optimal sample allocation for normal discrimination and logistic regression under stratified sampling abstract for two multivariate normal populations with a common covariance matrix and stratified sampling we consider two methods of estimation fisher s linear discriminant function and logistic regression intuition suggests that taking half of the observations from each population is a reasonable design choice based on minimizing the expected error regret asymptotic optimal sample allocations are found the results indicate that the differences in the expected error regret for optimal versus balanced allocation are generally quite small it is recommended that equal sample sizes for the two populations be used for these problems

inference based sensitivity analysis for mixed integer linear programming a new method of sensitivity analysis for mixed integer linear programming milp is derived from the idea of inference duality the inference dual of an optimization problem asks how the optimal value can be deduced from the constraints in milp a deduction based on the resolution method oftheorem proving can be obtained from the branch and cut tree that solves the primal problem one can then investigate which perturbations ofthe problem leave this proof intact on this basis it is shown that in a minimization problem any perturbation that satisfies a certain system of linear inequalities will reduce the optimal value no more than a prespecified amount one can also give an upper bound on the increase in the optimal value that results from a given perturbation the method is illustrated on two realistic problems

a posteriori error estimation based on potential and flux reconstruction for the heat equation we derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous galerkin various finite volume and mixed finite element methods in space and the backward euler scheme in time extensions to conforming and nonconforming finite element spatial discretizations are also outlined our estimates are based on a h 1 conforming reconstruction of the potential continuous and piecewise affine in time and a locally conservative mathbf h mathrm div conforming reconstruction of the flux piecewise constant in time they yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative local in time lower bounds are also derived for nonconforming methods on time varying meshes the lower bounds require a mild parabolic type constraint on the meshsize

allocation and scheduling of conditional task graphs we propose an original complete and efficient approach to the allocation and scheduling of conditional task graphs ctgs in ctgs nodes represent activities some of them are branches and are labeled with a condition arcs rooted in branch nodes are labeled with condition outcomes and a corresponding probability a task is executed at run time if the condition outcomes that label the arcs in the path to the task hold at schedule execution time this can be captured off line by adopting a stochastic model tasks need for their execution either unary or cumulative resources and some tasks can be executed on alternative resources the solution to the problem is a single assignment of a resource and of a start time to each task so that the allocation and schedule is feasible in each scenario and the expected value of a given objective function is optimized for this problem we need to extend traditional constraint based scheduling techniques in two directions i compute the probability of sets of scenarios in polynomial time in order to get the expected value of the objective function ii define conditional constraints that ensure feasibility in all scenarios we show the application of this framework on problems with objective functions depending either on the allocation of resources to tasks or on the scheduling part also we present the conditional extension to the timetable global constraint experimental results show the effectiveness of the approach on a set of benchmarks taken from the field of embedded system design comparing our solver with a scenario based solver proposed in the literature we show the advantages of our approach both in terms of execution time and solution quality

a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors conforming approximations this paper derives a posteriori error estimates for conforming numerical approximations of the laplace eigenvalue problem with a homogeneous dirichlet boundary condition in particular upper and lower bounds for an arbitrary simple eigenvalue are given these bounds are guaranteed fully computable and converge with optimal speed to the given exact eigenvalue they are valid without restrictions on the computational mesh or on the approximate eigenvector we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones which can be checked in practice guaranteed fully computable optimally convergent and polynomial degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well under the same hypotheses remarkably there appears no unknown solution regularity or polynomial degree dependent constant in our theory and no convexity regularity assumption on the computational domain exact eigenvector s is needed two improvements of the multiplicative constant appearing in our estimates are presented first it is reduced by a fixed factor under an explicit a posteriori calculable condition on the mesh and on the approximate eigenvector eigenvalue pair second when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants the multiplicative factor can be brought to the optimal value of one inexact algebraic solvers are taken into account the estimates are valid on each iteration and can serve for the design of adaptive stopping criteria the application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided along with a numerical illustration on a set of test problems

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

forecasting technological substitutions with concurrent short time series abstract it has been shown in the literature that the data of technological substitutions exhibit a strong correlation across different time periods that is a strong serial correlation significant improvement in predicting such substitutions has been achieved by incorporating serial correlation and power transformation parameters into four growth curve models the modified models or data based transformed models however break down when the number of time points is small this article proposes a generalized growth curve model for forecasting technological substitutions with concurrent short time series the model combines the concepts of power transformations and repeated measurements with a common serial covariance structure concurrent time series for several cases provide the repeated measurement requirement of the model improvement in forecasting accuracy by using this model is demonstrated with a set of telephone switching data

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

integer programming duality price functions and sensitivity analysis recently a duality theory for integer programming has been developed here we examine some of the economic implications of this theory in particular the necessity of using price functions in place of prices and the possibility of carrying out sensitivity analysis of optimal solutions in addition we consider the form of price functions that are generated by known algorithms for integer programming

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

a branch and check algorithm for minimizing the weighted number of late jobs on a single machine with release dates in this paper we consider the scheduling problem of minimizing the weighted number of late jobs on a single machine 1 rj wjuj a branch and check algorithm is proposed where a relaxed integer programming formulation is solved by branch and bound and infeasible solutions are cut off using infeasibility cuts we suggest two ways to generate cuts first tightened no good cuts are derived using a modification of the algorithm by carlier 1982 ejor v 11 42 47 which was developed for the problem of minimizing maximum lateness on a single machine secondly we show how to create cuts by using constraint propagation the proposed algorithm is implemented in the mosel modelling and optimization language computational experiments on instances with up to 140 jobs are reported a comparison is presented with the exact approach of peridy at al 2003 ejor v 148 591 603

eigenvalue approximation by mixed and hybrid methods note univ maryland dept math college pk md 20742 ecole polytech ctr math appl f 91128 palaiseau france univ paris 6 f 75230 paris 05 france mercier b cea limeil f 94190 villeneuve france isi document delivery no lq584times cited 29cited reference count 44 reference asn article 1981 005doi 10 2307 2007651 record created on 2006 08 24 modified on 2017 05 12

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

sensitivity analysis for branch and bound integer programming after an integer program is solved we may be interested in the effect of small changes in the right hand side or objective function coefficients without modifying the underlying branch and bound or cutting plane algorithm we show what additional data to collect during the solution of the original integer program to give useful bounds on the solution of the modified problem

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

fully reliable localized error control in the fem if the first task in numerical analysis is the calculation of an approximate solution the second is to provide a guaranteed error bound and is often of equal importance the standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property for coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds the aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well the main argument is a localization via a partition of unity which leads to problems on small domains two fully reliable estimates are established the sharper one solves an analytical interface problem with residuals following babuska and rheinboldt siam j numer anal 15 1978 pp 736 754 the second estimate is a modification of the standard residual based a posteriori estimate with explicit constants from local analytical eigenvalue problems for some class of triangulations we show that the efficiency constant is smaller than 2 5 according to our numerical experience the overestimation of our computable estimates proved to be reasonably small with an overestimation by a factor between 2 5 and 4 only

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

an elementary survey of general duality theory in mathematical programming we survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming the idea of replacing dual variables prices by price functions suggested by everett and developed by gould is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming the dual problem adopted has a traditional economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality in addition we attempt to make precise the relationship between primal dual and saddlepoint results in both the traditional lagrangean and the more general duality theories and to see the implications of passing from prices to price functions finally and perhaps surprisingly it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value i e by satisfying generalised optimality conditions

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

estimating and localizing the algebraic and total numerical errors using flux reconstructions this paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization and an arbitrary iterative algebraic solver the derived bounds are based on the flux reconstruction techniques do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain we also discuss bounds on the discretization error their application for constructing mathematically justified stopping criteria for iterative algebraic solvers global and local efficiency of the total error upper bound and the relationship to the previously published estimates on the algebraic error theoretical results are illustrated on numerical experiments for higher order finite element approximations and the preconditioned conjugate gradient method they in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over thecomputational domain and illustrate the associate cost

joint optimization of product family configuration and scaling design by stackelberg game abstract product family design is generally characterized by two types of approaches module based and scale based while the former aims to enable product variety based on module configuration the latter is to variegate product design by scaling up or down certain design parameters the prevailing practice is to treat module configuration and scaling design as separate decisions or aggregate two design problems as a single level all in one optimization problem in practice optimization of scaling variables is always enacted within a specific modular platform and meanwhile an optimal module configuration depends on how design parameters are to be scaled the key challenge is how to deal with explicitly the coupling of these two design optimization problems this paper formulates a stackelberg game theoretic model for joint optimization of product family configuration and scaling design in which a bilevel decision structure reveals coupled decision making between module configuration and parameter scaling a bilevel mixed 0 1 non linear programming model is developed and solved comprising an upper level optimization problem and a lower level optimization problem the upper level seeks for an optimal configuration of modules and module attributes by maximizing the shared surplus of an entire product family the lower level entails parametric optimization of attribute values for optimal technical performance of each individual module a case study of electric motors demonstrates that the bilevel joint optimization model excels in leveraging optimal scaling in conjunction with optimal module configuration which is advantageous over the existing paradigm of product family scaling design that cannot change the product family configuration

discontinuous galerkin approximation of the laplace eigenproblem in this paper we analyse the problem of computing eigenvalues and eigenfunctions of the laplace operator by means of discontinuous galerkin dg methods it results that several dg methods actually provide a spectrally correct approximation of the laplace operator we present here the convergence theory which applies to a wide class of dg methods as well as numerical tests demonstrating the theoretical results

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

minimal inequalities and subadditive duality in this note we use a duality theorem for mixed integer programs first explicitly stated by johnson 1973 for the one row group problem to characterize minimal inequalities this characterization extends earlier results which assumed either a rational or a bounded constraint set blair 1978 jeroslow 1979 johnson 1976 by relaxing these assumptions either entirely or almost so it also extends results given first by gomory and johnson 1969 for the group problem

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

a posteriori error analysis for locally conservative mixed methods in this work we present a theoretical analysis for a residual type error estimator for locally conservative mixed methods this estimator was first introduced by braess and verfurth for the raviart thomas mixed finite element method working in mesh dependent norms we improve and extend their results to cover any locally conservative mixed method under minimal assumptions in particular avoiding the saturation assumption made by braess and verfurth our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries the main results are applied to the p1 nonconforming finite element method and the interior penalty discontinuous galerkin method as well as the mixed finite element method

explicit error bounds in a conforming finite element method the goal of this paper is to define a procedure for bounding the error in a conforming finite element method the new point is that this upper bound is fully explicit and can be computed locally numerical tests prove the efficiency of the method it is presented here for the case of the poisson equation and a first order finite element approximation

a benders approach for the constrained minimum break problem this paper presents a hybrid ip cp algorithm for designing a double round robin schedule with a minimal number of breaks both mirrored and non mirrored schedules with and without place constraints are considered the algorithm uses benders cuts to obtain feasible home away pattern sets in few iterations and this approach leads to significant reductions in computation time for hard instances furthermore the algorithm is capable of solving a number of previously unsolved benchmark problems for the traveling tournament problem with constant distances

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

prediction and estimation of growth curves with special covariance structures abstract the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitutions it is defined as yp n xp m m rar n p n where is unknown and x and a are known design matrices of ranks m p and r n respectively further the columns of are independent p variate normal with mean vector 0 and common covariance matrix in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial and r is the number of groups this article focuses mainly on predicting future observations including partially observed vectors and future values on a given number of cases using their past observations these prediction problems as well as the estimation of parameters are considered for two parsimonious covariance structures that are very important in practice the results are illustrated with three real data sets som

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

guaranteed a posteriori bounds for eigenvalues and eigenvectors multiplicities and clusters this paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second order self adjoint elliptic linear operators with compact resolvent given a cluster of eigenvalues we estimate the error in the sum of the eigenvalues as well as the error in the eigenvectors represented through the density matrix i e the orthogonal projector on the associated eigenspace this allows us to deal with degenerate multiple eigenvalues within the framework all the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues we show how this assumption can be numerically checked our bounds are guaranteed and converge with the same speed as the exact errors they can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available which is presented in two particular cases the laplace eigenvalue problem discretized with conforming finite elements and a schrodinger operator with periodic boundary conditions of the form v discretized with planewaves for these two cases numerical illustrations are provided on a set of test problems

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

planning and scheduling by logic based benders decomposition we combine mixed integer linear programming milp and constraint programming cp to solve an important class of planning and scheduling problems tasks are allocated to facilities using milp and scheduled using cp and the two are linked via logic based benders decomposition tasks assigned to a facility may run in parallel subject to resource constraints cumulative scheduling we solve problems in which the objective is to minimize cost makespan or total tardiness we obtain significant computational speedups of several orders of magnitude for the first two objectives relative to the state of the art in both milp and cp we also obtain better solutions and bounds for problems than cannot be solved to optimality

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

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

needed an empirical science of algorithms deductive algorithmic science has reached a high level of sophistication but its worst case and average case results seldom tell us how well an algorithm is actually going to work in practice i argue that an empirical science of algorithms is a viable alternative i respond to misgivings about an empirical approach including the prevalent notion that only a deductive treatment can be theoretical or sophisticated np completeness theory for instance is interesting partly because it has significant if unacknowledged empirical content an empirical approach requires not only rigorous experimental design and analysis but also the invention of empirically based explanatory theories i give some examples of recent work that partially achieves this aim

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

hp adaptation driven by polynomial degree robust a posteriori error estimates for elliptic problems we devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions the adaptation is based on equilibrated flux estimates these estimates are presented here for inhomogeneous dirichlet and neumann boundary conditions for spatially varying polynomial degree and for mixed rectangular triangular grids possibly containing hanging nodes they deliver a global error upper bound with constant one and up to data oscillation error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound we numerically asses the estimates and several hp adaptive strategies using the interior penalty discontinuous galerkin method asymptotic exactness is observed for all the symmetric nonsymmetric odd degrees and incomplete variants on non nested unstructured triangular grids for a smooth solution and uniform refinement exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

generalized benders decomposition for one class of minlps with vector conic constraint in this paper we mainly study one class of mixed integer nonlinear programming problems minlps with vector conic constraint in banach spaces duality theory of convex vector optimization problems applied to this class of minlps is deeply investigated with the help of duality we use the generalized benders decomposition method to establish an algorithm for solving this minlp several convergence theorems on the algorithm are also presented the convergence theorems generalize and extend the existing results on minlps in finite dimensional spaces

explicit error bounds for a nonconforming finite element method let u be the solution of the following model left begin array l mbox find u in h 1 0 omega such that 3pt delta u f quad mbox in omega end array right where f is a given function in l2 omega and omega a bounded open set in mathbb r 2 with a polygonal boundary let uh be a finite element approximation of u the goal of this paper is to suggest an explicit bound of the error u uh furthermore this bound is obtained without complex computations the basic idea is to define locally an admissible vector field for the dual model