approximate max min resource sharing for structured concave optimization we present a lagrangian decomposition algorithm which uses logarithmic potential reduction to compute an varepsilon approximate solution of the general max min resource sharing problem with m nonnegative concave constraints on a convex set b we show that this algorithm runs in o m varepsilon 2 rm ln m iterations a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy varepsilon in 0 1 in the block angular case b is the product of k convex sets blocks and each constraint function is block separable for such models an iteration of our method requires a theta varepsilon approximate solution of k independent block maximization problems which can be computed in parallel

approximate minimum cost multicommodity flows in o e 2 knm time we show that an e approximate solution of the cost constrainedk commodity flow problem on ann nodem arc network g can be computed by sequentially solving o k 2 loggk loggm log g 1 gk single commodity minimum cost flow problems on the same network in particular an approximate minimum cost multicommodity flow can be computed in tilde o g 2 gknm running time where the notation o means up to logarithmic factors this result improves the time bound mentioned by grigoriadis and khachiyan 4 by a factor ofm n and that developed more recently by karger and plotkin 8 by a factor of 1 we also provide a simple tilde o nm time algorithm for single commodity budget constrained minimum cost flows which is tilde o 3 times faster than the algorithm developed in the latter paper

coordination complexity of parallel price directive decomposition the general block angular convex resource sharing problem in k blocks and m nonnegative block separable coupling constraints is considered applications of this model are in combinatorial optimization network flows scheduling communication networks engineering design and finance this paper studies the coordination complexity of approximate price directive decomposition pdd for this problem i e the number of iterations required to solve the problem to a fixed relative accuracy as a function of k and m first a simple pdd method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in m in the class of all pdd methods that work with the original unrestricted blocks it is then shown that logarithmic and exponential potentials generate a polylogarithmically optimal algorithm for a wider class of pdd methods which can restrict the blocks by the coupling constraints as an application the fastest currently known deterministic approximation algorithm for minimum cost multicommodity flows is obtained

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

using central prices in the decomposition of linear programs abstract this paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical dantzig wolfe algorithm this approach exploits a the relationship between the master program and the minimization of a convex piecewise linear function and b a recently proposed cutting plane algorithm for convex nondifferentiable optimization the new algorithm seeks to achieve deep cuts by generating them at or near to the analytic centre of a set of localization containing the solution therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence this algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems its implementation as a decomposition algorithm for large scale structured linear program seems to be a promising alternative to the standard dantzig wolfe approach which suffers from very slow convergence in many practical problems the new approach differs from the dantzig wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem but rather central prices i e analytic centres which balance them all this property seems to explain the good behaviour of the new algorithm

fast approximation schemes for two stage two dimensional bin packing we present an asymptotic fully polynomial time approximation scheme for the two dimensional generalization of bin packing which requires packing or cutting a given set of rectangles from the minimum number of square bins with the further restriction that packing the rectangles in the bins is done in two stages as is frequently the case in real world applications to the best of our knowledge this is the first approximation scheme for a nontrivial two dimensional and real world generalization of a classical one dimensional packing problem in which rectangles have to be packed in finite squares in addition to the approximability result we obtain as a byproduct of our analysis an interesting structural result namely the asymptotic optimality of the lower bound provided by the natural linear programming relaxation of the problem having one variable for each bin pattern that can be solved effectively by column generation techniques besides its practical relevance a main reason to concentrate on the two stage version of the problem is that its study continues to shed light on the general version in particular we strongly conjecture that the approximation guarantee of our algorithm for two dimensional bin packing without the two stage restriction is better than the best known ones a simplification of the method leads to an approximation scheme for the two stage version of two dimensional strip packing

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

how many invariant polynomials are needed to decide local unitary equivalence of qubit states given l qubit states with the fixed spectra of reduced one qubit density matrices we find a formula for the minimal number of invariant polynomials needed for solving local unitary lu equivalence problem that is problem of deciding if two states can be connected by local unitary operations interestingly this number is not the same for every collection of the spectra some spectra require less polynomials to solve lu equivalence problem than others the result is obtained using geometric methods i e by calculating the dimensions of reduced spaces stemming from the symplectic reduction procedure

on the geometry and quantization of symplectic howe pairs we study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold if the pair of actions fulfils a symplectic howe condition we show that there is a canonical correspondence between the orbit spaces of the respective moment images furthermore we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group in the kahler case we show that the linear representation of a pair of compact connected lie groups on the geometric quantization of the manifold is then equipped with a representation theoretic howe duality

fast approximation schemes for convex programs with many blocks and coupling constraints this paper presents block coordinate descent algorithms for the approximate solution of large structured convex programming problems the constraints of such problems consist of k disjoint convex compact sets b k called blocks and m nonnegative valued convex block separable inequalities called coupling or resource constraints the algorithms are based on an exponential potential function reduction technique it is shown that feasibility as well as min mix resource sharing problems for such constraints can be solved to a relative accuracy varepsilon in o k ln m varepsilon 2 ln k iterations each of which solves k block problems to a comparable accuracy either sequentially or in parallel the same bound holds for the expected number of iterations of a randomized variant of the algorithm which uniformly selects a random block to process at each iteration an extension to objective and constraint functions of arbitrary sign is also presented the above results yield fast approximatio

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

parameterized approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a well known generalization of the classical knapsack problem we are given a set a of n items and set b of m bins knapsacks such that each item a in a has a size size a and a profit value profit a and each bin b in b has a capacity c b the goal is to find a subset u subset a of maximum total profit such that u can be packed into b without exceeding the capacities the decision version of mkp is strongly np complete since it is a generalization of the classical knapsack and bin packing problem furthermore mkp does not admit a fully time polynomial time approximation scheme fptas even if the number m of bins is two kellerer gave a polynomial time approximation scheme ptas for mkp with identical capacities and chekuri and khanna presented a ptas for mkp with general capacities with running time n o log 1 epsilon epsilon 8 in this paper we propose an efficient ptas eptas with parameterized running time 2 o log 1 epsilon epsilon 5 cdot poly n o m for mkp this also solves an open question by chekuri and khanna

lagrangian submanifolds and moment convexity we consider a hamiltonian torus action t x m m on a compact connected symplectic manifold m and its associated momentum map for certain lagrangian submanifolds q c m we show that q is convex the submanifolds q arise as the fixed point set of an involutive diffeomorphism t m m which satisfies several compatibility conditions with the torus action but which is in general not anti symplectic as an application we complete a symplectic proof of kostant s non linear convexity theorem

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

a finite difference method for a conservative allencahn equation on non flat surfaces we present an efficient numerical scheme for the conservative allencahn cac equation on various surfaces embedded in a narrow band domain in the three dimensional space we apply a quasi neumann boundary condition on the narrow band domain boundary using the closest point method this boundary treatment allows us to use the standard cartesian laplacian operator instead of the laplacebeltrami operator we apply a hybrid operator splitting method for solving the cac equation first we use an explicit euler method to solve the diffusion term second we solve the nonlinear term by using a closed form solution third we apply a spacetime dependent lagrange multiplier to conserve the total quantity the overall scheme is explicit in time and does not need iterative steps therefore it is fast a series of numerical experiments demonstrate the accuracy and efficiency of the proposed hybrid scheme

color image segmentation by the vector valued allen cahn phase field model a multigrid solution we present an efficient numerical solution of a pde driven model for color image segmentation and give numerical examples of the results the method combines the vector valued allen cahn phase field equation with initial data fitting terms with prescribed interface width and fidelity constants efficient numerical solution is achieved using a multigrid splitting of a finite element space thereby producing an efficient and robust method for the segmentation of large images we also present the use of adaptive mesh refinement to further speed up the segmentation process

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

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

convexity properties of gradient maps abstract we consider the action of a real reductive group g on a kahler manifold z which is the restriction of a holomorphic action of a complex reductive group h we assume that the action of a maximal compact subgroup u of h is hamiltonian and that g is compatible with a cartan decomposition of h we have an associated gradient map p z p where g k p is the cartan decomposition of g for a g stable subset y of z we consider convexity properties of the intersection of p y with a closed weyl chamber in a maximal abelian subspace a of p our main result is a convexity theorem for real semi algebraic subsets y of z p v where v is a unitary representation of u

a new approximation method for set covering problems with applications to multidimensional bin packing in this paper we introduce a new general approximation method for set covering problems based on the combination of randomized rounding of the near optimal solution of the linear programming lp relaxation leading to a partial integer solution and the application of a well behaved approximation algorithm to complete this solution if the value of the solution returned by the latter can be bounded in a suitable way as is the case for the most relevant generalizations of bin packing the method leads to improved approximation guarantees along with a proof of tighter integrality gaps for the lp relaxation for d dimensional vector packing we obtain a polynomial time randomized algorithm with asymptotic approximation guarantee arbitrarily close to ln d 1 for d 2 this value is 1 693 dots i e we break the natural 2 barrier for this case moreover for small values of d this is a notable improvement over the previously known o ln d guarantee by chekuri and khanna siam j comput 33 2004 pp 837 851 for two dimensional bin packing with and without rotations we obtain polynomial time randomized algorithms with asymptotic approximation guarantee 1 525 dots improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by jansen and van stee on strip packing with rotations in proceedings of the 37th annual acm symposium on the theory of computing 2005 pp 755 761 for the problem with rotations and 1 691 ldots by caprara math oper res 33 2008 pp 203 215 for the problem without rotations the previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by fernandez de la vega and lueker combinatorica 1 1981 pp 349 355 we prove that their approximation scheme is subset oblivious which leads to numerous applications

approximation algorithm for the mixed fractional packing and covering problem we propose an approximation algorithm based on the lagrangian or price directive decomposition method to compute an epsilon approximate solution of the mixed fractional packing and covering problem find x in b such that f x le 1 epsilon a g x ge 1 epsilon b where f x g x are vectors with m nonnegative convex and concave functions a and b are m dimensional nonnegative vectors and b is a convex set that can be queried by an optimization oracle we propose an algorithm that needs only o m epsilon 2 ln m epsilon 1 iterations or calls to the oracle the main contribution is that the algorithm solves the general mixed fractional packing and covering problem in contrast to pure fractional packing and covering problems and to the special mixed packing and covering problem with b rz n and runs in time independent of the so called width of the problem

an unconditionally gradient stable numerical method for solving the allen cahn equation abstract we consider an unconditionally gradient stable scheme for solving the allen cahn equation representing a model for anti phase domain coarsening in a binary mixture the continuous problem has a decreasing total energy we show the same property for the corresponding discrete problem by using eigenvalues of the hessian matrix of the energy functional we also show the pointwise boundedness of the numerical solution for the allen cahn equation we describe various numerical experiments we performed to study properties of the allen cahn equation

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

semistable points with respect to real forms we consider actions of real lie subgroups g of complex reductive lie groups on kahlerian spaces our main result is the openness of the set of semistable points with respect to a momentum map and the action of g

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

stability of measures on kahler manifolds abstract let m be a kahler manifold and let k be a compact group that acts on m in a hamiltonian fashion we study the action of k c on probability measures on m first of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability semi stability and polystability next we apply this setting to the action of k c on measures we get various stability criteria for measures on kahler manifolds the same circle of ideas gives a very general surjectivity result for a map originally studied by hersch and bourguignon li yau

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

improved approximation algorithms for minimum power covering problems given an undirected graph with edge costs the power of a node is the maximum cost of an edge incident to it and the power of a graph is the sum of the powers of its nodes motivated by applications in wireless networks we consider two network design problems under the power minimization criteria in both problems we are given a graph g v e with edge costs and a set t subseteq v of terminals the goal is to find a minimum power edge subset f subseteq e such that the graph h v f satisfies some prescribed requirements in the min power edge cover problem h should contain an edge incident to every terminal using the iterative randomized rounding irr method we give an algorithm with expected approximation ratio 1 41 the ratio is reduced to 73 60 1 217 when t is an independent set in g in the case of unit costs we also achieve ratio 73 60 and in addition give a simple efficient combinatorial algorithm with ratio 5 4 for all these np hard problems the previous best known ratio was 3 2 in the related min power terminal backup problem h should contain a path from every t in t to some node in t setminus t we obtain ratio 3 2 for this np hard problem improving the trivial ratio of 2

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

kahlerian potentials and convexity properties of the moment map since kostant proved his convexity theorems for torus actions on ag varieties ko there have been numerous contributions to this subject in the more general symplectic and k ahlerian setting for example let k be a compact lie group acting in a hamiltonian fashion on a connected compact symplectic manifold x then the intersection x of the image of the moment map x liek with a positive weyl chamber t is convex thus x k x where x is a natural convex section this was proved by atiyah and guillemin sternberg a g s in the abelian case i e where k t is a compact torus by mumford m for x projective algebraic with an integral k ahlerian structure and k not necessarily abelian and in its nal form in the compact symplectic case by kirwan k a precise description of x x t in the compact k ahlerian framework can be given in terms of the image of the components of the set of t xed points in x in the projective case where the k ahlerian structure comes from a projective embedding there are explicit connections to the representations theory of g kc which are due to brion and to mumford ness b n the purpose of the present paper is to prove convexity properties of in the setting of non compact k ahlerian spaces note that by removing an appropriate k invariant subset from x one can easily construct non convex images from convex ones on the other hand hilgert neeb plank proved h n p that convexity holds for a proper moment map x liek another convexity result has been obtained by sjamaar for x a ne and the restriction of the moment map of a representation s2 here the representation theoretical approach of brion is used now let x be an irreducible complex space endowed with a holomorphic action of a complex reductive group g assume that a maximal

an efficient algorithm for solving the phase field crystal model we present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal pfc model this algorithm allows for an arbitrarily large algorithmic time step as the basis for our analysis of the accuracy of this algorithm we determine an effective time step in fourier space we then compare our calculations with a set of representative numerical results and demonstrate that this algorithm is an effective approach for the study of the pfc models yielding a time step effectively 180 times larger than the euler algorithm for a representative set of material parameters as the pfc model is just a simple example of a wide class of density functional theories we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities