the effects of rounding errors in the nodes on barycentric interpolation we analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation these errors are particularly relevant for the first barycentric formula with the chebyshev points of the second kind here we propose a method for reducing them

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

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

finite bruck loops bruck loops are bol loops satisfying the automorphic inverse property we prove a structure theorem for finite bruck loops x showing that x is essentially the direct product of a bruck loop of odd order with a 2 element bruck loop the former class of loops is well understood we identify the minimal obstructions to the conjecture that all finite 2 element bruck loops are 2 loops leaving open the question of whether such obstructions actually exist

on the lebesgue constant of barycentric rational interpolation at equidistant nodes recent results reveal that the family of barycentric rational interpolants introduced by floater and hormann is very well suited for the approximation of functions as well as their derivatives integrals and primitives especially in the case of equidistant interpolation nodes these infinitely smooth interpolants offer a much better choice than their polynomial analogue a natural and important question concerns the condition of this rational approximation method in this paper we extend a recent study of the lebesgue function and constant associated with berrut s rational interpolant at equidistant nodes to the family of floater hormann interpolants which includes the former as a special case

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

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

the stability of barycentric interpolation at the chebyshev points of the second kind we present a new analysis of the stability of the first and second barycentric formulae for interpolation at the chebyshev points of the second kind our theory shows that the second formula is more stable than previously thought and our experiments confirm its stability in practice we also extend our current understanding regarding the accuracy problems of the first barycentric formula

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

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

on bruck loops of 2 power exponent the goal of this paper is two fold first we provide the information needed to study bol ar or bruck loops by applying group theoretic methods this information is used in this paper as well as in baumeister and stein 2010 bs3 and in stein 2009 s moreover we determine the groups associated to bruck loops of 2 power exponent under the assumption that every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power in a separate paper it is proven that indeed every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power stein 2009 s the results obtained here are used in baumeister and stein 2010 bs3 where we determine the structure of the groups associated to the finite bruck loops

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

a 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

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

the stability of extended floater hormann interpolants we present a new analysis of the stability of extended floater hormann interpolants in which both noisy data and rounding errors are considered contrary to what is claimed in the current literature we show that the lebesgue constant of these interpolants can grow exponentially with the parameters that define them and we emphasize the importance of using the proper interpretation of the lebesgue constant in order to estimate correctly the effects of noise and rounding errors we also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants our experiments show that extended interpolants mentioned in the literature satisfy this condition and therefore the formula used to implement them is not backward stable finally we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation

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

on the lebesgue constant of berrut s rational interpolant at equidistant nodes it is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting in this paper we confirm this observation by proving that the lebesgue constant of berrut s rational interpolant grows only logarithmically in the number of interpolation nodes moreover the numerical results suggest that the lebesgue constant behaves similarly for interpolation at chebyshev as well as logarithmically distributed nodes

an extension of the floater hormann family of barycentric rational interpolants the barycentric rational interpolants introduced by floater and hormann in 2007 are blends of polynomial interpolants of fixed degree d in some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants which e g are ill conditioned and lead to runge s phenomenon if the interpolation nodes are equispaced for such nodes however the condition of floater hormann interpolation deteriorates exponentially with increasing d in this paper an extension of the floater hormann family with improved condition at equispaced nodes is presented and investigated the efficiency of its applications such as the approximation of derivatives integrals and primitives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants math subject classification 65d05 65l12 65d32 41a05 41a20 41a25

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

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

the finite bruck loops abstract we continue the work of aschbacher kinyon and phillips 2006 akp06 as well as of glauberman 1964 1968 g64 g68 by describing the structure of the finite bruck loops we show that a finite bruck loop x is the direct product of a bruck loop of odd order with either a soluble bruck loop of 2 power order or a product of loops related to the groups psl 2 q q 9 or q 5 a fermat prime the latter possibility does occur as is shown in nagy 2008 n08 and baumeister and stein 2010 bs10 as corollaries we obtain versions of sylow s lagrange s and hall s theorems for loops

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

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

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

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