capacity constraints across nests in assortment optimization under the nested logit model we consider assortment optimization problems when customers choose according to the nested logit model and there is a capacity constraint limiting the total capacity consumption of all products offered in all nests when each product consumes one unit of capacity our capacity constraint limits the cardinality of the offered assortment for the cardinality constrained case we develop an efficient algorithm to compute the optimal assortment when the capacity consumption of each product is arbitrary we give an algorithm to obtain a 4 approximate solution we show that we can compute an upper bound on the optimal expected revenue for an individual problem instance by solving a linear program in our numerical experiments we consider problem instances involving products with arbitrary capacity consumptions comparing the expected revenues from the assortments obtained by our 4 approximation algorithm with the upper bounds on the optimal expected revenues our numerical results indicate that the 4 approxim

min max min robust combinatorial optimization the idea of k adaptability in two stage robust optimization is to calculate a fixed number k of second stage policies here and now after the actual scenario is revealed the best of these policies is selected this idea leads to a min max min problem in this paper we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function we investigate the complexity of this special case for convex uncertainty sets we first show that the min max min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time we also provide an exact and practical oracle based algorithm to solve the latter problem for any underlying combinatorial problem on the other hand we prove that the min max min problem is np hard for every fixed number k even when the uncertainty set is a polyhedron given by an inner description for the case that k is smaller or equal to the number of variables we finally propose a fast heuristic algorithm and evaluate its performance

a novel adaptive finite element method for the ground state solution of bose einstein condensates abstract in this study we propose a novel adaptive finite element method for the ground state solution of bose einstein condensates bec different from the classical adaptive scheme applied to bec which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low dimensional correction space further the linear elliptic boundary value problem is solved by adaptive multigrid iterations since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces the solving efficiency can be improved evidently in addition the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically

domains with non compact automorphism group a survey abstract we survey results arising from the study of domains in c n with non compact automorphism group beginning with a well known characterization of the unit ball we develop ideas toward a consideration of weakly pseudoconvex and even non pseudoconvex domains with particular emphasis on characterizations of i smoothly bounded domains with non compact automorphism group and ii the levi geometry of boundary orbit accumulation points particular attention will be paid to results derived in the past ten years

solving multiobjective multiconstraint knapsack problems using mathematical programming and evolutionary algorithms abstract in this paper we solve instances of the multiobjective multiconstraint or multidimensional knapsack problem momckp from the literature with three objective functions and three constraints we use exact as well as approximate algorithms the exact algorithm is a properly modified version of the multicriteria branch and bound mcbb algorithm which is further customized by suitable heuristics three branching heuristics and a more general purpose composite branching and construction heuristic are devised comparison is made to the published results from another exact algorithm the adaptive e constraint method laumanns m thiele l zitzler e 2006 an efficient adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method european journal of operational research 169 932 942 using the same data sets furthermore the same problems are solved using standard multiobjective evolutionary algorithms moea namely the spea2 and the nsgaii the results from the exact case show that the branching heuristics greatly improve the performance of the mcbb algorithm which becomes faster than the adaptive e constraint regarding the performance of the moea algorithms in the specific problems spea2 outperforms nsgaii in the degree of approximation of the pareto front as measured by the coverage metric especially for the largest instance

an improved version of the augmented e constraint method augmecon2 for finding the exact pareto set in multi objective integer programming problems generation or a posteriori methods in multi objective mathematical programming momp is the most computationally demanding category among the momp approaches due to the dramatic increase in computational speed and the improvement of mathematical programming algorithms the generation methods become all the more attractive among today s decision makers in the current paper we present the generation method augmecon2 which is an improvement of our development augmecon although augmecon2 is a general purpose method we will demonstrate that augmecon2 is especially suitable for multi objective integer programming moip problems specifically augmecon2 is capable of producing the exact pareto set in moip problems by appropriately tuning its running parameters in this context we compare the previous and the new version in a series of new and old benchmarks found in the literature we also compare augmecon2 s performance in the generation of the exact pareto sets with established methods and algorithms based on specific moip problems knapsack set packing and on published results except from other mathematical programming methods augmecon2 is found to be competitive also with multi objective meta heuristics momh in producing adequate approximations of the pareto set in multi objective combinatorial optimization moco problems

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

analysis of a multiobjective evolutionary algorithm on the 0 1 knapsack problem multiobjective evolutionary algorithms moeas are increasingly being used for effectively solving many real world problems and many empirical results are available however theoretical analysis is limited to a few simple toy functions in this work we select the well known knapsack problem for the analysis the multiobjective knapsack problem in its general form is np complete moreover the size of the set of pareto optimal solutions can grow exponentially with the number of items in the knapsack thus we formalize a 1 e approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a moea to obtain the formalized set the algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population we call the algorithm a restricted evolutionary multiobjective optimizer remo we also analyze the running time of remo on a special bi objective linear function known as lotz leading ones trailing zeros whose pareto set is shown to be a subset of the knapsack an extension of the analysis to the simple evolutionary multiobjective optimizer semo is also presented a strategy based on partitioning of the decision space into fitness layers is used for the analysis

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

feedback and adaptive finite element solution of one dimensional boundary value problems this paper examines the concepts of feedback and adaptivity for the finite element method the model problem concernsc 0 elements of arbitrary fixed degree for a one dimensional two point boundary value problem three different feedback methods are introduced and a detailed analysis of their adaptivity is given

coupled and k sided placements generalizing generalized assignment in modern data centers and cloud computing systems jobs often require resources distributed across nodes providing a wide variety of services motivated by this we study the coupled placement problem in which we place jobs into computation and storage nodes with capacity constraints so as to optimize some costs or profits associated with the placement the coupled placement problem is a natural generalization of the widely studied generalized assignment problem which concerns the placement of jobs into single nodes providing one kind of service we also study a further generalization the k sided placement problem in which we place jobs into k tuples of nodes each node in a tuple offering one of k services for both the coupled and k sided placement problems we consider minimization and maximization versions in the minimization versions mincpand min k ksp the goal is to achieve minimum placement cost while incurring a minimum blowup in the capacity of the individual nodes our first main result is an algorithm for min k ksp that achieves optimal cost while increasing capacities by at most a factor of k 1 k 1 also yielding the first constant factor approximation for mincp in the maximization versions maxcp and max k ksp the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints max k ksp can be expressed as a k column sparse integer program and can be approximated to within a factor of o k factor using randomized rounding of a linear program relaxation we consider alternative combinatorial algorithms that are much more efficient in practice our second main result is a local search based combinatorial algorithm that yields a 15 approximation and o k 2 o k2 approximation for maxcp and max k ksp respectively finally we consider an online version of max k ksp and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions

a multigrid method for eigenvalue problem a multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem in this scheme solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space besides the multigrid scheme all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems the total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem therefore this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving some numerical experiments are presented to validate the efficiency of the method

single approximation for the biobjective max tsp we mainly study the max tsp with two objective functions we propose an algorithm which returns a single hamiltonian cycle with performance guarantee on both objectives the algorithm is analyzed in three cases when both respectively at least one objective function s fulfill s the triangle inequality the approximation ratio is 512 e 0 41 respectively 38 e when the triangle inequality is not assumed on any objective function the algorithm is 1 2214 e 0 27 approximate

it cemop an iterative co evolutionary algorithm for multiobjective optimization problem with nonlinear constraints over the past few years researchers have developed a number of multiobjective evolutionary algorithms moeas although most studies concentrate on solving unconstrained optimization problems there exit a few studies where moeas have been extended to solve constrained optimization problems most of them were based on penalty functions for handling nonlinear constraints by genetic algorithms however the performance of these methods is highly problem dependent many methods require additional tuning of several parameters in this paper we present a new optimization algorithm which is based on concept of co evolution and repair algorithm for handling nonlinear constraints the algorithm maintains a finite sized archive of nondominated solutions which gets iteratively updated in the presence of new solutions based on the concept of e dominance the use of e dominance also makes the algorithms practical by allowing a decision maker to control the resolution of the pareto set approximation by choosing an appropriate e value which guarantees convergence and diversity the results provided by the proposed algorithm for six benchmark problems are promising when compared with exiting well known algorithms also our results suggest that our algorithm is better applicable for solving real world application problems

approximation of min max and min max regret versions of some combinatorial optimization problems this paper investigates for the first time in the literature the approximation of min max regret versions of classical problems like shortest path minimum spanning tree and knapsack for a constant number of scenarios we establish fully polynomial time approximation schemes for the min max versions of these problems using relationships between multi objective and min max optimization using dynamic programming and classical trimming techniques we construct a fully polynomial time approximation scheme for min max regret shortest path we also establish a fully polynomial time approximation scheme for min max regret spanning tree and prove that min max regret knapsack is not at all approximable for a non constant number of scenarios in which case min max and min max regret versions of polynomial time solvable problems usually become strongly np hard non approximability results are provided for min max regret versions of shortest path and spanning tree

convergence and optimal complexity of adaptive finite element eigenvalue computations in this paper an adaptive finite element method for elliptic eigenvalue problems is studied both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved the analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper

a multilevel correction adaptive finite element method for kohn sham equation in this paper an adaptive finite element method is proposed for solving kohn sham equation with the multilevel correction technique in the method the kohn sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes a main feature of the method is that solving large scale kohn sham system is avoided effectively and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method hence the significant acceleration can be obtained on solving kohn sham equation with the proposed multilevel correction technique the performance of the method is examined by a variety of numerical experiments

stochastic convergence of random search methods to fixed size pareto front approximations in this paper we investigate to what extent random search methods equipped with an archive of bounded size to store a limited amount of solutions and other data are able to obtain good pareto front approximations we propose and analyze two archiving schemes that allow for maintaining a sequence of solution sets of given cardinality that converge with probability one to an pareto set of a certain quality under very mild assumptions on the process used to sample new solutions the first algorithm uses a hierarchical grid to define a family of approximate dominance relations to compare solutions and solution sets acceptance of a new solution is based on a potential function that counts the number of occupied boxes on various levels and thus maintains a strictly monotonous progress to a limit set that covers the pareto front with non overlapping boxes at finest resolution possible the second algorithm uses an adaptation scheme to modify the current value of based on the information gathered during the run this way it will be possible to achieve convergence to the best smallest value and to a corresponding solution set of k solutions that dominate all other solutions which is probably the best possible result regarding the limit behavior of random search methods or metaheuristics for obtaining pareto front approximations

fuzzy approach to multilevel knapsack problems this study proposes a fuzzy approach for solving the multiobjective and multilevel knapsack problems kps the problem was first formulated as a multilevel programming problem with multiple decision makers dms then the degree of satisfaction of each dm was established and represented by their individual membership functions the recursive formulation of dynamic programming was used to solve the decisions of the interrelated stages the overall satisfaction of the decision was obtained through this stage wise operation on the hierarchical structure capacity allocation was developed and a step by step solution procedure was illustrated a detailed comparison between multiobjective and multilevel kps was also carried out finally the possible use of turnpike theorem in kps was scrutinized in the fuzzy domain

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

approximation schemes for knapsack problems with shelf divisions given a knapsack of size k non negative values d and and a set s of items each item e s with size se and value ve we define a shelf as a subset of items packed inside a bin with total items size at most two subsequent shelves must be separated by a shelf divisor of size d the size of a shelf is the total size of its items plus the size of the shelf divisor the shelf knapsack problem sk is to find a subset s s partitioned into shelves with total shelves size at most k and maximum value the class constrained shelf knapsack ccsk is a generalization of the problem sk where each item in s has a class and each shelf in the solution must have only items of the same class we present approximation schemes for the sk and the ccsk problems to our knowledge these are the first approximation results where shelves of non null size are used in knapsack problems

the characterization of holomorphic vector fields vanishing at an infinite type point abstract this paper consider a rotationally symmetric real hypersurface in c 2 around the origin we also assume that the origin is a point of infinite type in the sense of d angelo we prove that a holomorphic vector field defined on a neighborhood of origin which is tangent to the real hypersurface and vanishing at the origin is indeed a vector field generating rotational transformations

the fast adaptive composite grid method fac algorithms for advanced computers fac is a fast systematic process for the numerical solution of partial differential equations on adapted grids this paper reports on numerical studies with fac in advancing computing environments including simulations with multiprocessor array architectures and actual tests on the hep we also analyze the parallel complexity of fac for general local grid applications

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

a fully polynomial approximation algorithm for the 0 1 knapsack problem abstract a fully polynomial approximation algorithm is developed for the 0 1 knapsack problem the algorithm uses results of lawler and ibarra and kim a pseudo polynomial dynamic programming algorithm is first suggested which solves the problem in o nb log n time and o b space

on the automorphism group of a certain infinite type domain in c2 abstract in this study we consider an infinite type domain p in c 2 the aim of this study is to investigate the holomorphic vector fields tangent to an infinite type model in c 2 vanishing at an infinite type point and to give an explicit description of the automorphism group of p

a multi level correction scheme for eigenvalue problems abstractin this paper a new type of multi level correction scheme is proposed forsolving eigenvalue problems by nite element method with this new scheme the accuracy of eigenpair approximations can be improved after each correc tion step which only needs to solve a source problem on ner nite elementspace and an eigenvalue problem on the coarsest nite element space thiscorrection scheme can improve the e ciency of solving eigenvalue problemsby nite element method keywords eigenvalue problem multi level correction scheme nite ele ment method multi space multi grid ams subject classi cations 65n30 65b99 65n25 65l15 1 introduction the purpose of this paper is to propose a new type of multi level correction schemebased on nite element discretization to solve eigenvalue problems the two gridmethod for solving eigenvalue problems has been proposed and analyzed by xu andzhou in 21 the idea of the two grid comes from 19 20 for nonsymmetric orinde nite problems and nonlinear elliptic equations since then there have existedmany numerical methods for solving eigenvalue problems based on the idea of two grid method 1 6 17

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

optimal multilevel methods for graded bisection grids we design and analyze optimal additive and multiplicative multilevel methods for solving h 1 problems on graded grids obtained by bisection we deal with economical local smoothers after a global smoothing in the finest mesh local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices we show that our methods lead to optimal complexity for any dimensions and polynomial degree the theory hinges on a new decomposition of bisection grids in any dimension which is of independent interest and yields a corresponding decomposition of spaces we use the latter to bridge the gap between graded and quasi uniform grids for which the multilevel theory is well established

priority based e dominance dominance is a fundamental concept in multiobjective optimization moo where a solution is said to dominate the other if it is better in at least one objective and same as or better in the other objectives given a pair of solutions if neither dominates the other then the solutions are said to be non dominated in real spaces a large number of non dominated solutions can exist in principle making convergence of moo techniques rather slow relaxed forms of dominance have been proposed in the literature for faster convergence one such form is the dominance which in principle partitions the space into sized grids the value of is often user defined and the performance of dominance based multiobjective evolutionary algorithms moeas critically depends on the value of in this article we propose a novel way of determining the value of which is different for each objective function based on the correlation between them we call this approach priority based pbe as the method ranks the objectives according to their priority before calculating the values pbe is incorporated in an archived simulated annealing based moo technique called amosa amosa has been earlier shown to outperform several well known moo techniques especially for many objective optimization pbe based amosa referred to as pbe amosa is found to comprehensively outperform amosa moea d de the conventional based version amosa and moea both in case of benchmark test problems and 0 1 knapsack problem

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a multilevel correction type of adaptive finite element method for eigenvalue problems an adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme different from the standard adaptive finite element method which requires solving ei

approximating multiobjective knapsack problems for multiobjective optimization problems it is meaningful to compute a set of solutions covering all possible trade offs between the different objectives the multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values for this problem efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions the pareto frontier are studied for the multiobjective one dimensional knapsack problem a practical fully polynomial time approximation scheme fptas is derived it is based on a new approach to the single objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length for the multiobjectivem dimensional knapsack problem the first known polynomial time approximation scheme ptas based on linear programming is presented

the automorphism group of a domain with an exponentially flat boundary point abstract we study the automorphisms group action on a bounded domain in c n having a boundary point that is exponentially flat such a domain typically has a compact automorphism group our results enable us to generate many new examples

implementing an efficient fptas for the 0 1 multi objective knapsack problem abstract in the present work we are interested in the practical behavior of a new fully polynomial time approximation schemes fptas to solve the approximation version of the 0 1 multi objective knapsack problem the proposed methodology makes use of very general techniques such as dominance relations in dynamic programming and thus may be applicable in the implementation of fptas for other problems as well extensive numerical experiments on various types of instances in the bi and tri objective cases establish that our method performs very well both in terms of cpu time and size of solved instances we point out some reasons for the good practical performance of our algorithm a comparison with an exact method and the fptas proposed in erlebach t kellerer h pferschy u 2002 approximating multiobjective knapsack problems management science 48 12 1603 1612 is also performed