generation approach to deal with its dimensionality is developed the associated pricing problem determines in each iteration one or more apertures to be added to our problem several variants of this pricing problem are discussed each corresponding to a particular set of constraints that the apertures must satisfy in one or more of the currently available types of commercial imrt equipment polynomial time algorithms are presented for solving each of these variants of the pricing problem to optimality finally the effectiveness of our approach is demonstrated on clinical data

location on tree networks p centre and n dispersion problems we consider two problems of locating facilities on trees in the first we wish to determine the location of n points so as to maximize the minimum distance between any pair in the second we wish to locate p centers of a single type of facility so as to minimize the maximum distance between any point and the center nearest to it we provide polynomial algorithms for both problems of o n 2 log n log n for the first problem and of o n 2 log p for the second where n is the number of nodes in the tree

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

on anstreicher s combined phase i phase ii projective algorithm for linear programming anstreicher has proposed a variant of karmarkar s projective algorithm that handles standard form linear programming problems nicely we suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm much of the analysis depends on a two constraint linear programming problem that is a relaxation of the scaled original problem

a kinship function approach to robust and probabilistic optimization under polynomial uncertainty in this paper we study a class of robust design problems with polynomial dependence on the uncertainty one of the main motivations for considering these problems comes from robust controller design where one often encounters systems that depend polynomially on the uncertain parameters this paper can be seen as integrated in the emerging area of probabilistic robustness where a probabilistic relaxation of the original robust problem is adopted thus requiring the satisfaction of the constraints not for all possible values of the uncertainty but for most of them different from the randomized approach for tackling probabilistic relaxations which is only guaranteed to provide soft bounds on the probability of satisfaction we present a deterministic approach based on the novel concept of kinship function introduced in this paper this allows the development of an original framework which leads to easily computable deterministic convex relaxations of the probabilistic problem in particular optimal polynomial kinship functions are introduced which can be computed a priori and once for all and provide the best convex bound on the probability of constraint violation more importantly it is proven that the solution of the relaxed problem converges to that of the original robust optimization problem as the degree of the optimal polynomial kinship function increases furthermore by relying on quadrature formulas for computation of integrals of polynomials it is shown that the computational complexity of the proposed approach is polynomial in the number of uncertain parameters finally unlike other deterministic approaches to robust polynomial optimization the number of variables in the ensuing optimization problem is not increased by the proposed approximation an important feature of this approach is that a significant amount of the computational burden is shifted to a one time offline computation whose results can be stored and provided to the end user

generation of anisotropic smoothness regularization filters for eit in the inverse conductivity problem as in any ill posed inverse problem regularization techniques are necessary in order to stabilize inversion a common way to implement regularization in electrical impedance tomography is to use tikhonov regularization the inverse problem is formulated as a minimization of two terms the mismatch of the measurements against the model and the regularization functional most commonly differential operators are used as regularization functionals leading to smooth solutions whenever the imaged region presents discontinuities in the conductivity distribution such as interorgan boundaries the smoothness prior is not consistent with the actual situation in these cases the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity in this paper we derive a method for generating gaussian anisotropic regularization filters the filters are generated on the basis of the prior structural information allowing a better reconstruction of conductivity profiles matching these priors when incorporating prior information into a reconstruction algorithm the risk is of biasing the inverse solutions toward the assumed distributions simulations show that with a careful selection of the regularization parameters the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information a generalized singular value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

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

a geometrical solution for quadratic bicriteria location models this paper deals with a single facility unweighted quadratic bicriteria location problem defined in the continuous space first we describe an analytical method for constructing the set of pareto optimal locations for this problem based on the farthest point voronoi diagram then we present an analytical method for finding the solution to the scalarized location problems based on the pareto optimal locations the analytical results enable planners to carry out some sensitivity analyses since both methods are based entirely on geometrical analysis they can be understood intuitively

discrete convexity and polynomial solvability in minimum 0 extension problems a 0 0 extension of graph vargamma is a metric d d on a set v v containing the vertex set v vargamma v of vargamma such that d d extends the shortest path metric of vargamma and for all x in v x v there exists a vertex s s in vargamma with d x s 0 d x s 0 the minimum 0 0 extension problem 0 ext vargamma on vargamma is given a set v supseteq v vargamma v v and a nonnegative cost function c c defined on the set of all pairs of v v find a 0 0 extension d d of vargamma with sum xy c xy d x y xyc xy d x y minimum the 0 0 extension problem generalizes a number of basic combinatorial optimization problems such as minimum s t s t cut problem and multiway cut problem karzanov proved the polynomial solvability of 0 ext vargamma for a certain large class of modular graphs vargamma and raised the question what are the graphs vargamma for which 0 ext vargamma can be solved in polynomial time he also proved that 0 ext vargamma is np hard if vargamma is not modular or not orientable in a certain sense in this paper we prove the converse if vargamma is orientable and modular then 0 ext vargamma can be solved in polynomial time this completes the classification of graphs vargamma for which 0 ext vargamma is tractable to prove our main result we develop a theory of discrete convex functions on orientable modular graphs analogous to discrete convex analysis by murota and utilize a recent result of thapper and ivn on valued csp

locating an absolute center on graphs that are almost trees abstract physical models that exhibit a treelike graph structure frequently arise in telecommunication networks interstate highway systems and computer communication networks treelike graphs are characterized and properties related to the absolute center problem are analyzed that are preserved under two transformations of the graph utilizing these preserved properties a linear time algorithm is presented for the absolute center problem whenever the underlying graph is homeomorphic to a treelike 3 cactus graph

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

locating centers on a tree with discontinuous supply and demand regions consider a tree t n e with supply and demand regions and each composed of a finite number of disjoint closed and connected subregions of t some of which may possibly consist of just one point given an integer p we seek a collection of p centers x1 xp which minimize the expression maxy mini 1 pd y xi we present a polynomial algorithm for this problem its running time is bounded by o n log2n if either or is discrete and by o n min p log2n n log p if both sets contain at least one full edge

technical note equivalence between the direct search approach and the cut approach to the rectilinear distance location problem picard and ratliff recently proposed a cut approach to the rectilinear distance location problem and claimed it is fundamentally different from the direct search approach as developed by pritsker and ghare rao juel and love and sherali and shetty our objective is to show that the cut approach is essentially a direct search approach

the discrete and mixed minimax 2 center problems abstract letting p be a set of n points in the plane the discrete minimax 2 center problem d m m 2 c p is to find two disks centered at p 1 p 2 p that minimize the maximum of two terms namely the euclidean distance between two centers and the distance of any other point to the closer center the mixed minimax 2 center problem m m m 2 c p is when one of the two centers is not in p we present algorithms solving the d m m 2 c p and m m m 2 c p the time complexities of solving the d m m 2 c p and m m m 2 c p are o n 2 log n and o n 2 log 2 n respectively furthermore we consider two steiner minimum sum dipolar spanning tree problems in which one of the two dipoles is a steiner point and the dipoles are both steiner points these two problems are shown to be solvable in o n log n and o n time respectively

half integrality of node capacitated multiflows and tree shaped facility locations on trees in this paper we establish a novel duality relationship between node capacitated multiflows and tree shaped facility locations we prove that the maximum value of a tree distance weighted maximum node capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree and the maximum is attained by a half integral multiflow utilizing this duality we show that a half integral optimal multiflow and an optimal location can be found in strongly polynomial time these extend previously known results in the maximum free multiflow problems we also show that the set of tree distance weights is the only class having bounded fractionality in maximum node capacitated multiflow problems

two unconstrained optimization approaches for the euclidean κ centrum location problem abstract consider the single facility euclidean centrum location problem in r n this problem is a generalization of the classical euclidean 1 median problem and 1 center problem in this paper we develop two efficient algorithms that are particularly suitable for problems where n is large by using unconstrained optimization techniques the first algorithm is based on the neural networks smooth approximation for the plus function and reduces the problem to an unconstrained smooth convex minimization problem the second algorithm is based on the fischer burmeister merit function for the second order cone complementarity problem and transforms the kkt system of the second order cone programming reformulation for the problem into an unconstrained smooth minimization problem our computational experiments indicate that both methods are extremely efficient for large problems and the first algorithm is able to solve problems of dimension n up to 10 000 efficiently

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

group centre and group median of a tree abstract we consider the problem of identifying group centre and group median of a tree algorithms are presented that are better than the available algorithms which solve these problems on general networks for the group median problem an alternate approach is suggested which runs in linear time on specially structured problems properties of the set of all optimal solutions of these problems are also investigated

rotating maxrs queries given a set of weighted objects in a data space the maxrs problem in spatial databases studied in a vldb 2012 paper is to find a location for a rectangular region of a given size such that the weighted sum of all the objects covered by the rectangular region centered at the optimal location is maximized this problem is useful in lots of location based service applications such as finding the location for a new fast food restaurant with a limited delivery range attracting the greatest number of customers the existing maxrs problem assumes that the rectangular region is always placed horizontally and is non rotatable however under this assumption the weighted sum of all the covered objects may not be the greatest when the rectangular region is rotatable in this paper we propose a generalized maxrs problem called rotating maxrs without this assumption in rotating maxrs the rectangular region is rotatable and can be associated with an inclination angle the goal of our problem is to find a location and an inclination angle such that the weighted sum of all the objects covered by the rectangular region of a given size centered at this location with this inclination angle is the greatest we also present an efficient algorithm for the problem extensive experiments were conducted to verify the efficiency of our algorithms based on the real and synthetic datasets the experimental results show that the weighted sum of all the objects in the rotating maxrs queries can be increased with up to 300 on the synthetic datasets compared with existing non rotating maxrs queries which shows the significance of the new rotating maxrs queries

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

an overview of representative problems in location research we present a survey of over 50 representative problems in location research our goal is not to review all variants of different location models or to describe solution results but rather to provide a broad overview of major location problems that have been studied indicating briefly how they are formulated and how they relate to one another we review standard problems such as median center and warehouse location problems as well as less traditional location problems which have emerged in recent years our primary focus is on problems for which operations research type models have been developed most of the problems we review have been formulated as optimization problems

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

k eccentricity and absolute k centrum of a probabilistic tree abstract the k eccentricity evaluated at a point x of a graph g is the sum of the weighted distances from x to the k vertices farthest from it the k centrum is the set of vertices for which the k eccentricity is a minimum the concept of k centrum includes as a particular case that of center and that of centroid or median of a graph the absolute k centrum is the set of points not necessarily vertices for which the k eccentricity is a minimum in this paper it will be proven that for a weighted tree both deterministic and probabilistic the k eccentricity is a convex function and that the absolute k centrum is a connected set and is contained in an elementary path hints will be given for the construction of an algorithm to find the k centrum and the absolute k centrum

in vivo impedance imaging with total variation regularization we show that electrical impedance tomography eit image reconstruction algorithms with regularization based on the total variation tv functional are suitable for in vivo imaging of physiological data this reconstruction approach helps to preserve discontinuities in reconstructed profiles such as step changes in electrical properties at interorgan boundaries which are typically smoothed by traditional reconstruction algorithms the use of the tv functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation this cannot be efficiently solved with traditional optimization techniques such as the newton method we explore two implementations methods for regularization with the tv functional the lagged diffusivity method and the primal dual interior point method pd ipm first we clarify the implementation details of these algorithms for eit reconstruction next we analyze the performance of these algorithms on noisy simulated data finally we show reconstructed eit images of in vivo data for ventilation and gastric emptying studies in comparison to traditional quadratic regularization tv regulariza tion shows improved ability to reconstruct sharp contrasts

impedance imaging with first order tv regularization eit problem is a typical inverse problem with serious ill posedness in general regularization techniques are necessary for such ill posed inverse problems to overcome ill posedness the total variation tv regularization is widely used and it is also successfully applied to eit for realtime monitoring a fast and robust image reconstruction algorithm is required by exploiting recent advances in optimization we propose a first order tv algorithm for eit which simply consists of matrix vector multiplications and in which the sparse structure of the system can be easily exploited furthermore a typical smoothing parameter to overcome nondifferentibility of the tv term is not needed and a closed form solution can be applied in part using soft thresholding it shows a fast reconstruction in the beginning numerical experiments using simulated data and real experimental data support our claim

an inverse problems approach to mr ept image reconstruction magnetic resonance electrical properties tomography mr ept is an imaging modality that maps the spatial distribution of the electrical conductivity and permittivity using standard mri systems the presence of a body within the scanner alters the rf field and by mapping these alterations it is possible to recover the electrical properties the field is time harmonic and can be described by the helmholtz equation approximations to this equation have been previously used to estimate conductivity and permittivity in terms of first or second derivatives of rf field data using these same approximations an inverse approach to solving the mr ept problem is presented here that leverages a forward model for describing the magnitude and phase of the field within the imaging domain and a fitting approach for estimating the electrical properties distribution the advantages of this approach are that 1 differentiation of the measured data is not required thus reducing noise sensitivity and 2 different regularization schemes can be adopted depending on prior knowledge of the distribution of conductivity or permittivity leading to improved image quality to demonstrate the developed approach both quadratic qr and total variation tv regularization methods were implemented and evaluated through numerical simulation and experimentally acquired data the proposed inverse approach to mr ept reconstruction correctly identifies contrasts and accurately reconstructs the geometry in both simulations and experiments the tv regularized scheme reconstructs sharp spatial transitions which are difficult to reconstruct with other more traditional approaches

state of the art location on networks a survey part ii exploiting tree network structure the discussion of the fundamental p center and p median problems in part i of this paper provides the basis for the work surveyed in part ii part ii deals with the minimax and minisum location problems with mutual communication location problems involving multiple objectives the distance constraints problem and problems involving the location of paths in addition convexity issues in network location problems are discussed virtually all of this work exploits network structure specifically it is based on the assumption that the network is a tree the conclusion section gives a brief discussion of the state of the art and of current trends in network location research

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

algorithms for the robust 1 center problem on a tree abstract we consider the weighted 1 center problem on a network with uncertainty in node weights and edge lengths uncertainty is modelled by means of interval estimates for parameters specifically each uncertain parameter is assumed to be random with unknown distribution and can take on any value within a corresponding prespecified interval it is required to find a robust minmax regret solution that is a location which is e optimal for any possible realization of parameters with e as small as possible the problem on a general network is known to be np hard for the problem on a tree we present a polynomial algorithm

fuzzy facility location problem with preference of candidate sites models so far considered as facility location problems treat either min max or max min criterion that is the facility is either desirable or undesirable we propose the following model considering the satisfaction degree with respect to the distance from the facility for each customer residents and preference of the site in an urban area the objective is to find the site of the facility which maximizes the minimal satisfaction degree among all demand points and maximizes the preference of the site since generally speaking there exists no site that maximizes both criteria we seek non dominated sites after defining non domination

distributional robustness analysis for nonlinear uncertainty structures one of the main objectives of this note is to address the question what is the worst case expected value of a continuous function worst case performance over a class of admissible distributions in this note the class of symmetric and non increasing distributions is considered and results are provided for the class of so called semi algebraic functions the first part of the note shows that for the class of distributions considered it suffices to solve a convex optimization problem for which efficient linear matrix inequality lmi relaxations are available secondly the proposed approach is applied to estimate hard bounds on the worst case probability of a semi algebraic function being negative several numerical examples are presented which illustrate the effectiveness of the approach presented

hyper2d a numerical code for hyperinterpolation on rectangles hyperinterpolation at morrow patterson xu cubature points for the product chebyshev measure provides a simple and powerful polynomial approximation method on rectangles here we present an accurate and efficient matlab octave implementation of the hyperinterpolation formula accompanied by several numerical tests

an o n log 2 n algorithm for the kth longest path in a tree with applications to location problems many known algorithms are based on selection in a set whose cardinality is superlinear in terms of the input length it is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set this paper presents a successful development in this direction the methods developed here are applied to improve the previously known upper bounds for the time complexity of various location problems

group centre and group median of a network abstract the absolute centre and median of a network can be computed in polynomial time we use a partition of the nodes to define group centre and group median of the network these ideas can be used to unify the concepts of absolute centre and median we give polynomially bounded algorithms to identify group centre and group median of a network it may be observed that the introduction of partitions in some other combinatorial optimization problems e g the assignment problem changes the polynomial nature of the problem

on the complexity of the elzinga hearn algorithm for the 1 center problem we show that the complexity of the elzinga hearn algorithm for the 1 center single facility minimax problem is n2 we also show that the algorithm reaches a solution within any given accuracy in on time

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

efficient algorithms for two generalized 2 median problems and the group median problem on trees the p median problem on a tree t is to find a set s of p vertices on that minimizes the sum of distances from t s vertices to s in this paper we study two generalizations of the 2 median problem which are obtained by imposing constraints on the two vertices selected as a 2 median one is to limit their distance while the other is to limit their eccentricity previously both the best upper bounds of these two generalizations were o n2 a tamir d perez brito j a moreno perez a polynomial algorithm for the p centdian problem on a tree networks 32 1998 255 262 b f wang s c ku k h shi cost optimal parallel algorithms for the tree bisector problem and applications ieee transactions on parallel and distributed systems 12 9 2001 888 898 in this paper we solve both in o nlogn time we also study cases when linear time algorithms exist for the two generalizations for example we solve both in linear time when edge lengths and vertex weights are all polynomially bounded integers furthermore we consider the relaxation of the two generalized problems by allowing 2 medians on any position of edges instead of just on vertices and we give o nlogn time algorithms for them a problem named the tree marker problem arises several times in our approaches to the two generalized 2 median problems and we give an o nlogn time algorithm for this problem we also use this algorithm to speedup an algorithm of gupta and punnen s k gupta a p punnen group center and group median of a tree european journal of operational research 65 1993 400 406 for the group median problem improving the running time from o kn to o n klogn where k is the number of groups in the input

a deep cut ellipsoid algorithm for convex programming theory and applications this paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems in each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided moreover for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed finally some computational results of the deep and central cut version of the algorithm applied to a min max stochastic queue location problem are reported

polynomially bounded algorithms for locatingp centers on a tree we discuss several forms of thep center location problems on an undirected tree network our approach is based on utilizing results for rigid circuit graphs to obtain polynomial algorithms for solving the model duality theory on perfect graphs is used to define and solve the dual location model

tikhonov regularization and prior information in electrical impedance tomography the solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method the generalized tikhonov regularization methods have been popular in the solution of many inverse problems the regularization matrices that are usually used with the tikhonov method are more or less ad hoc and the implicit prior assumptions are thus in many cases inappropriate in this paper the authors propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution the approach is based on the construction of an approximating subspace for the expected impedance distributions it is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object on the other hand when the prior is incompatible with the true object the method will still give reasonable estimates

inverse eigenvalue problem of jacobi matrix with mixed data in this paper the inverse eigenvalue problem of reconstructing a jacobi matrix from its eigenvalues its leading principal submatrix and part of the eigenvalues of its submatrix is considered the necessary and sufficient conditions for the existence and uniqueness of the solution are derived furthermore a numerical algorithm and some numerical examples are given

linear time algorithms for two and three variable linear programs o n time algorithms for linear programming problems with two or three variables and n constraints are described the approach uses convexity dominance of linear functions and linear time median finding algorithms the algorithms improve the previously known best bounds of o n log n time for both of these problems

on solving the planar k centrum problem with euclidean distances this paper presents a solution procedure based on a gradient descent method for the k centrum problem in the plane the particular framework of this problem for the euclidean norm leads to bisector lines whose analytical expressions are easy to handle this allows us to develop different solution procedures which are tested on different problems and compared with existing procedures in the literature of location analysis the computational analysis reports that our procedures provide better results than the existing ones for the k centrum problem

an efficient algorithm for the euclidean r centrum location problem in this paper we consider the single facility euclidean r centrum location problem in r n which generalizes and unifies the classical 1 center and 1 median problem specifically we reformulate this problem as a nonsmooth optimization problem only involving the maximum function and then develop a smoothing algorithm that is shown to be globally convergent the method transforms the original nonsmooth problem with certain combinatorial property into the solution of a deterministic smooth unconstrained optimization problem numerical results are presented for some problems generated randomly indicating that the algorithm proposed here is extremely efficient for large problems

efficient algorithms for center problems in cactus networks efficient algorithms for solving the center problems in weighted cactus networks are presented in particular we have proposed the following algorithms for the weighted cactus networks of size n an o nlogn time algorithm to solve the 1 center problem and an o nlog 3n time algorithm to solve the weighted continuous 2 center problem we have also provided improved solutions to the general p center problems in cactus networks the developed ideas are then applied to solve the obnoxious 1 center problem in weighted cactus networks

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

minmax regret 1 facility location on uncertain path networks abstract let p be an undirected path graph of n vertices each edge of p has a positive length and a constant capacity every vertex has a nonnegative supply which is an unknown value but is known to be in a given interval the goal is to find a point on p to build a facility and move all vertex supplies to the facility such that the maximum regret is minimized the previous best algorithm solves the problem in o n log 2 n time and o n log n space in this paper we present an o n log n time and o n space algorithm and our approach is based on new observations and algorithmic techniques

one dimensional k center on uncertain data problems on uncertain data have attracted significant attention due to the imprecise nature of many measurement data in this paper we consider the k center problem on one dimensional uncertain data the input is a set p of weighted uncertain points on a real line and each uncertain point is specified by its probability density function pdf which is a piecewise uniform function i e a histogram the goal is to find a set q of k points on the line to minimize the maximum expected distance from the uncertain points of p to their expected closest points in q we present efficient algorithms for this uncertain k center problem and their running times almost match those for the deterministic k center problem

existence and uniqueness for electrode models for electric current computed tomography the following experiment is considered to a body of given conductivity a number of electrodes are attached through which current is sent on the same electrodes the resulting voltages are measured this experiment can be described by a number of mathematical models k s cheng et al ieee transactions on biomedical engineering 36 1989 pp 918 924 these models are discussed and their predictions compared with experiment in particular a model is exhibited that is capable of predicting the experimentally measured voltages to within 0 1 percent for this model existence and uniqueness of the associated electrical potential is proved

out of roundness problem revisited the properties and computation of the minimum radial separation mrs standard for out of roundness are discussed another standard out of roundness measurement called the minimum area difference mad center is introduced although the two centers have different characteristics the approach to finding both centers shares many commonalities an o n log n k time algorithm which is used to compute the mrs center is presented it also computes the mad center of a simple polygon g where n is the number of vertices of g and k is the number of intersection points of the medial axis and the farthest neighbor voronoi diagram of g the relationship between mrs and mad is discussed

an efficient algorithm for the smallest enclosing ball problem in high dimensions abstract consider the problem of computing the smallest enclosing ball of a set of m balls in r n this problem arises in many applications such as location analysis military operations and pattern recognition etc in this paper we reformulate this problem as an unconstrained convex optimization problem involving the maximum function max 0 t and then develop a simple algorithm particularly suitable for problems in high dimensions this algorithm could efficiently handle problems of dimension n up to 10 000 under a moderately large m as well as problems of dimension m up to 10 000 under a moderately large n numerical results are given to show the efficiency of algorithm

the partial sum criterion for steiner trees in graphs and shortest paths many optimization problems are defined in a weighted graph generally the criterion is min sum i e the total weight of all edges in the solution here we consider a different criterion