mixed integer programming model formulations for dynamic multi item multi level capacitated lotsizing abstract recently there have been two new proposals to model the dynamic multi item multi level capacitated lotsizing problem by variable redefinitions based on either a shortest route or a simple plant location representation of the underlying decision problem here we will introduce an extended version of the well known inventory and lotsize model followed by a new model formulation based on modeling the changes of end of period inventory levels explicitly secondly we will compare the suitability of different model formulations when solving problem instances with up to 40 items and 16 periods utilizing state of the art standard mip software on a personal computer

a hybrid optimization approach for multi level capacitated lot sizing problems abstract solving multi level capacitated lot sizing problems is still a challenging task in spite of increasing computational power and faster algorithms in this paper a new approach combining an ant based algorithm with an exact solver for mixed integer linear programs is presented a max min ant system is developed to determine the principal production decisions a lp mip solver is used to calculate the corresponding production quantities and inventory levels two different local search methods and an improvement strategy based on reduced mixed integer problems are developed and integrated into the ant algorithm this hybrid approach provides superior results for small and medium sized problems in comparison to the existing approaches in the literature for large scale problems the performance of this method is among the best

the structure of qubit unextendible product bases unextendible product bases upbs have been shown to have many important uses in quantum information theory particularly in the qubit case however very little is known about their mathematical structure beyond three qubits we present several new results about qubit upbs including a complete characterization of all four qubit upbs which we show there are exactly 1446 of we also show that there exist p qubit upbs of almost all sizes less than 2p this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

lagrangean relaxation based heuristics for lot sizing with setup times we consider a lot sizing problem with setup times where the objective is to minimize the total inventory carrying cost only the demand is dynamic over time and there is a single resource of limited capacity we show that the approaches implemented in the literature for more general versions of the problem do not perform well in this case we examine the lagrangean relaxation lr of demand constraints in a strong reformulation of the problem we then design a primal heuristic to generate upper bounds and combine it with the lr problem within a subgradient optimization procedure we also develop a simple branch and bound heuristic to solve the problem computational results on test problems taken from the literature show that our relaxation procedure produces consistently better solutions than the previously developed heuristics in the literature

multilevel capacitated lotsizing complexity and lp based heuristics abstract this paper presents the first heuristics capable of solving multilevel lotsizing problems with capacity constraints on more than one level moreover the form of the heuristics is quite general so that they can easily be extended to solve a variety of problems if one wants to solve these problems on a routine basis in a real environment one needs to find fast and easy algorithms however we show that for certain problem classes this is a very difficult task far more difficult than has been suggested in the literature for problems with setup times we show that finding a feasible solution is np complete even without setup times testing for feasibility can be very difficult just how time consuming such heuristics must be is demonstrated this leaves little chance to build fast and easy heuristics except for the most simple cases our exploration of the complexity issues points to mathematical programming as a potential source of heuristics for these problems this paper presents a new and general approach based on rounding an lp solution for the problem without setup times the methods use different information and patterns evident in the lp solution are explored the approach is tested on a large set of problems the main contributions of this paper are the way in which we distinguish between the easy and hard lotsizing problems the lp based heuristics and the test set of capacitated multilevel lotsizing problems

asymptotic state discrimination and a strict hierarchy in distinguishability norms in this paper we consider the problem of discriminating quantum states by local operations and classical communication locc when an arbitrarily small amount of error is permitted this paradigm is known as asymptotic state discrimination and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic locc we use this new criterion to prove a gap in the locc and separable distinguishability norms we then turn to the operational advantage of using two way classical communication over one way communication in locc processing with a simple two qubit product state ensemble we demonstrate a strict majorization of the two way locc norm over the one way norm

high dimensional entanglement in states with positive partial transposition genuine high dimensional entanglement i e the property of having a high schmidt number constitutes an instrumental resource in quantum communication overcoming limitations of low dimensional systems states with a positive partial transpose ppt are generally considered weakly entangled as they can never be distilled into pure entangled states this naturally raises the question of whether high schmidt numbers are possible for ppt states volume estimates suggest that optimal i e linear scaling in the local dimension should be possible albeit without providing insight into the possible slope we provide the first explicit construction of a family of ppt states that achieves linear scaling in the local dimension and we prove that random ppt states typically share this feature our construction also allows us to prove a recent conjecture of chen et al on the existence of ppt states whose schmidt number increases by an arbitrarily large amount upon partial transposition finally we link the schmidt number to entangled sub block matrices of a quantum state we use this connection to prove that quantum states that are either i invariant under partial transposition on the smallest of their two subsystems or ii absolutely ppt cannot have a maximal schmidt number overall our findings shed new light on some fundamental problems in entanglement theory

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

a lagrangian based heuristic for the capacitated lot sizing problem in parallel machines this paper addresses the capacitated lot sizing problem involving the production of multiple items on unrelated parallel machines a production plan should be determined in order to meet the forecast demand for the items without exceeding the capacity of the machines and minimize the sum of production setup and inventory costs a heuristic based on the lagrangian relaxation of the capacity constraints and subgradient optimization is proposed initially the heuristic is tested on instances of the single machine problem and results are compared with heuristics from the literature for parallel machines and small problems the heuristic performance is tested against optimal solutions and for larger problems it is compared with the lower bound provided by the lagrangian relaxation

a hybrid adaptive large neighborhood search heuristic for lot sizing with setup times abstract this paper presents a hybrid of a general heuristic framework and a general purpose mixed integer programming mip solver the framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched a mixed integer programming solver and its built in feasibility heuristics is used to search a neighborhood for improving solutions the general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods the hybrid heuristic framework is applied to the multi item capacitated lot sizing problem with setup times where experiments have been conducted on a series of instances from the literature and a newly generated extension of these on average the presented heuristic outperforms the best heuristics from the literature and the upper bounds found by the commercial mip solver ilog cplex using state of the art mip formulations furthermore we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature

construction of 3 3 entangled edge states with positive partial transposes we construct a class of 3 3 entangled edge states with positive partial transposes using indecomposable positive linear maps this class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

dimensions lengths and separability in finite dimensional quantum systems many important sets of normalized states in a multipartite quantum system of finite dimension d such as the set s of all separable states are real semialgebraic sets we compute dimensions of many such sets in several low dimensional systems by using dimension arguments we show that there exist separable states which are not convex combinations of d or less pure product states for instance such states exist in bipartite m n systems when m 2 n 2 1 this solves an open problem proposed by divincenzo terhal and thapliyal about 12 years ago we prove that there exist a separable state and a pure product state whose mixture has smaller length than that of we show that any real s which is invariant under all partial transpose operations is a convex sum of real pure product states in the case of the 2 n system the number r of product states can be taken to be r rank we also show that the general multipartite separability problem can be reduced to the case of real states regarding

grasp heuristic with path relinking for the multi plant capacitated lot sizing problem this paper addresses the independent multi plant multi period and multi item capacitated lot sizing problem where transfers between the plants are allowed this is an np hard combinatorial optimization problem and few solution methods have been proposed to solve it we develop a grasp greedy randomized adaptive search procedure heuristic as well as a path relinking intensification procedure to find cost effective solutions for this problem in addition the proposed heuristics is used to solve some instances of the capacitated lot sizing problem with parallel machines the results of the computational tests show that the proposed heuristics outperform other heuristics previously described in the literature the results are confirmed by statistical tests

lot sizing in general assembly systems with setup costs setup times and multiple constrained resources we introduce a heuristic method for finding good feasible solutions for multiproduct lot sizing problems with general assembly structures multiple constrained resources and nonzero setup costs and setup times we evaluate the performance of this heuristic by comparing its solutions to optimal solutions of small randomly generated problems and to time truncated optimization subroutine library osl solutions of medium sized randomly generated problems in the first case the heuristic locates solutions averaging 4 percent worse than optimal in less than 1 percent of time required by osl the heuristic solutions to medium sized problems are approximately 26 percent better than solutions osl finds after 10 000 cpu seconds and the heuristic finds these solutions in approximately 10 percent of osl time

limitations on separable measurements by convex optimization we prove limitations on locc and separable measurements in bipartite state discrimination problems using techniques from convex optimization specific results that we prove include an exact formula for the optimal probability of correctly discriminating any set of either three or four bell states via locc or separable measurements when the parties are given an ancillary partially entangled pair of qubits an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements and an optimal bound on the success probability for any locc or separable measurement for the recently proposed state discrimination problem of yu duan and ying

unextendible product bases and the construction of inseparable states abstract let h n denote the tensor product of n finite dimensional hilbert spaces h r a state of h n is separable if 1 n where the states r are in h r an orthogonal unextendible product basis is a finite set b of separable orthonormal states k 1 k m such that the non empty space b the set of vectors orthogonal to b contains no separable state examples of orthogonal upb sets were first constructed by bennett et al 1 and other examples and references appear for example in 3 if f f b denotes the set of convex combinations of k k 1 k m then f is a face in the set s of separable densities in this note we show how to use f to construct families of positive partial transform states ppt which are not separable we also show how to make an analogous construction when the condition of orthogonality is dropped the analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable ppt states from s

rank two bipartite bound entangled states do not exist we explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement we show a relation between the rank marginal ranks and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n n hilbert space a direct consequence of this result is that there are no bipartite bound entangled states of rank two we also show that a separability condition in terms of a quantum entropy inequality is associated with the above results we explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a schmidt rank n pure state and provide a lower bound of n 1 we also prove that a mixture of a non zero amount of any pure entangled state with a pure product state is distillable

the octomorphic criterion for real parameter uncertainty real m bounds without circles and d n scales abstract in this paper we introduce new bounds for robust stability analysis with real parameter uncertainty the approach is based on an absolute stability criterion that excludes the nyquist plot from a paraboloidal region containing the point 1 j0 transformation of this criterion to the case of norm bounded uncertainty leads to a stability criterion in terms of the octomorphic or figure eight shaped region the requirement that the nyquist plot lie inside the octomorphic region thus yields a bound on the allowable real parameter uncertainty this stability criterion is distinct from recent bounds for real which involve frequency dependent scales having a frequency dependent off axis circle interpretation since the octomorphic region includes both upper and lower halves it is able to encompass the entire nyquist plot without using frequency dependent scales

solving uncapacitated multilevel lot sizing problems using a particle swarm optimization with flexible inertial weight the multilevel lot sizing mlls problem is a key production planning problem in materials requirements planning mrp system the mlls problem deals with determining the production lot sizes of various items appearing in the product structure over a given finite planning horizon to minimize the production cost the inventory carrying cost the back ordering cost and etc this paper proposed a particle swarm optimization pso algorithm for solving the uncapacitated mlls problem with assembly structure all the mathematical operators in our algorithm are redefined and the inertial weight parameter can be either a negative real number or a positive one the feasibility and effectiveness of our algorithm are investigated by comparing the experimental results with those of a genetic algorithm ga

unextendible product bases uncompletable product bases and bound entanglement we report new results and generalizations of our work on unextendible product bases upb uncompletable product bases and bound entanglement we present a new construction for bound entangled states based on product bases which are only completable in a locally extended hilbert space we introduce a very useful representation of a product basis an orthogonality graph using this representation we give a complete characterization of unextendible product bases for two qutrits we present several generalizations of upbs to arbitrary high dimensions and multipartite systems we present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators we prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement

meta heuristics for dynamic lot sizing a review and comparison of solution approaches proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve because these problems are so difficult various solution techniques have been proposed to solve them in the past decade meta heuristics such as tabu search genetic algorithms and simulated annealing have become popular and efficient tools for solving hard combinatorial optimization problems we review the various meta heuristics that have been specifically developed to solve lot sizing problems discussing their main components such as representation evaluation neighborhood definition and genetic operators further we briefly review other solution approaches such as dynamic programming cutting planes dantzig wolfe decomposition lagrange relaxation and dedicated heuristics this allows us to compare these techniques understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms finally we discuss general guidelines for computational experiments and illustrate these with several examples

the minimum size of unextendible product bases in the bipartite case and some multipartite cases a long standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite hilbert space a partial solution was found by alon and lovasz j comb theory ser a 95 169 179 2001 but since then only a few other cases have been solved we solve all remaining bipartite cases as well as a large family of multipartite cases

extensions of mixed µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory in this paper we make explicit connections between classical absolute stability theory and modern mixed analysis and synthesis specifically using the parameter dependent lyapunov function framework of haddad and bernstein and the frequency dependent off axis circle interpretation of how and hall we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty an immediate application of this framework is the generalization and reformulation of mixed analysis and synthesis in terms of lyapunov functions and riccati equations this observation is exploited to provide robust reduced order controller synthesis while avoiding the standard d n k iteration and curve fitting procedures

feynman graphs and the large dimensional limit of multipartite entanglement in this paper we extend the analysis of multipartite entanglement based on techniques from classical statistical mechanics to a system composed of n d level parties qudits we introduce a suitable partition function at a fictitious temperature with the average local purity of the system as hamiltonian in particular we analyze the high temperature expansion of this partition function prove the convergence of the series and study its asymptotic behavior as d we make use of a diagrammatic technique classify the graphs and study their degeneracy we are thus able to evaluate their contributions and estimate the moments of the distribution of the local purity

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

parameter dependent lyapunov functions and the popov criterion in robust analysis and synthesis many practical applications of robust feedback control involve constant real parameter uncertainty whereas small gain or norm bounding techniques guarantee robust stability against complex frequency dependent uncertainty thus entailing undue conservatism since conventional lyapunov bounding techniques guarantee stability with respect to time varying perturbations they possess a similar drawback in this paper we develop a framework for parameter dependent lyapunov functions a less conservative refinement of fixed lyapunov functions an immediate application of this framework is a reinterpretation of the classical popov criterion as a parameter dependent lyapunov function this result is then used for robust controller synthesis with full order and reduced order controllers

product vectors in the ranges of multi partite states with positive partial transposes and permanents of matrices in this paper we consider a system of homogeneous algebraic equations in complex variables and their conjugates which arise naturally from the range criterion for separability of ppt states we examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions this gives us possible upper bounds of ranks of ppt entangled edge states and their partial transposes we will focus on the multi partite cases which are much more delicate than the bi partite cases we use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion

the multi item capacitated lot size problem error bounds of manne s formulations we discuss an approximation scheme for the multi item lot size problem it is based on an optimal basic solution of a linear programming problem derived from the original problem the approximate solution is obtained by taking a linear convex combination of the optimal solution of the linear programming problem we express error bounds of the approximation as a function of some parameters that can be easily estimated in practice when set up times are positive the approximation may result in an infeasible solution we take the same approach to show that the infeasibility of the approximation is small the analysis is extended to a variable capacity problem with overtime as an auxiliary result we provide a bound on the duality gap of the lagrangian dual problem

multilevel lot sizing with setup times and multiple constrained resources internally rolling schedules with lot sizing windows in this paper a new time oriented decomposition heuristic is proposed to solve the dynamic multi item multilevel lot sizing problem in general product structures with single and multiple constrained resources as well as setup times while lot sizing decisions are made sequentially within an internally rolling planning interval or lot sizing window capacities are always considered over the entire planning horizon for each submodel a model formulation based on the simple plant location representation is developed these mixed integer linear submodels are solved by standard mathematical programming software even for relatively large test instances extensive computational tests show that the heuristic proposed provides a better solution quality than a well known special purpose heuristic

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

separability problem for multipartite states of rank at most 4 one of the most important problems in quantum information is the separability problem which asks whether a given quantum state is separable we investigate multipartite states of rank at most 4 which are ppt i e all their partial transposes are positive semidefinite we show that any ppt state of rank 2 or 3 is separable and has length at most 4 for separable states of rank 4 we show that they have length at most 6 it is six only for some qubit qutrit or multiqubit states it turns out that any ppt entangled state of rank 4 is necessarily supported on a 3 3 or a 2 2 2 subsystem we obtain a very simple criterion for the separability problem of the ppt states of rank at most 4 such a state is entangled if and only if its range contains no product vectors this criterion can be easily applied since a four dimensional subspace in the 3 3 or 2 2 2 system contains a product vector if and only if its plucker coordinates satisfy a homogeneous polynomial equation the chow form of the corresponding segre variety we have computed an explicit determinantal expression for the chow form in the former case while such an expression was already known in the latter case

a lagrangean based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times in this paper a heuristic approach for the dynamic multilevel multiitem lotsizing problem in general product structures with multiple constrained resources and setup times is proposed with the help of lagrangean relaxation the capacitated multilevel multiitem lotsizing problem is decomposed into several uncapacitated single item lotsizing problems from the solutions of these single item problems lower bounds on the minimum objective function value are derived upper bounds are generated by means of a heuristic finite scheduling procedure the quality of the approach is tested with reference to various problem groups of differing sizes

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

on characterizing integral zeros of krawtchouk polynomials by quantum entanglement abstract we propose a new characterization of integral zeros of binary krawtchouk polynomials in terms of the range criterion for entangled states in quantum information theory

classification of bi qutrit positive partial transpose entangled edge states by their ranks we construct 3 3 ppt entangled edge states with maximal ranks to complete the classification of 3 3 ppt entangled edge states by their types the ranks of the states and their partial transposes are 8 and 6 respectively these examples also disprove claims in the literature

entangled edge states of corank one with positive partial transposes we construct a parameterized family of n n ppt positive partial transpose states of corank one for each n 3 with a suitable choice of parameters we show that they are n n ppt entangled edge states of corank one for 3 n 1000 they violate the range criterion for separability in the most extreme way note that corank one is the smallest possible corank for such states the corank of the partial transpose is given by 2n 3 which is also the smallest possible corank for the partial transposes of ppt entangled edge states of corank one they provide the first explicit examples of such states for n 4

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables

real m bounds based on fixed shapes in the nyquist plane parabolas hyperbolas cissoids nephroids and octomorphs in this paper we introduce new bounds for the real structured singular value the approach is based on absolute stability criteria with plant dependent multipliers that exclude the nyquist plot from fixed plane curve shapes containing the critical point jo unlike half plane and circle based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty since the plant dependent multipliers have the same functional form at all frequencies the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency dependent off axis circles that arise in standard real bounds