heuristics and reduction methods for multiple constraints 0 1 linear programming problems abstract this work extends the efficient results relative to the 0 1 knapsack problem to the multiple inequality constraints 0 1 linear programming problems the two crucial phases for the solving of this type of problems are presented i two linear expected time complexity greedy algorithms are proposed for the determination of a lower bound on the optimal value by using a cascade of surrogate relaxations of the original problem whose sizes are decreasing step by step a comparative study with the best known heuristic methods is reported it concerned the accuracy of the approximate solutions and the practical computational times ii this greedy algorithm is inserted in an efficient reduction framework variables and constraints are eliminated by the conjunction of tests applied to lagrangean and surrogate relaxations of the original problem a lot of computational results are summarized by considering test problems of the literature

lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

an efficient tabu search approach for the 0 1 multidimensional knapsack problem abstract in this paper we describe a new approach to tabu search ts based on strategic oscillation and surrogate constraint information that provides a balance between intensification and diversification strategies new rules needed to control the oscillation process are given for the 0 1 multidimensional knapsack 0 1 mkp based on a portfolio of test problems from the literature our method obtains solutions whose quality is at least as good as the best solutions obtained by previous methods especially with large scale instances these encouraging results confirm the efficiency of the tunneling concept coupled with surrogate information when resource constraints are present

finding a maximum clique in an arbitrary graph we describe a new type of branch and bound procedure for finding a maximum clique in an arbitrary graph g v e the two main ingredients both of o v e time complexity are i an algorithm for finding a maximal triangulated induced subgraph of g and ii an algorithm for finding a maximal k chromatic induced subgraph of g we discuss computational experience on randomly generated graphs with up to 400 vertices and 30 000 edges

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

s function series revisited a systematic and readily programmable computational method is given to extract the s function content of any arbitrary symmetric polynomial function applications to the derivation of new series and the outer product or skew division of an s function by a series are shown

spectrum generating functions for strings and superstrings the spectrum generating functions for a variety of strings and superstrings are derived these are expressed in terms of characters of irreducible representations of the appropriate transverse spacetime symmetry group and in the case of heterotic strings the relevant gauge group e8 e8 or spin 32 z2 the expansions involve principal specialisations of characters of both gl n and gl m n in the limit as m and n tend to infinity by using schur function methods modification rules and various theta function identities a number of spectrum generating functions are derived and tabulated the equivalence of the gso projected neveu schwarz ramond superstring and the green schwarz superstring is proved and the connection between the lattice formulation and the gso projected neveu schwarz ramond formulation of heterotic strings is spelled out

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

solving zero one mixed integer programming problems using tabu search we describe a tabu search ts approach for solving general zero one mixed integer programming mip problems that exploits the extreme point property of zero one solutions specialized choice rules and aspiration criteria are identified for the problems expressed as functions of integer infeasibility measures and objective function values the first level ts mechanisms are then extended with advanced level strategies and learning we also look at probabilistic measures in this framework and examine how the learning tool target analysis ta can be applied to identify better control structures and decision rules computational results are reported on a portfolio of multiconstraint knapsack problems our approach is designed to solve thoroughly general 0 1 mip problems and thus contains no problem domain specific knowledge yet it obtains solutions for the multiconstraint knapsack problem whose quality rivals and in some cases surpasses the best solutions obtained by special purpose methods that have been created to exploit the special structure of these problems

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

an approximate dynamic programming approach to multidimensional knapsack problems we present an approximate dynamic programming adp approach for the multidimensional knapsack problem mkp we approximate the value function a using parametric and nonparametric methods and busing a base heuristic we propose a new heuristic which adaptively rounds the solution of the linear programming relaxation our computational study suggests athe new heuristic produces high quality solutions fast and robustly bstate of the art commercial packages like cplex require significantly larger computational time to achieve the same quality of solutions c the adp approach using the new heuristic competes successfully with alternative heuristic methods such as genetic algorithms dthe adp approach based on parametric and nonparametric approximations while producing reasonable solutions is not competitive overall this research illustrates that the base heuristic approach is a promising computational approach for mkps worthy of further investigation

new optimization heuristics in a former paper we introduced a very effective new general purpose optimization principle we compared this method which we called threshold accepting ta with the well known simulated annealing sa method for discrete optimization the empirical results demonstrated the superiority of the ta algorithm in further experiments with the ta principle we discovered two new powerful optimization heuristics the great deluge algorithm gda and the record to record travel rrt these algorithms resemble in their structure the formerly studied ta and the sa method the differences lie in their acceptance rules for worse intermediate solutions both the gda and the rrt are essentially one parameter algorithms i e for the achievement of best possible performance a good choice is necessary only for a single parameter this is in contrast for instance to the classical as algorithm where it is necessary to choose carefully a certain sequence of parameters the so called annealing schedule the quality of the computational results obtained so far by rrt and gda shows that the new algorithms behave equally well as ta and thus a fortiori better than sa

efficient algorithms for solving multiconstraint zero one knapsack problems to optimality the multiconstraint 0 1 knapsack problem is encountered when one has to decide how to use a knapsack with multiple resource constraints even though the single constraint version of this problem has received a lot of attention the multiconstraint knapsack problem has been seldom addressed

new s function series and non compact lie groups a number of new infinite series of s functions are described in terms of their generating functions and s function content applications to the character theory of non compact lie groups are noted

a multi kp modeling for the maximum clique problem abstract in this paper the problem of finding the maximum clique of an arbitrary graph g x a is studied this problem is formulated as a 0 1 multidimensional knapsack problem and a greedy algorithm for solving it is presented finally a branch and bound method based on the 0 1 multi kp model and some computational results are given

slinkies and the s function content of certain generating functions a variety of methods are described for expanding particular generating functions in terms of their s function content plethysms are used to establish conjugacy relations between s function series and the task of evaluating certain products of classical s function series introduced by littlewood 1950 is then completed slinky diagrams are introduced to represent non standard s functions and their modification to give standard s functions the role of slinkies in determining the s function content of other generating functions is then explained and exemplified

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

convex normality of rational polytopes with long edges abstract we introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths guaranteeing the property as an application we show that if every edge of a lattice d polytope p has lattice length 4 d d 1 then p is normal this answers in the positive a question raised in 2007 if p is a lattice simplex whose edges have lattice lengths d d 1 then p is even covered by lattice parallelepipeds for the approach developed here it is necessary to involve rational polytopes even for the application to lattice polytopes

an approximate algorithm for multidimensional zero one knapsack problems a new approximate algorithm for multidimensional zero one knapsack problems with all positive coefficients is presented the procedure is controlled by three parameters which affect the tradeoff between solution quality and computation time and whose values are set by the users for 48 test problems with 5 to 20 constraints and 6 to 500 variables the solution found was on the average within 0 34 of the optimum and the computation time was the shortest compared with three other well known heuristics

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

normality and covering properties of affine semigroups s x gp s mx s for some m 0 one calls s normal if s s for simplicity we will often assume that gp s z this is harmless because we can replace z by gp s if necessary the rank of s is the rank of gp s we will only be interested in the case in which s s 0 such affine semigroups will be called positive the positivity of s is equivalent to the pointedness of the cone c s r s generated by s in r one has ranks dimc s all the cones appearing in this paper have their apex at the origin the normality of s can now be characterized geometrically one obviously has s c s z and so s is normal if and only if s c s z conversely every convex pointed finitely generated rational cone c r yields a normal affine semigroup s c c z this semigroup is finitely generated by gordan s lemma if we simply speak of a cone c in the following then it is always assumed that c is convex pointed finitely generated and rational since dimc s ranks we often call the dimension of a cone its rank each positive affine semigroup s can be embedded into z m ranks one uses m linearly independent integral linear forms representing support hyperplanes of c s therefore every element can be written as a sum of irreducible elements and the set of irreducible elements of s is finite since s is finitely generated for a cone c the set of irreducible elements of s c is often called the hilbert basis hilb c of c in the combinatorial literature a convention we will follow more generally we also call the set of irreducible elements of a positive affine semigroup s its hilbert basis and denote it by hilb s this paper is devoted to a discussion of sufficient and potentially necessary conditions for the normality of s in terms of combinatorial properties of hilb s as far as the necessity is concerned these conditions can of course be formulated in terms of hilbert bases of rational cones the first property under consideration is the existence of a unimodular hilbert cover

a heuristic ceiling point algorithm for general integer linear programming this paper first examines the role of ceiling points in solving a pure general integer linear programming problem p several kinds of ceiling points are defined and analyzed and one kind called feasible 1 ceiling points proves to be of special interest we demonstrate that all optimal solutions for a problem p whose feasible region is nonempty and bounded are feasible 1 ceiling points consequently such a problem may be solved by enumerating just its feasible 1 ceiling points the paper then describes an algorithm called the heuristic ceiling point algorithm hcpa which approximately solves p by searching only for feasible 1 ceiling points relatively near the optimal solution for the lp relaxation such solutions are apt to have a high possibly even optimal objective function value the results of applying the hcpa to 48 test problems taken from the literature indicate that this approach usually yields a very good solution with a moderate amount of computational effort

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given