heuristics for the container loading problem abstract the knapsack container loading problem is the problem of loading a subset of rectangular boxes into a rectangular container of fixed dimensions such that the volume of the packed boxes is maximized a new heuristic based on the wall building approach is proposed which decomposes the problem into a number of layers which again are split into a number of strips the packing of a strip may be formulated and solved optimally as a knapsack problem with capacity equal to the width or height of the container the depth of a layer as well as the thickness of each strip is decided through a branch and bound approach where at each node only a subset of branches is explored several ranking rules for the selection of the most promising layer depths and strip widths are presented and the performance of the corresponding algorithms is experimentally compared for homogeneous and heterogeneous instances the best ranking rule is then used in a comprehensive computational study involving large sized instances these computational results show that instances with a total box volume up to 90 easily may be solved to optimality and that average fillings of the container volume exceeding 95 may be obtained for large sized instances

a dynamic programming based reduction procedure for the multidimensional 0 1 knapsack problem abstract this paper presents a preprocessing procedure for the 0 1 multidimensional knapsack problem first a non increasing sequence of upper bounds is generated by solving lp relaxations then a non decreasing sequence of lower bounds is built using dynamic programming the comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal or to fix a subset of variables to their optimal values in addition a heuristic solution is obtained computational experiments with a set of large scale instances show the efficiency of our reduction scheme particularly it is shown that our approach allows to reduce the cpu time of a leading commercial software

local and global lifted cover inequalities for the 0 1 multidimensional knapsack problem abstract the 0 1 multidimensional knapsack problem 0 1 mkp is a well known and strongly np hard combinatorial optimization problem with many applications up to now the majority of upper bounding techniques for the 0 1 mkp have been based on lagrangian or surrogate relaxation we show that good upper bounds can be obtained by a cutting plane method based on lifted cover inequalities lcis as well as using traditional lcis we use some new global lcis which take the whole constraint matrix into account

large deviation principle for benedicks carleson quadratic maps since the pioneering works of jakobson and benedicks carleson and others it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to lebesgue these measures allow one to statistically predict the asymptotic fate of lebesgue almost every initial condition estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space we use the sub exponential slow recurrence condition of benedicks carleson to build induced markov maps of arbitrarily small scale and associated towers to which the absolutely continuous measures can be lifted these various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle for the absolutely continuous measures this result encompasses dynamics far from equilibrium and thus significantly extends presently known local large deviations results for quadratic maps

lagrangean decomposition a model yielding stronger lagrangean bounds given a mixed integer programming problem with two matrix constraints it is possible to define a lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems each having one of the two matrices of the original problem as its constraints there is one lagrangean multiplier per variable we prove that the optimal value of this new lagrangean dual dominates the optimal value of the lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement we show on an example that the resulting bound improvement can be substantial we show on a complex practical problem how lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology

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

dynamic programming based algorithms for the discounted 0 1 knapsack problem abstract the discounted 0 1 knapsack problem dkp is an extension of the classical 0 1 knapsack problem kp that consists of selecting a set of item groups where each group includes three items and at most one of the three items can be selected the dkp is more challenging than the kp because four choices of items in an item group diversify the selection of the items consequently it is not possible to solve the dkp based on a classical definition of a core consisting of a small number of relevant variables this paper partitions the dkp into several easier sub problems to achieve problem reductions by imitating the core concept of the kp to derive an alternative core for the dkp numerical experiments with dp based algorithms are conducted to evaluate the effectiveness of the problem partition by solving the partitioned problem and the original problem based on different types of dkp instances

a branch and price and cut approach for sustainable crop rotation planning in this paper we study a multi periodic production planning problem in agriculture this problem belongs to the class of crop rotation planning problems which have received increased attention in the literature in recent years crop cultivation and fallow periods must be scheduled on land plots over a given time horizon so as to minimize the total surface area of land used while satisfying crop demands every period this problem is proven strongly np hard we propose a 0 1 linear programming compact formulation based on crop sequence graphs an extended formulation is then provided with a polynomial time pricing problem and a branch and priceand cut bpc algorithm is presented with adapted branching rules and cutting planes the numerical experiments on instances varying the number of crops periods and plots show the effectiveness of the bpc for the extended formulation compared to solving the compact formulation even though these two formulations have the same linear relaxation bound

a family of composite discrete bivariate distributions with uniform marginals for simulating realistic and challenging optimization problem instances abstract we consider a family of composite bivariate distributions or probability mass functions pmfs with uniform marginals for simulating optimization problem instances for every possible population correlation except the extreme values there are an infinite number of valid joint distributions in this family we quantify the entropy for all member distributions including the special cases under independence and both extreme correlations greater variety is expected across optimization problem instances simulated based on a high entropy pmf we present a closed form solution to the problem of finding the joint pmf that maximizes entropy for a specified population correlation and we show that this entropy maximizing pmf belongs to our family of pmfs we introduce the entropy range as a secondary indicator of the variety of instances that may be generated for a particular correlation finally we discuss how to systematically control entropy and correlation to simulate a set of synthetic problem instances that includes challenging examples and examples with realistic characteristics

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

separation algorithms for 0 1 knapsack polytopes valid inequalities for 0 1 knapsack polytopes often prove useful when tackling hard 0 1 linear programming problems to generate such inequalities one needs separation algorithms for them i e routines for detecting when they are violated we present new exact and heuristic separation algorithms for several classes of inequalities namely lifted cover extended cover weight and lifted pack inequalities moreover we show how to improve a recent separation algorithm for the 0 1 knapsack polytope itself extensive computational results on miplib and or library instances show the strengths and limitations of the inequalities and algorithms considered

an exact approach for the bilevel knapsack problem with interdiction constraints and extensions we consider the bilevel knapsack problem with interdiction constraints an extension of the classic 0 1 knapsack problem formulated as a stackelberg game with two agents a leader and a follower that choose items from a common set and hold their own private knapsacks first the leader selects some items to be interdicted for the follower while satisfying a capacity constraint then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits the goal of the leader is to minimize the follower s total profit we derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower s problem the approach strongly outperforms the current state of the art algorithms designed for the problem we extend the same algorithmic framework to the interval min max regret knapsack problem after providing a novel bilevel programming reformulation also for this problem the proposed approach outperforms the exact algorithms available in the literature

yet another induction scheme for non uniformly expanding transformations abstract we introduce a new induction scheme for non uniformly expanding maps f of compact riemannian manifolds relying upon ideas of 33 and 10 we use this induction approach to prove that the existence of a gibbs markov young structure is a necessary condition for f to preserve an absolutely continuous probability with all its lyapunov exponents positive

detectability in conventional and adaptive sampling in this paper a simple but very general method is given for estimating a population total with any sampling design when objects in sampled units are observed with imperfect detectability a problem characteristic of many surveys of natural and human populations in the most general case the method consists of dividing the value of the variable of interest associated with each detected object by the detection probability for that object and then proceeding to use the estimation method that would ordinarily be used under the design if there were no detectability problems examples illustrating the method include simple random sampling conventional unequal probability sampling and adaptive cluster sampling

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

thermodynamic formalism for contracting lorenz flows we study the expansion properties of the contracting lorenz flow introduced by rovella via thermodynamic formalism specifically we prove the existence of an equilibrium state for the natural potential hat varphi t x y z t log j x y z cu for the contracting lorenz flow and for t in an interval containing 0 1 we also analyse the lyapunov spectrum of the flow in terms of the pressure

easily computable facets of the knapsack polytope it is known that facets and valid inequalities for the knapsack polytope p can be obtained by lifting a simple inequality derived from each minimal cover we study the computational complexity of such lifting in particular we show that the task of computing a lifted facet can be accomplished in o ns where s n is the cardinality of the minimal cover also for a lifted inequality with integer coefficients we show that the dual tasks of recognizing whether the inequality is valid for p or is a facet of p can be done within the same time bound

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

equilibrium states of weakly hyperbolic one dimensional maps for holder potentials there is a wealth of results in the literature on the thermodynamic formalism for potentials that are in some sense hyperbolic we show that for a sufficiently regular one dimensional map satisfying a weak hyperbolicity assumption every holder continuous potential is hyperbolic a sample consequence is the absence of phase transitions the pressure function is real analytic on the space of holder continuous functions another consequence is that every holder continuous potential has a unique equilibrium state and that this measure has exponential decay of correlations

an exact algorithm for constrained two dimensional two staged cutting problems theconstrained two dimensional cutting c tdc problem consists of determining acutting pattern of a set ofn small rectangular piece types on a rectangular stock plates with lengthl and widthw to maximize the sum of the profits of the pieces to be cut each piece typei i 1 n is characterized by a lengthl i a widthw i a profit or weight c i and an upper demand valueb i the upper demand value is the maximum number of pieces of typei that can be cut ons in this paper we study the two staged c tdc problem noted c 2tdc it is a classical variant of the c tdc where each piece is produced in the final cutting pattern by at most two cuts we solve the c 2tdc problem using an exact algorithm that is mainly based on a bottom up strategy we introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns we evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm

local connectivity and quasi conformal rigidity of non renormalizable polynomials we prove that topologically conjugate non renormalizable polynomials are quasi conformally conjugate from this we derive that each such polynomial can be approximated by a hyperbolic polynomial as a by product we prove that the julia set of a non renormalizable polynomial with only hyperbolic periodic points is locally connected and the branner hubbard conjecture the main tools are the enhanced nest construction developed in a previous joint paper with rigidity for real polynomials ann of math 2 165 2007 749 841 and a lemma of kahn and lyubich for which we give an elementary proof in the real case

weak expansion properties and large deviation principles for expanding thurston maps abstract in this paper we prove that an expanding thurston map f s 2 s 2 is asymptotically h expansive if and only if it has no periodic critical points and that no expanding thurston map is h expansive as a consequence for each expanding thurston map without periodic critical points and each real valued continuous potential on s 2 there exists at least one equilibrium state for such maps we also establish large deviation principles for iterated preimages and periodic points it follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding thurston map without periodic critical points and a potential that is holder continuous with respect to a visual metric on s 2

fenchel cutting planes for integer programs a technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program the theoretical properties of the cuts and their relationship to lagrangian relaxation are discussed the cut generation procedure is described and computational results are presented

measures with positive lyapunov exponent and conformal measures in rational dynamics ergodic properties of rational maps are studied generalising the work of f ledrappier a new construction allows for simpler proofs of stronger results very general conformal measures are considered equivalent conditions are given for an ergodic invariant probability measure with positive lyapunov exponent to be absolutely continuous with respect to a general conformal measure if they hold we can construct an induced expanding markov map with integrable return time which generates the invariant measure

an expanding core algorithm for the exact 0 1 knapsack problem abstract a new branch and bound algorithm for the exact solution of the 0 1 knapsack problem is presented the algorithm is based on solving an expanding core which intially only contains the break item but which is expanded each time the branch and bound algorithm reaches the border of the core computational experiments show that most data instances are optimally solved without sorting or preprocessing a great majority of the items detailed program sketches are provided and computational experiments are reported indicating that the algorithm presented not only is shorter but also faster and more stable than any other algorithm hitherto proposed

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

equivalence and topological invariance of conditions for non uniform hyperbolicity in the iteration of rational maps we show equivalence of several standard conditions for non uniform hyperbolicity of complex rational functions including the topological collet eckmann condition tce uniform hyperbolicity on periodic orbits exponential shrinking of components of pre images of small discs backward collet eckmann condition at one point positivity of the infimum of lyapunov exponents of finite invariant measures on the julia set the condition tce is stated in purely topological terms so we conclude that all these conditions are invariant under topological conjugacy for rational maps with one critical point in julia set all the conditions above are equivalent to the usual collet eckmann and backward collet eckmann conditions thus the latter ones are invariant by topological conjugacy in the unicritical setting we also prove that neither part of this stronger statement is valid in the multicritical case

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

adaptive cluster sampling designs with primary and secondary units summary in adaptive cluster sampling designs an initial probability sample is selected and whenever the observed value of the variable of interest satisfies a given condition units in the neighborhood of that observation are added to the sample in this paper the initial design is selected in terms of primary units while subsequent sampling is in terms of secondary units such initial designs include systematic sampling strip sampling and other forms of classical cluster sampling but because of the subsequent addition to the sample of secondary units in the neighborhood of any secondary unit that satisfies the condition of interest the final clusters of units obtained through the procedure may be quite different in shape from the initial primary units the methods described in this paper apply to such sampling situations as whale surveys in which the research vessel temporarily leaves the selected transect to close in on sighted whales surveys of rare bird species in which initial observations are made at systematically selected sites and additional observations are made in the vicinity of any site at which sufficiently high abundance is observed and aerial walrus surveys in which the aircraft searches to either side of the preselected transect line whenever a congregation of animals is encountered because conventional estimators of the population mean and total are biased with such a procedure estimators that are unbiased under the adaptive designs are presented in this paper variance formulae and unbiased estimators of variance are also given the designs are illustrated using a point pattern representing locations of individuals or objects in a spatially aggregated population for such a population the adaptive designs can be substantially more efficient than their conventional counterparts

spectral properties of a class of operators associated with conformal maps in two dimensions iff is a rational map of the riemann sphere define the transfer operator by open image in new window

a branch and cut algorithm for the two echelon capacitated vehicle routing problem with grouping constraints abstract in this paper we present a branch and cut algorithm for the two echelon capacitated vehicle routing problem with grouping constraints 2e cvrpgc which is a new problem deriving from the classical two echelon capacitated vehicle routing problem 2e cvrp by considering grouping constraints in the second echelon customers in the 2e cvrpgc are divided into several disjoint groups and the grouping constraints ensure that customers from the same group are served by vehicles from the same satellite we formulate the problem as a mixed 0 1 linear program and propose five families of valid inequalities to strengthen the model based on the model and the inequalities we implement a branch and cut algorithm to solve the problem the proposed branch and cut algorithm was evaluated on two classes of randomly generated instances the computational results show that the five families of valid inequalities can substantially improve the lower bounds yielded by the lp relaxation of the model and the branch and cut algorithm can solve more instances to optimality than cplex we also conduct additional experiments to analyze the impacts of the grouping constraints on the problem and illustrate the differences between the 2e cvrpgc and the 2e cvrp

performance prediction and preselection for optimization and heuristic solution procedures the operations research literature contains numerous studies on the design and application of optimization and heuristic solution procedures these studies identify a particular optimization problem suggest a general solution procedure and then customize that procedure to improve its efficiency and or accuracy in contrast this paper shows how to use existing solution procedures more effectively we develop a methodology for predicting the relative performance of alternative procedures using easily computed problem characteristics this methodology enables us for any given data set to preselect a solution procedure we apply this preselection methodology to the 0 1 knapsack problem for which two successful optimization procedures dynamic programming and branch and search are available extensive computational testing indicates that substantial savings in average computation time are achieved the benefits of our work include faster and cheaper identification of effective solution procedures as well as an improved understanding of the relationship between problem characteristics and the performance of various procedures our methodology can be applied to many optimization problems to develop easily implemented guidelines for selecting appropriate solution procedures

a minimal algorithm for the multiple choice knapsack problem abstract the multiple choice knapsack problem is defined as a 0 1 knapsack problem with the addition of disjoined multiple choice constraints as for other knapsack problems most of the computational effort in the solution of these problems is used for sorting and reduction but although o n algorithms which solve the linear multiple choice knapsack problem without sorting have been known for more than a decade such techniques have not been used in enumerative algorithms in this paper we present a simple o n partitioning algorithm for deriving the optimal linear solution and show how it may be incorporated in a dynamic programming algorithm such that a minimal number of classes are enumerated sorted and reduced computational experiments indicate that this approach leads to a very efficient algorithm which outperforms any known algorithm for the problem

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

a new algorithm for the 0 1 knapsack problem we present a new algorithm for the optimal solution of the 0 1 knapsack problem which is particularly effective for large size problems the algorithm is based on determination of an appropriate small subset of items and the solution of the corresponding core problem from this we derive a heuristic solution for the original problem which with high probability can be proved to be optimal the algorithm incorporates a new method of computation of upper bounds and efficient implementations of reduction procedures the corresponding fortran code is available we report computational experiments on small size and large size random problems comparing the proposed code with all those available in the literature

a cut and solve based algorithm for the single source capacitated facility location problem abstract in this paper we present a cut and solve cs based exact algorithm for the single source capacitated facility location problem sscflp at each level of cs s branching tree it has only two nodes corresponding to the sparse problem sp and the dense problem dp respectively the sp whose solution space is relatively small with the values of some variables fixed to zero is solved to optimality by using a commercial mip solver and its solution if it exists provides an upper bound to the sscflp meanwhile the resolution of the lp of dp provides a lower bound for the sscflp a cutting plane method which combines the lifted cover inequalities and fenchel cutting planes to separate the 0 1 knapsack polytopes is applied to strengthen the lower bound of sscflp and that of dp these lower bounds are further tightened with a partial integrality strategy numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the cs approach in searching an optimal solution in a reasonable time computational results on large sized instances are also presented

differentiability of thermodynamical quantities in non uniformly expanding dynamics abstract in this paper we study the ergodic theory of a robust non uniformly expanding maps where no markov assumption is required we prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal lyapunov exponents with respect to the dynamics we also prove a local large deviations principle and central limit theorem and show that the rate function mean and variance vary continuously with respect to observables potentials and dynamics finally we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is c 1 convergent to zero in addition precise formulas for the derivatives of thermodynamical quantities are given

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

robust efficiency measures for linear knapsack problem variants abstract numerous combinatorial optimization applications such as the mobile retailer product mix problem utilize the multidemand multidimensional knapsack problem variant in which there are multiple knapsack constraints requiring a weighted summation of the variables to be less than or equal to a nonnegative value and multiple demand constraints requiring a weighted summation of the variables to be greater than or equal to a nonnegative value the purpose of this paper is to demonstrate that core variables and efficiency measures concepts used in the most efficient solvers to date for binary knapsack problems and some of its variants can be extended to the multidemand multidimensional knapsack problem variant specifically new efficiency measure calculations are provided and their properties are mathematically proven and experimentally demonstrated the contribution of such measures to knapsack problem research is that these measures are applicable to all knapsack problem variants with a single linear objective function and linear constraints of any quantity the applicability of these new measures is demonstrated through the development of three heuristic procedures a fixed core heuristic a genetic algorithm and a kernel search heuristic the results from these tests are compared with the results from a commercial solver and an existing heuristic the findings from these tests demonstrate that the fixed core and kernel search heuristics developed for this paper are the most efficient solvers to date for hard multidemand multidimensional knapsack problems

approximating polyhedra with sparse inequalities in this paper we study how well one can approximate arbitrary polytopes using sparse inequalities our motivation comes from the use of sparse cutting planes in mixed integer programing mip solvers since they help in solving the linear programs encountered during branch bound more efficiently however how well can we approximate the integer hull by just using sparse cutting planes in order to understand this question better given a polyope p p e g the integer hull of a mip let p k pk be its best approximation using cuts with at most k non zero coefficients we consider text d p p k max x in p k left min y in p vert x y vert right d p pk maxx pkminy p x y as a measure of the quality of sparse cuts in our first result we present general upper bounds on text d p p k d p pk which depend on the number of vertices in the polytope our bounds imply that if p p has polynomially many vertices using half sparsity already approximates it very well second we present a lower bound on text d p p k d p pk for random polytopes that show that the upper bounds are quite tight third we show that for a class of hard packing ips sparse cutting planes do not approximate the integer hull well that is d p p k d p pk is large for such instances unless k is very close to n finally we show that using sparse cutting planes in extended formulations is at least as good as using them in the original polyhedron and give an example where the former is actually much better

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

hitting and escaping statistics mixing targets and holes abstract there is a natural connection between two types of recurrence law hitting times to shrinking targets and hitting times to a fixed target usually seen as escape through a hole we show that for systems which mix exponentially fast one can move through a natural parameter space from one to the other on the other hand if the mixing is subexponential there is a phase transition between the hitting times law and the escape law

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

solving multiple knapsack problems by cutting planes in this paper we consider the multiple knapsack problem which is defined as follows given a set n of items with weights f i i in n a set m of knapsacks with capacities f k k in m and a profit function c ik i in n k in m find an assignment of a subset of the set of items to the set of knapsacks that yields minimum cost we consider the multiple knapsack problem from a polyhedral point of view the inequalities that we describe here serve as the theoretical basis for a cutting plane algorithm we present some of the implementation details of this algorithm including a discussion and evaluation of different separation and primal heuristics our algorithm is applied to practical problem instances arising in the design of main frame computers in the layout of electronic circuits and in sugar cane alcohol production

a cooperative local search based algorithm for the multiple scenario max min knapsack problem the purpose of this article is to present a novel method to approximately solve the multiple scenario max min knapsack problem msm2kp this problem models many real world situations e g when for many scenarios noted pi in mathcal p 1 ldots p the aim is to identify the one offering a better alternative in term of maximizing the worst possible outcome herein is presented a cooperative approach based on two local search algorithms i a limited area local search applied in the elite neighborhood and which accepts the first solution with some deterioration threshold of the current solution ii a wide range local search is applied to perform a sequence of paths exchange to improve the current solution results have been analyzed by means state of the art methods and via problem instances obtained by a generator code taken from the literature the tests were executed in compeltely comparable scenarios to those of the literature the results are promising and the efficiency of the proposed approach is also shown

low temperature phase transitions in the quadratic family abstract we give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function this implies that several dimension spectra and large deviation rate functions associated to this map are not expected to be real analytic in contrast to the uniformly hyperbolic case the quadratic map we study has a non recurrent critical point so it is non uniformly hyperbolic in a strong sense

renormalisation induced phase transitions for unimodal maps the thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential t log df multiple and indeed infinitely many phase transitions at positive t can occur for some quadratic maps all unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum

robust branch cut and price for the capacitated minimum spanning tree problem over a large extended formulation this paper presents a robust branch cut and price algorithm for the capacitated minimum spanning tree problem cmst the variables are associated to q arbs a structure that arises from a relaxation of the capacitated prize collecting arborescence problem in order to make it solvable in pseudo polynomial time traditional inequalities over the arc formulation like capacity cuts are also used moreover a novel feature is introduced in such kind of algorithms powerful new cuts expressed over a very large set of variables are added without increasing the complexity of the pricing subproblem or the size of the lps that are actually solved computational results on benchmark instances from the or library show very significant improvements over previous algorithms several open instances could be solved to optimality

an approach to the asymmetric multi depot capacitated arc routing problem despite the fact that the capacitated arc routing problems carps received substantial attention in the literature most of the research concentrates on the symmetric and single depot version of the problem in this paper we fill this gap by proposing an approach to solving a more general version of the problem and analysing its properties we present an milp formulation that accommodates asymmetric multi depot case and consider valid inequalities that may be used to tighten its lp relaxation a symmetry breaking scheme for a single depot case is also proposed an extensive numerical study is carried to investigate the properties of the problem and the proposed solution approach

upper bounds and algorithms for hard 0 1 knapsack problems it is well known that many instances of the 0 1 knapsack problem can be effectively solved to optimality also for very large values of n the number of binary variables while other instances cannot be solved for n equal to only a few hundreds we propose upper bounds obtained from the mathematical model of the problem by adding valid inequalities on the cardinality of an optimal solution and relaxing it in a lagrangian fashion we then introduce a specialized iterative technique for determining the optimal lagrangian multipliers in polynomial time a branch and bound algorithm is finally developed computational experiments prove that several classes of hard instances are effectively solved even for large values of n

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

ray projection for optimizing polytopes with prohibitively many constraints in set covering column generation a recurrent task in mathematical programming consists of optimizing polytopes with prohibitively many constraints e g the primal polytope in cutting planes methods or the dual polytope in column generation cg we present a method to optimize such polytopes using the following intersection sub problem given ray mathbf r in mathbb z n r zn find the maximum t ge 0 t 0 such that t mathbf r t r is feasible we interpret mathbf 0 n rightarrow mathbf r 0n r as a direction of growth from the origin mathbf 0 n 0n t mathbf r t r is the intersection point between mathbf 0 n rightarrow mathbf r 0n r and the polytope boundary to ease the exposition the first part of the paper is devoted to the general context of a primal or dual polytope with rational data such that i the intersection sub problem can be effectively solved in practice for input rays mathbf r in mathbb z n r zn and ii mathbf 0 n 0n is feasible by iterating over such rays mathbf r r our method produces a sequence of feasible solutions t mathbf r t r that we prove to be finitely convergent from section 3 on we focus on dual polytopes in cg formulations typically if the cg separation sub problem can be solved by dynamic programming dp so can be the intersection sub problem the proposed integer ray method irm only uses integer rays and so the intersection sub problem can be solved using a dp scheme based on states indexed by integer ray values we show that under such conditions the intersection sub problem can be even easier than the cg sub problem especially when no other integer data is available to index states in dp i e if the cg sub problem input consists of fractional or large range values as such the irm can tackle scaled instances with large range weights of capacitated problems that seem prohibitively hard for classical cg this is confirmed by numerical experiments on various capacitated set covering problems capacitated arc routing cutting stock and other three versions of elastic cutting stock i e a problem class that includes variable size bin packing

strip adaptive cluster sampling probability proportional to size selection of primary units summary strip adaptive cluster sampling is extended to situations where the primary units have unequal numbers of secondary units primary units are selected with replacement and with probabilities proportional to the number of secondary units the horvitz thompson estimator of the parametric total is used as the basis for estimation two cases are considered basic strip adaptive cluster sampling and strip adaptive cluster sampling under stratification of primary units a numerical example is presented in which the goal is to estimate the number of objects on a site the sampling strategy is useful for ecological studies where a rectangular grid of secondary units is not adequate to cover the study site

surrogate upper bound sets for bi objective bi dimensional binary knapsack problems the paper deals with the definition and the computation of surrogate upper bound sets for the bi objective bi dimensional binary knapsack problem it introduces the optimal convex surrogate upper bound set which is the tightest possible definition based on the convex relaxation of the surrogate relaxation two exact algorithms are proposed an enumerative algorithm and its improved version this second algorithm results from an accurate analysis of the surrogate multipliers and the dominance relations between bound sets based on the improved exact algorithm an approximated version is derived the proposed algorithms are benchmarked using a dataset composed of three groups of numerical instances the performances are assessed thanks to a comparative analysis where exact algorithms are compared between them the approximated algorithm is confronted to an algorithm introduced in a recent research work

on the lyapunov spectrum for rational maps we study the dimension spectrum for lyapunov exponents for rational maps on the riemann sphere

an efficient algorithm for the collapsing knapsack problem the collapsing knapsack problem ckp is a type of nonlinear knapsack problem in which the knapsack size is a non increasing function of the number of items included this paper proposes an exact algorithm for ckp by partitioning ckp to some subproblems then solving them with the improved expanding core technique the proposed algorithm solves the subproblems in the special processing order resulting in the reduction of computing time experimental results show that the proposed algorithm is an efficient approach for various random instances of size up to 1000

robust optimization approach for a chance constrained binary knapsack problem we consider a certain class of chance constrained binary knapsack problem where each item has a normally distributed random weight that is independent of the other items for this problem we propose an efficient pseudo polynomial time algorithm based on the robust optimization approach for finding a solution with a theoretical bound on the probability of satisfying the knapsack constraint our algorithm is tested on a wide range of random instances and the results demonstrate that it provides qualified solutions quickly in contrast a state of the art mip solver is only applicable for instances of the problem with a restricted number of items

approximate and exact algorithms for the fixed charge knapsack problem abstract the subject of this paper is the formulation and solution of a variation of the classical binary knapsack problem the variation that is addressed is termed the fixed charge knapsack problem in which sub sets of variables activities are associated with fixed costs these costs may represent certain set ups and or preparations required for the associated sub set of activities to be scheduled several potential real world applications as well as problem extensions generalizations are discussed the efficient solution of the problem is facilitated by a standard branch and bound algorithm based on 1 a non iterative polynomial algorithm to solve the lp relaxation 2 various heuristic procedures to obtain good candidate solutions by adjusting the lp solution and 3 powerful rules to peg the variables computational experience shows that the suggested branch and bound algorithm shows excellent potential in the solution of a wide variety of large fixed charge knapsack problems

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

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

new trends in exact algorithms for the 0 1 knapsack problem abstract while the 1980s were focused on the solution of large sized easy knapsack problems kps this decade has brought several new algorithms which are able to solve hard large sized instances we will give an overview of the recent techniques for solving hard kps with special emphasis on the addition of cardinality constraints dynamic programming and rudimentary divisibility computational results comparing all recent algorithms are presented

an optimization algorithm for the ordered open end bin packing problem the ordered open end bin packing problem is a variant of the bin packing problem in which the items to be packed are sorted in a given order and the capacity of each bin can be exceeded by the last item packed into the bin we present a branch and price algorithm for its exact optimization the pricing subproblem is a special variant of the binary knapsack problem in which the items are ordered and the last one does not consume capacity we present a specialized optimization algorithm for this subproblem the speed of the column generation algorithm is improved by subgradient optimization steps allowing for multiple pricing and variable fixing computational results are presented on instances of different sizes and items with different correlations between their size and their position in the given order

natural equilibrium states for multimodal maps this paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps this class contains but it is larger than collet eckmann for a map in this class we prove existence and uniqueness of equilibrium states for the geometric potentials t log df for the largest possible interval of parameters t we also study the regularity and convexity properties of the pressure function completely characterising the first order phase transitions results concerning the existence of absolutely continuous invariant measures with respect to the lebesgue measure are also obtained

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

cutting planes for mixed integer knapsack polyhedra an algorithm for generating cutting planes for mixed integer knapsack polyhedra is described the algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper computational results are presented 1998 the mathematical programming society inc published by elsevier science b v

stratified adaptive cluster sampling summary stratified adaptive cluster sampling refers to designs in which following an initial stratified sample additional units are added to the sample from the neighbourhood of any selected unit with an observed value that satisfies a condition of interest if any of the added units in turn satisfies the condition still more units are added to the sample estimation of the population mean or total with the stratified adaptive cluster designs is complicated by the possibility that a selection in one stratum may result in the addition of units from other strata to the sample so that observations in separate strata are not independent since conventional estimators such as the stratified sample mean are biased with the adaptive designs of this paper several types of estimators are developed which are unbiased for the population mean or total with stratified adaptive cluster sampling

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