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

asymptotic behavior of non linear eigenvalue problems involving p laplacian type operators we study the asymptotic behaviour of two nonlinear eigenvalue problems which involve p laplacian type operators in the first problem we consider the limit as p to infty of the sequences of the k th eigenvalues of the p laplacian operators the second problem we study is the homogenization of nonlinear eigenvalue problems for some p laplacian type operators with p fixed our asymptotic analysis relies on a convergence result for particular critical values of a class of rayleigh quotients stated in a unified framework and on the notion of vargamma convergence

convergence of interval type algorithms for generalized fractional programming the purpose of this paper is to analyze the convergence of interval type algorithms for solving the generalized fractional program they are characterized by an interval lb k ub k including and the length of the interval is reduced at each iteration a closer analysis of the bounds lb k and ub k allows to modify slightly the best known interval type algorithm newmodm accordingly to prove its convergence and derive convergence rates similar to those for a dinkelbach type algorithm maxmodm under the same conditions numerical results in the linear case indicate that the modifications to get convergence results are not obtained at the expense of the numerical efficiency since the modified version bfii is as efficient as newmodm and more efficient than maxmodm

on the exact separation of mixed integer knapsack cuts during the last decades much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets which we call knapsack cuts bixby et al the sharpest cut the impact of manfred padberg and his work mps siam series on optimization pp 309 326 2004 empirically observe that within the context of branch and cut algorithms to solve mixed integer programming problems the most important inequalities are knapsack cuts derived by the mixed integer rounding mir procedure in this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set the main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships the exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them using these benchmarks on miplib 3 0 and miplib 2003 instances we analyze the performance of mir inequalities our computations which are performed in exact arithmetic are surprising in the vast majority of the instances in which knapsack cuts yield bound improvements mir cuts alone achieve over 87 of the observed gain

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

homogenization of a class of nonlinear eigenvalue problems in this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form a e div a e x u which are sub differentials of even positively homogeneous convex functionals under the assumption that the operators g converge to an operator a hom div a hom x u we show that any limit point of a sequence of eigenvalues e is an eigenvalue of the limit operator a hom where e is an eigenvalue corresponding to the operator a e we also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator

revenue utility tradeoff in assortment optimization under the multinomial logit model with totally unimodular constraints we examine the revenue utility assortment optimization problem with the goal of finding an assortment that maximizes a linear combination of the expected revenue of the firm and the expected utilit

e optimality and duality for multiobjective fractional programming abstract using the scalar e parametric approach we establish the karush kuhn tucker which we call kkt necessary and sufficient conditions for an e pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints these optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems subsequently we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem under this case we derive the kkt necessary and sufficient conditions without a constraint qualification for e pareto optimality of multiobjective fractional programming

complexity of some parametric integer and network programming problems two examples of parametric cost programming problems one in network programming and one in np hard 0 1 programming are given in each case the number of breakpoints in the optimal cost curve is exponential in the square root of the number of variables in the problem

the fredholm alternative at the first eigenvalue for the one dimensionalp laplacian abstract in this work we study the range of the operatoru u p 2 u 1 u p 2 u u 0 u t 0 p 1 we prove that all functionsh c1 0 t satisfying t0 h t sinp pt t dt 0 lie in the range but that ifp 2 andh 0 the solution set is bounded here sin pt t is a first eigenfunction associated to 1 we also show that in this case the associated energy functionalu 1 p t0 u p 1 p t0 u p t0 huis unbounded from below if 1 2 onw1 p0 0 t 1corresponds precisely to the best constant in thelp poincare inequality moreover we show that unlike the linear casep 2 forp 2 the range contains a nonempty open set inl 0 t

fractional programming with convex quadratic forms and functions this article is concerned with two global optimization problems p1 and p2 each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function where at least one of the convex functions is a quadratic form first the article presents and validates a number of theoretical properties of these problems included among these properties is the result that under a mild assumption any globally optimal solution for problem p1 must belong to the boundary of its feasible region also among these properties is a result that shows that problem p2 can be reformulated as a convex maximization problem second the article presents for the first time an algorithm for globally solving problem p2 the algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems convergence properties of the algorithm are presented and computational issues that arise in implementing the algorithm are discussed preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem p2 provided that the number of variables is not too large

convergence rate for some quasilinear eigenvalues homogenization problems abstract in this work we study the homogenization problem for nonlinear eigenvalues of some quasilinear elliptic operators we obtain an explicit order of convergence in k and in e for the variational eigenvalues

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

analysis of a class fractional programming problems we propose a solution strategy for fractional programming problems of the form max x x g x u x where the function satisfies certain convexity conditions it is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max x x g x u x where is an exogenous parameter the proposed strategy combines fractional programming andc programming techniques a maximal mean standard deviation ratio problem is solved to illustrate the strategy in action

algorithms for generalized fractional programming a generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of n the purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis the conclusion includes results and comments on the numerical efficiency of these algorithms

volume functions of linear series the volume of a cartier divisor is an asymptotic invariant which measures the rate of growth of sections of powers of the divisor it extends to a continuous homogeneous and log concave function on the whole neron severi space thus giving rise to a basic invariant of the underlying projective variety analogously one can also define the volume function of a possibly non complete multigraded linear series in this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties in the multigraded setting inspired by the work of lazarsfeld and musta ann inst fourier grenoble 56 6 1701 1734 2006 on okounkov bodies we show that any continuous homogeneous and log concave function appears as the volume function of a multigraded linear series by contrast we show that there exists countably many functions which arise as the volume functions of projective varieties we end the paper with an example where the volume function of a projective variety is given by a transcendental formula emphasizing the complicated nature of the volume in the classical case

structural and algorithmic properties for parametric minimum cuts we consider the minimum s t cut problem in a network with parametrized arc capacities following the seminal work of gallo et al siam j comput 18 1 30 55 1989 classes of this parametric problem have been shown to enjoy the nice structural property that minimum cuts are nested and the nice algorithmic property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever flow update step to move from one value of the parameter to the next we present a general framework for parametric minimum cuts that extends and unifies such results we define two conditions on parametrized arc capacities that are necessary and sufficient for strictly decreasing differences of the parametric cut function known results in parametric submodular optimization then imply the structural property we show how to construct appropriate flow updates in linear time under the above conditions implying that the algorithmic property also holds under these conditions we then consider other classes of parametric minimum cut problems without decreasing differences for which we establish the structural and or the algorithmic property as well as other cases where nested minimum cuts arise

arithmetic linear series with base conditions in this note we study the volume of arithmetic linear series with base conditions as an application we consider the problem of zariski decompositions on arithmetic varieties

a minimal algorithm for the 0 1 knapsack problem several types of large sized 0 1 knapsack problems kp may be easily solved but in such cases most of the computational effort is used for sorting and reduction in order to avoid this problem it has been proposed to solve the so called core of the problem a knapsack problem defined on a small subset of the variables the exact core cannot however be identified before kp is solved to optimality thus previous algorithms had to rely on approximate core sizes in this paper we present an algorithm for kp where the enumerated core size is minimal and the computational effort for sorting and reduction also is limited according to a hierarchy the algorithm is based on a dynamic programming approach where the core size is extended by need and the sorting and reduction is performed in a similar lazy way computational experiments are presented for several commonly occurring types of data instances experience from these tests indicate that the presented approach outperforms any known algorithm for kp

a new algorithm for generalized fractional programs a new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach the resulting algorithm can be seen as dual to the dinkelbach type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual primal problem from below convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established moreover under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem we also consider a variant of this new algorithm based on scaling the dual parametric function the numerical results in case of quadratic linear ratios and linear constraints show that the performance of the new algorithm and its scaled version is superior to that of the dinkelbach type algorithms from the computational results it also appears that contrary to the primal approach the dual approach is less influenced by scaling

approximation algorithms for combinatorial fractional programming problems we are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function this type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function because of its combinatorial nature however it is often difficult to solve the auxiliary problem exactly in this paper we propose an algorithm which assumes that the auxiliary problems are solved only approximately and prove that it gives an approximate solution to the original problem of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems it is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem as an example of the proposed algorithm we present a fully polynomial time approximation scheme for the fractional 0 1 knapsack problem

computational study of a column generation algorithm for bin packing and cutting stock problems this paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem the main focus of the research is to study the extend to which standard branch and bound enhancement features such as variable fixing the tightening of the formulation with cutting planes early branching and rounding heuristics can be usefully incorporated in a branch and price algorithm we review and compare lower bounds for the cutting stock problem we propose a pseudo polynomial heuristic we discuss the implementation of the important features of the integer programming column generation algorithm and in particular the implementation of the branching scheme our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems

convergence rates in l 2 for elliptic homogenization problems we study rates of convergence of solutions in l2 and h1 2 for a family of elliptic systems mathcal l varepsilon with rapidly oscillating coefficients in lipschitz domains with dirichlet or neumann boundary conditions as a consequence we obtain convergence rates for dirichlet neumann and steklov eigenvalues of mathcal l varepsilon most of our results which rely on the recently established uniform estimates for the l2 dirichlet and neumann problems in kenig and shen math ann 350 867 917 2011 commun pure appl math 64 1 44 2011 are new even for smooth domains

parametric approaches to fractional programs the fractional program p is defined by maxf x g x subject tox x a class of methods for solving p is based on the auxiliary problem q with a parameter maxf x g x subject tox x starting with two classical methods in this class the newton method and the binary search method a number of variations are introduced and compared among the proposed methods the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value bar lambda computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs the interpolated binary search method seems to be most efficient while other methods also behave surprisingly well

approximating a class of combinatorial problems with rational objective function in the late seventies megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 a 1 x 1 a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form a 0 a 1 x 1 a n x n b 0 b 1 x 1 b n x n subject to the same set of constraints assuming that the denominator is always positive using a rather strong assumption hashizume et al extended megiddo s result to include approximation algorithms their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the linear objective function which is rather unusual for most combinatorial problems in this paper we present an alternative extension of megiddo s result for approximations that avoids this issue and applies to a large class of optimization problems specifically we show that if there is an approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an approximation 1 approximation for the problem of minimizing maximizing a nonnegative rational function subject to the same constraints our framework applies to covering problems and network design problems among others

low frequency asymptotic analysis of a string with rapidly oscillating density we consider the eigenvalue problem associated to the vibrations of a string with a rapidly oscillating bounded periodic density it is well known that when the size of the microstructure epsilon is small enough with respect to the wavelength of the eigenfunctions 1 sqrt lambda epsilon eigenvalues and eigenfunctions can be approximated by those of the limit system where the oscillating density is replaced by its average on the other hand it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions epsilon sim 1 sqrt lambda epsilon singular phenomena may occur in this paper we study the behavior of the eigenvalues and eigenfunctions when 1 sqrt lambda epsilon approaches the critical size epsilon to do this we use the wkb approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to epsilon in particular our analysis provides all order correction formulas fo

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

on the robust knapsack problem we consider an uncertain variant of the knapsack problem that arises when the exact weight of each item is not exactly known in advance but belongs to a given interval and the number of items whose weight differs from the nominal value is bounded by a constant we analyze the worsening of the optimal solution value with respect to the classical problem and exactly determine its worst case performance depending on uncertainty for all parameter configurations we perform the same analysis for the fractional version of the problem in which one is allowed to pack any fraction of the items in addition we derive the worst case performance ratio with respect to the optimal solution value for both the fractional problem and for a variant of the well known greedy algorithm finally we consider a relevant special case and provide a combinatorial algorithm for solving the fractional problem in an efficient way

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

newton okounkov bodies on projective bundles over curves in this article we study newton okounkov bodies on projective vector bundles over curves inspired by wolfe s estimates used to compute the volume function on these varieties we compute all newton okounkov bodies with respect to linear flags moreover we characterize semi stable vector bundles over curves via newton okounkov bodies

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

optimization problems with algebraic solutions quadratic fractional programs and ratio games a mathematical program with a rational objective function may have irrational algebraic solutions even when the data are integral we suggest that for such problems the optimal solution will be represented as follows if denotes the optimal value there will be given an intervali and a polynomialp such thati contains and is the unique root ofp ini it is shown that with this representation the solutions to convex quadratic fractional programs and ratio games can be obtained in polynomial time

fractional programming applications and algorithms abstract the paper deals with decision problems that give rise to the optimization of ratios subject to constraints these so called fractional programs have been treated in a considerable number of papers it is attempted to survey applications as well as solution methods in linear quadratic and concave convex fractional programming in the first part we give a detailed outline on the major areas of applications of fractional programming following a brief review on some basic theoretical results like duality relations in the second part we then discuss different algorithmic approaches a primal a dual and a parametric solution method is considered here we try to determine for each method the kind of fractional programs that are solved by it suitably

log concavity of multiplicities with application to characters ofu the log concavity of the reduction multiplicities for the classical groups of typean bn cnis proved moreover the skew schur functionss are shown to be log concave coefficient by coefficient the results are applied to the calculation of the characters of the infinite dimensional classical groups the log concavity of density of the push forward of the liouville measure on coadjoint orbits under moment map is proved

strong continuity of eigen pairs of the schrodinger operator with integrable potentials abstract in this paper we consider the eigenvalue problem of the schrodinger operator with an integrable potential our first result states that each eigenvalue is continuously depending on the potential in the weak topology of some lebesgue space furthermore if potentials are convergent weakly in the lebesgue space then for each m 1 the m th normalized eigenfunction is strongly convergent to the eigen space of the m th eigenvalue our second result obtains the convergence rate of eigenvalues provided the potential is a cubic periodic integrable function

the inverse parametric knapsack problem abstract this paper deals with parametric knapsack problems where the costs resp weights are replaced by linear functions depending on a parameter t the aim is to find the smallest parameter t such that the optimal solution value of the knapsack problem is equal to a prespecified solution value for this inverse parametric problem pseudopolynomial algorithms are developed and search methods making use of the special properties of the parametric value function are constructed using computational experiments the behaviour of these algorithms is investigated and the favourable practical performance of different search methods is exhibited

strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs we consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded rank hypergraphs with multiple edge cost functions for a fixed number of edge cost functions we show that the total number of supported non dominated snd cuts is bounded by a polynomial in the numbers of nodes and edges i e is strongly polynomial this bound also applies to the combinatorial facet complexity of the problem i e the maximum number of facets linear pieces of the parametric curve for the parametrized linear combination objective over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive we sharpen this bound in the case of two objectives the bicriteria problem for which we also derive a strongly polynomial upper bound on the total number of non dominated pareto optimal cuts in particular the bicriteria global minimum cut problem in an n node graph admits o n 3 log n o n3logn snd cuts and o n 5 log n o n5logn pareto optimal cuts these results significantly improve on earlier graph cut results by mulmuley siam j comput 28 4 1460 1509 1999 and armon and zwick algorithmica 46 1 15 26 2006 they also imply that the parametric curve and all snd cuts and for the bicriteria problems all pareto optimal cuts can be computed in strongly polynomial time when the number of objectives is fixed

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

okounkov bodies on projectivizations of rank two toric vector bundles abstract the global okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety in the case of a rank two toric vector bundle e on a smooth projective toric variety we use its klyachko filtrations to give an explicit description of the global okounkov body of p e in particular we show that this is a rational polyhedral cone and that p e is a mori dream space

special cases of the quadratic assignment problem abstract this paper presents a result of transforming a class of non convex programs into parametric programs parametric linear programming algorithms are then developed to solve several special cases of the quadratic assignment problem qap lower and upper bounds of the parameters are derived to further accelerate the parameter search numerical experiments are conducted to show the efficiency of the algorithm

an efficient algorithm for globally minimizing sum of quadratic ratios problem with nonconvex quadratic constraints in this paper a new branch and bound algorithm based on the rectangular partition and the lagrangian relaxation for solving sum of quadratic ratios problem with nonconvex quadratic constraints qsrp is proposed firstly the problem qsrp is converted into an equivalent sum of linear ratios problem with quadratic constraints lsrp then the bounding procedure is investigated by minimizing the restricted lagrangian function of lsrp with a given estimate of the lagrange multipliers minimizing a linear relaxation of the restricted lagrangian overcomes the difficulty that the restricted lagrangian is often nonconvex and facilitates the use of lagrangian duality within a global optimization framework in the algorithm the interval newton method is used to facilitate convergence in the neighborhood of the global solution the proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems finally numerical experiments are reported to show the efficiency of our algorithm

convex bodies appearing as okounkov bodies of divisors abstract based on the work of okounkov okounkov 1996 15 2003 16 lazarsfeld and musta 2009 13 and kaveh and khovanskii preprint 10 have independently associated a convex body called the okounkov body to a big divisor on a smooth projective variety with respect to a complete flag in this paper we consider the following question what can be said about the set of convex bodies that appear as okounkov bodies we show first that the set of convex bodies appearing as okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable we then give a complete characterisation of the set of convex bodies that arise as okounkov bodies of r divisors on smooth projective surfaces such okounkov bodies are always polygons satisfying certain combinatorial criteria finally we construct two examples of non polyhedral okounkov bodies in the first one the variety we deal with is fano and the line bundle is ample in the second one we find a mori dream space variety such that under small perturbations of the flag the okounkov body remains non polyhedral

integrable systems toric degenerations and okounkov bodies let x be a smooth projective variety of dimension n over mathbb c equipped with a very ample hermitian line bundle mathcal l in the first part of the paper we show that if there exists a toric degeneration of x satisfying some natural hypotheses which are satisfied in many settings then there exists a surjective continuous map from x to the special fiber x 0 which is a symplectomorphism on an open dense subset u from this we are then able to construct a completely integrable system on x in the sense of symplectic geometry more precisely we construct a collection of real valued functions h 1 ldots h n on x which are continuous on all of x smooth on an open dense subset u of x and pairwise poisson commute on u moreover our integrable system in fact generates a hamiltonian torus action on u in the second part we show that the toric degenerations arising in the theory of newton okounkov bodies satisfy all the hypotheses of the first part of the paper in this case the image of the moment map mu h 1 ldots h n x rightarrow mathbb r n is precisely the newton okounkov body delta delta r v associated to the homogeneous coordinate ring r of x and an appropriate choice of a valuation v on r our main technical tools come from algebraic geometry differential kahler geometry and analysis specifically we use the gradient hamiltonian vector field and a subtle generalization of the famous ojasiewicz gradient inequality for real valued analytic functions since our construction is valid for a large class of projective varieties x this manuscript provides a rich source of new examples of integrable systems we discuss concrete examples including elliptic curves flag varieties of arbitrary connected complex reductive groups spherical varieties and weight varieties

fractional programming revisited abstract a new format is proposed for fractional programming problems this format gives full expression to the fact that the parametric approach to fractional programming problems is rooted in a first order necessary and sufficient optimality condition it is thus shown that although traditionally it has not been construed as such the parametric approach is in fact classical par excellence

ratio combinatorial programs abstract we consider here a combinatorial optimization problem where the objective function is a ratio of two linear functions one way of solving this problem is to solve repeatedly an auxiliary problem with a parameterized linear objective function in this paper we relate this method to newton s method for solving equations and present a modification of this algorithm which solves the ratio problem approximate ly by repeatedly solving the auxiliary linear problems with the same accuracy we demonstrate by reporting results of extensive computational experiments that the above modified algorithm for the approximate knapsack problem performs extremely well in practice

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

a global optimization algorithm for sum of quadratic ratios problem with coefficients abstract in this paper a global optimization algorithm for solving sum of quadratic ratios problem with coefficients and nonconvex quadratic function constraints nsp is proposed first the problem nsp is converted into an equivalent sum of linear ratios problem with nonconvex quadratic constraints lsp using a linearization technique the linearization relaxation problem of lsp is obtained the original problem is then solvable using the branch and bound method in the algorithm lower bounds are derived by solving a sequence of linear lower bounding functions for the objective function and the constraint functions of the problem nsp over the feasible region the proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems the numerical examples demonstrate that the proposed algorithm can easily be applied to solve problem nsp

tolerance analysis for 0 1 knapsack problems abstract post optimal analysis is the task of understanding the behavior of the solution of a problem due to changes in the data frequently post optimal analysis is as important as obtaining the optimal solution itself post optimal analysis for linear programming problems is well established and widely used however for integer programming problems the task is much more computationally demanding and various approaches based on branch and bound or cutting planes have been presented in the present paper we study how much coefficients in the original problem can vary without changing the optimal solution vector the so called tolerance analysis we show how to perform exact tolerance analysis for the 0 1 knapsack problem with integer coefficients in amortized time o clog n for each item where n is the number of items and c is the capacity of the knapsack amortized running times report the time used for each item when calculating tolerance limits of all items exact tolerance limits are the widest possible intervals while approximate tolerance limits may be suboptimal we show how various upper bounds can be used to determine approximate tolerance limits in time o log n or o 1 per item using the dantzig bound and dembo hammer bound respectively the running times and quality of the tolerance limits of all exact and approximate algorithms are experimentally compared showing that all tolerance limits can be found in less than a second the approximate bounds are of good quality for large sized instances while it is worth using the exact approach for smaller instances

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

local okounkov bodies and limits in prime characteristic this article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic these sequences which we call p families of ideals are ubiquitous in prime characteristic commutative algebra e g they occur naturally in the theories of tight closure hilbert kunz multiplicity and f signature we associate to each p family of ideals an object in euclidean space that is analogous to the newton okounkov body of a graded family of ideals which we call a p body generalizing the methods used to establish volume formulas for the hilbert kunz multiplicity and f signature of semigroup rings we relate the volume of a p body to a certain asymptotic invariant determined by the corresponding p family of ideals we apply these methods to obtain new existence results for limits in positive characteristic an analogue of the brunn minkowski theorem for hilbert kunz multiplicity and a uniformity result concerning the positivity of a p family

testing integer knapsacks for feasibility abstract we present a new approach for determining whether there exist nonnegative integers x1 x2 xn satisfying a1x1 a2x2 anxn b where a1

asymptotic base loci via okounkov bodies abstract an okounkov body is a convex subset of euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag in this paper we recover the asymptotic base loci from the okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving seshadri constants consequently we obtain the nefness and ampleness criteria of divisors in terms of the okounkov bodies furthermore we compute the divisorial zariski decomposition by the okounkov bodies and find upper and lower bounds for moving seshadri constants given by the size of simplexes contained in the okounkov bodies

generalized fractional programming algorithms and numerical experimentation abstract several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case these algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case the most efficient algorithm is obtained by marrying the newton approach within the dinkelbach approach for fractional programming

core problems in knapsack algorithms since balas and zemel in the 1980s introduced the so called core problem as an efficient tool for solving the knapsack problem all the most successful algorithms have applied this concept balas and zemel proved that if the weights in the core are uniformly distributed then there is a high probability for finding an optimal solution in the core items outside the core may be fathomed because of reduction rules this paper demonstrates that generally it is not reasonable to assume a uniform distribution of the weights in the core and it is experimentally shown that the heuristic proposed by balas and zemel does not find as good solutions as expected also other algorithms that solve some kind of core problem may be stuck by difficult cores this behavior has apparently not been noticed before because of unsufficient testing capacities leading to difficult problems are identified for several categories of instance types and it is demonstrated that the hitherto applied test instances are easier than the average as a consequence we propose a series of new randomly generated test instances and show how recent algorithms behave when applied to these problems