non stability of paneitz branson type equations in arbitrary dimensions abstract let m g be a compact riemannian manifold of dimension n 5 we consider a paneitz branson type equation with general coefficients e g 2 u div g a g d u h u u 2 2 e u on m where a g is a smooth symmetric 2 0 tensor h c m 2 2 n n 4 and e is a small positive parameter assuming that there exists a positive nondegenerate solution of e when e 0 and under suitable conditions we construct solutions u e of type u 0 b b l e to e which blow up at one point of the manifold when e tends to 0 for all dimensions n 5

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems abstract we propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems these formulations are obtained by lifting using the reformulation linearization technique a novel polynomial size ordered path based formulation we show that the strongest proposed formulation is equivalent to the strongest multi commodity flow formulation presented so far in addition we propose polynomial size valid inequalities that aim at further tightening the proposed formulations also the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs we show that the new derived model is not comparable with a state of the art polynomial length formulation we present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities

requiem for the miller tucker zemlin subtour elimination constraints abstract the miller tucker zemlin mtz subtour elimination constraints secs and the improved version by desrochers and laporte dl have been and are still in regular use to model a variety of routing problems this paper presents a systematic way of deriving inequalities that are more complicated than the mtz and dl inequalities and that in a certain way generalize the underlying idea of the original inequalities we present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions this approach allows us to generate generalizations of the mtz and dl inequalities which are good in the sense that they define facets of these small polyhedra it is well known that dl inequalities imply a subset of dantzig fulkerson johnson dfj secs for two node subsets through the approach presented we describe a generalization of these inequalities which imply dfj secs for three node subsets and show that generalizations for larger subsets are unlikely to exist our study presents a similar analysis with generalizations of mtz inequalities and their relation with the lifted circuit inequalities for three node subsets

a bound for the symmetric travelling salesman problem through matroid formulation abstract a new bound for the symmetric traveling salesman problem is proposed using a matroid theoretic formulation of the problem the resulting bound is shown to present a link between the well studied 1 tree and 2 matching bounds its potential is investigated

an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts this paper presents a new exact algorithm for the capacitated vehicle routing problem cvrp based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities the exact algorithm uses a bounding procedure that finds a near optimal dual solution of the lp relaxation of the resulting mathematical formulation by combining three dual ascent heuristics the first dual heuristic is based on the q route relaxation of the set partitioning formulation of the cvrp the second one combines lagrangean relaxation pricing and cut generation the third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique the final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved the resulting problem is solved by an integer programming solver computational results over the main instances from the literature show the effectiveness of the proposed algorithm

the asymmetric travelling salesman problem and a reformulation of the miller tucker zemlin constraints abstract in this paper we compare the linear programming lp relaxations of several old and new formulations for the asymmetric travelling salesman problem atsp the main result of this paper is the derivation of a compact formulation whose lp relaxation is characterized by a set of circuit inequalities given by grotschel and padberg in lawler e l lenstra j k rinnooy kan a shmoys d b eds the travelling salesman problem a guided tour of combinatorial optimization wiley new york 1985 the new compact model is an improved and disaggregated version of a well known model for the atsp based on the subtour elimination constraints miller et al journal of acm 7 1960 326 329 the circuit inequalities are weaker than the subtour elimination constraints given by dantzig et al however each one of these circuit inequalities can be lifted into several different facet defining inequalities which are not dominated by the subtour elimination inequalities we show that some of the inequalities involved in the previously mentioned compact formulation can be lifted in such a way that by projection we obtain a small subset of the so called d k and d k inequalities this shows that the lp relaxation of our strongest model is not dominated by the lp relaxation of the model presented by dantzig et al operations research 2 1954 393 410 the new models motivate a new classification of formulations for the atsp

2d grushin type equations minimal time and null controllable data abstract we study internal null controllability for degenerate parabolic equations of grushin type g x x 2 x 2 y y 2 0 in the rectangle x y 1 1 0 1 previous works proved that null controllability holds for weak degeneracies small and fails for strong degeneracies large moreover in the transition regime and with strip shaped control domains a positive minimal time is required in this paper we work with controls acting on two strips symmetric with respect to the degeneracy we give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time t when the system is not null controllable their regularity depends on the control domain and the time t we also prove that with a control that acts on one strip touching the degeneracy line x 0 then grushin type equations are null controllable in any time t 0 and for any degeneracy 0 our approach is based on a precise study of the observability property for the one dimensional heat equations satisfied by the fourier coefficients in variable y this precise study is done through a transmutation process on the resulting one dimensional wave equations by lateral propagation of energy method

an analytical comparison of different formulations of the travelling sales man problem a transformation technique is proposed that permits one to derive the linear description of the imagex of a polyhedronz under an affine linear transformation from the given linear description ofz this result is used to analytically compare various formulations of the asymmetric travelling salesman problem to the standard formulation due to dantzig fulkerson and johnson which are all shown to be weaker formulations in a precise setting we also apply this transformation technique to symmetrize formulations and show in particular that the symmetrization of the standard asymmetric formulation results into the standard one for the symmetric version of the travelling salesman problem

a cutting plane approach to the sequential ordering problem with applications to job scheduling in manufacturing the sequential ordering problem sop finds a minimum cost hamiltonian path subject to certain precedence constraints the sop has a number of practical applications and arises for instance in production planning for flexible manufacturing systems this paper presents several 0 1 models of the sop and reports the authors computational experience in finding lower bounds of the optimal solution value of several real life instances of sop one of the most successful approaches is a cutting plane procedure that is based on polynomial time separation algorithms for large classes of valid inequalities for the associated polyhedron

a parallel branch and bound algorithm for solving large asymmetric traveling salesman problems a parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described the algorithm uses an assignment problem based lower bounding technique subtour elimination branching rules and a subtour patching algorithm as an upper bounding procedure the algorithm is organized around a data flow framework for parallel branch and bound the algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution computational results are presented for three different classes of problem instances 1 matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities 2 instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix and 3 instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics

hybrid optimization methods for time dependent sequencing problems abstract in this paper we introduce novel optimization methods for sequencing problems in which the setup times between a pair of tasks depend on the relative position of the tasks in the ordering our proposed methods rely on a hybrid approach where a constraint programming model is enhanced with two distinct relaxations one discrete relaxation based on multivalued decision diagrams and one continuous relaxation based on linear programming both relaxations are used to generate bounds and enhance constraint propagation experiments conducted on three variants of the time dependent traveling salesman problem indicate that our techniques substantially outperform general purpose methods such as mixed integer linear programming and constraint programming models

an additive bounding procedure for combinatorial optimization problems we know that the effectiveness of the branch and bound algorithms proposed for the solution of combinatorial optimization problems greatly depends on the tightness of the available bounds in this paper we consider optimization problems with a linear objective function we propose an additive approach for computing lower bounds that yields an increasing sequence of values an application to the traveling salesman problem with precedence constraints is presented to exemplify the technique

the single vehicle routing problem with toll by weight scheme a branch and bound approach abstract expressways in china make use of the toll by weight scheme in which expressway tolls are collected based on the weight and traveling distance of the vehicle most vehicle routing models assume that the cost of traversing each edge is equivalent to edge length or some constant as a result such models cannot be practically applied to the chinese expressway transportation system this study addresses a new single vehicle routing problem that takes the vehicle s laden and unladen weight into account to solve this problem exactly we provide a branch and bound algorithm with a provably valid lower bound measure along with five dominance checkers for additional pruning we analyze our algorithm using instances generated from standard tsp test cases as well as two new sets of test cases based on real expressway information from the gansu and jiangxi provinces in china the algorithm can be applied to any toll scheme in which the toll per unit distance monotonically increases with weight even if the toll function is non linear

nodal aggregation of resource constraints in a shortest path problem the shortest path problem with resource constraints consists of finding the minimum cost path between two specified points while respecting constraints on resource consumption its solving by a dynamic programming algorithm requires a computation time increasing with the number of resources with the aim of producing rapidly a good heuristic solution we propose to reduce the state space by aggregating resources our approach consists of projecting the resources on a vector of smaller dimension and then to dynamically adjust the projection matrix to get a better approximation of the optimal solution we propose an adjustment based on lagrangian and surrogate relaxations in a column generation framework in which the sub problems are shortest path problems with resource constraints we adjust the multipliers only one time at each column generation iteration this permit to obtain good solutions of the scheduling problem in few time

formulations and valid inequalities for the heterogeneous vehicle routing problem we consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips we develop six different formulations the first four based on miller tucker zemlin constraints and the last two based on flows we compare the linear programming bounds of these formulations we derive valid inequalities and lift some of the constraints to improve the lower bounds we generalize and strengthen subtour elimination and generalized large multistar inequalities

an efficient genetic algorithm for the traveling salesman problem with precedence constraints abstract the traveling salesman problem with precedence constraints tsppc is one of the most difficult combinatorial optimization problems in this paper an efficient genetic algorithm ga to solve the tsppc is presented the key concept of the proposed ga is a topological sort ts which is defined as an ordering of vertices in a directed graph also a new crossover operation is developed for the proposed ga the results of numerical experiments show that the proposed ga produces an optimal solution and shows superior performance compared to the traditional algorithms

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

interior controllability of semi linear degenerate wave equations abstract in this paper we prove the existence of interior controls for one dimensional semi linear degenerate wave equations by using a duality argument we reduce the problem to an observability estimate for the linear degenerate wave equation first the unique continuation for the degenerate wave equation is established by means of this and the multiplier method we obtain the observability estimate

a column generation approach to the multiple depot vehicle scheduling problem we give a new formulation to the multiple depot vehicle scheduling problem as a set partitioning problem with side constraints whose continuous relaxation is amenable to be solved by column generation we show that the continuous relaxation of the set partitioning formulation provides a much tighter lower bound than the additive bound procedure previously applied to this problem we also establish that the additive bound technique cannot provide tighter bounds than those obtained by lagrangian decomposition in the framework in which it has been used so far computational results that illustrate the robustness of the combined set partitioning column generation approach are reported for problems four to five times larger than the largest problems that have been exactly solved in the literature finally we show that the gap associated with the additive bound based on the assignment and shortest path relaxations can be arbitrarily bad in the general case and as bad as 50 in the symmetric case

solving the asymmetric travelling salesman problem with time windows by branch and cut many optimization problems have several equivalent mathematical models it is often not apparent which of these models is most suitable for practical computation in particular when a certain application with a specific range of instance sizes is in focus our paper addresses the asymmetric travelling salesman problem with time windows atsp tw from such a point of view the real world application we aim at is the control of a stacker crane in a warehouse we have implemented codes based on three alternative integer programming formulations of the atsp tw and more than ten heuristics computational results for real world instances with up to 233 nodes are reported showing that a new model presented in a companion paper outperforms the other two models we considered at least for our special application and that the heuristics provide acceptable solutions

a restricted lagrangean approach to the traveling salesman problem we describe an algorithm for the asymmetric traveling salesman problem tsp using a new restricted lagrangean relaxation based on the assignment problem ap the lagrange multipliers are constrained so as to guarantee the continued optimality of the initial ap solution thus eliminating the need for repeatedly solving ap in the process of computing multipliers we give several polynomially bounded procedures for generating valid inequalities and taking them into the lagrangean function with a positive multiplier without violating the constraints so as to strengthen the current lower bound upper bounds are generated by a fast tour building heuristic when the bound strengthening techniques are exhausted without matching the upper with the lower bound we branch by using two different rules according to the situation the usual subtour breaking disjunction and a new disjunction based on conditional bounds we discuss computational experience on 120 randomly generated asymmetric tsp s with up to 325 cities the maximum time used for any single problem being 82 seconds this is a considerable improvement upon earlier methods though the algorithm discussed here is for the asymmetric tsp the approach can be adapted to the symmetric tsp by using the 2 matching problem instead of ap

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

path planning on a cuboid using genetic algorithms the traveling salesman problem tsp is one of the most extensively studied problems in the fields of combinatorial optimization and global search heuristics a variety of heuristic algorithms are available for solving euclidean tsp and planar tsps however optimization on a cuboid has potential applications for areas like path planning on the faces of buildings rooms furniture books and products or simulating the behaviors of insects in this paper we address a variant of the tsp in which all points cities and paths solution are on the faces of a cuboid we develop an effective hybrid method based on genetic algorithms and 2 opt to adapt the euclidean tsp to the surface of a cuboid the method was tested on some benchmark problems from tsplib with satisfactory results a web based interactive visualization tool has also been developed using java 3d and optimization results for different point densities on the cuboid are presented

carleman estimates for a class of degenerate parabolic operators given alpha in 0 2 and f in l 2 0 t times 0 1 we derive new carleman estimates for the degenerate parabolic problem w t x alpha w x x f where t x in 0 t times 0 1 associated to the boundary conditions w t 1 0 and w t 0 0 if 0 leq alpha 1 or x alpha w x t 0 0 if 1 leq alpha 2 the proof is based on the choice of suitable weighted functions and hardy type inequalities as a consequence for all 0 leq alpha 2 and omega subset subset 0 1 we deduce null controllability results for the degenerate one dimensional heat equation u t x alpha u x x h chi omega with the same boundary conditions as above

regional null controllability of a linearized crocco type equation we are interested in controllability problems of equations coming from a boundary layer model we simplify the problem by considering only equations with constant coefficients the problem is described by a degenerate parabolic equation a linearized crocco type equation where phenomena of diffusion and transport are coupled first we give a geometric characterization of the influence domain of a locally distributed con trol then we prove regional null controllability results on this domain the proof is based on an adequate observability inequality for the homogeneous adjoint problem this inequality is obtained by decomposition of the space time domain and carleman type estimates along characteristics in the second part of this paper we treat the case of a boundary control

a branch and bound algorithm for the capacitated vehicle routing problem on directed graphs we consider the asymmetric capacitated vehicle routing problem cvrp a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed cvrp is known to be np hard and finds practical applications in distribution and scheduling we describe two new bounding procedures for cvrp based on the so called additive approach each procedure computes a sequence of nondecreasing lower bounds obtained by solving different relaxations of cvrp effective implementations of the procedures are also outlined which considerably reduce the computational effort the two procedures are combined into an overall bounding algorithm a branch and bound exact algorithm is then proposed whose performance is enhanced by means of reduction procedures dominance criteria and feasibility checks extensive computational results on both real world and random test problems are presented showing that the proposed approach favorably compares with previous algorithms from the literature

a threshold accepting heuristic with intense local search for the solution of special instances of the traveling salesman problem in real life scheduling variations of the standard traveling salesman problem are very often encountered the aim of this work is to present a new heuristic method for solving three such special instances with a common approach the proposed algorithm uses a variant of the threshold accepting method enhanced with intense local search while the candidate solutions are produced through an insertion heuristic scheme the main characteristic of the algorithm is that it does not require modifications and parameter tuning in order to cope with the three different problems computational results on a variety of real life and artificial problems are presented at the end of this work and prove the efficiency and the ascendancy of the proposed method over other algorithms found in the literature

a polyhedral approach to the asymmetric traveling salesman problem several branch and bound algorithms for the exact solution of the asymmetric traveling salesman problem atsp based on the assignment problem ap relaxation have been proposed in the literature these algorithms perform very well for some instances e g those with uniformly random integer costs but very poorly for others the aim of this paper is to evaluate the effectiveness of a branch and cut algorithm exploiting atsp specific facet defining cuts to be used to attack hard instances that cannot be solved by the ap based procedures from the literature we present new separation algorithms for some classes of facet defining cuts and a new variable pricing technique for dealing with highly degenerate primal lp problems a branch and cut algorithm based on these new results is designed and evaluated through computational analysis on several classes of both random and real world instances the outcome of the research is that on hard instances the branch and cut algorithm clearly outperforms the best ap based algorithms from the literature

the control transmutation method and the cost of fast controls in this paper the null controllability in any positive time t of the first order equation 1 dot x t e i theta ax t bu t theta pi 2 fixed is deduced from the null controllability in some positive time l of the second order equation 2 ddot z t az t bv t the differential equations 1 and 2 are set in a banach space b is an admissible unbounded control operator and a is a generator of cosine operator function the control transmutation method makes explicit the input function u of 1 in terms of the input function v of 2 u t int mathbb r k t s v s ds where the compactly supported kernel k depends only on t and l this method proves roughly that the norm of a u steering the system 1 from an initial state x 0 x 0 to the final state x t 0 grows at most like x 0 exp alpha l 2 t as the control time t tends to zero the rate alpha is characterized independently by a one dimensional controllability problem in applications to the cost of fast controls for the heat equation l is roughly the length of the longest ray of geometric optics which does not intersect the control region

a result on projection for the vehicle routing ptoblem abstract in this paper we present a result on projection for the vehicle routing problem vrp the vrp is closely related to delivery type problems and appears in a large number of practical situations concerning the distribution of commodities the present work focuses on a commodity flow formulation presented by gavish and graves this formulation includes two sets of variables and hence it also must include coupling constraints between the two sets of variables these coupling constraints can be defined in several ways the main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation the equivalent formulation using only the x ij variables satisfies the so called multistar constraints which for certain parameters induce facets of the non directed vrp polytope using an idea taken from gouveia we show how to derive a more compact representation in the number of constraints of the multistar constraints some consequences of our projection result are also discussed

optimal toll design a lower bound framework for the asymmetric traveling salesman problem we propose a framework of lower bounds for the asymmetric traveling salesman problem tsp based on approximating the dynamic programming formulation with different basis vector sets we discuss how several well known tsp lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between cities we then introduce an exact reformulation that generates a family of successively tighter lower bounds all solvable in polynomial time we show that the base member of this family yields a bound greater than or equal to the well known held karp bound obtained by solving the linear programming relaxation of the tsp s integer programming arc based formulation

a new approach to solving the multiple traveling salesperson problem using genetic algorithms the multiple traveling salesperson problem mtsp involves scheduling m 1 salespersons to visit a set of n m locations so that each location is visited exactly once while minimizing the total or maximum distance traveled by the salespersons the mtsp is similar to the notoriously difficult traveling salesperson problem tsp with the added complication that each location may be visited by any one of the salespersons previous studies investigated solving the mtsp with genetic algorithms gas using standard tsp chromosomes and operators this paper proposes a new ga chromosome and related operators for the mtsp and compares the theoretical properties and computational performance of the proposed technique to previous work computational testing shows the new approach results in a smaller search space and in many cases produces better solutions than previous techniques

facets of the asymmetric traveling salesman polytope in this paper we consider the asymmetric traveling salesman polytope pn defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices and its monotonization pi n several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of pn or pi n we describe a general technique which can often be used to prove that a given inequality defines a facet of pn the method is applied to prove that the so called dk dk c3 comb and c2 inequalities define facets of pn thus solving open problems as for the monotone polytope a necessary and sufficient condition for a facet defining inequality of pn to define a facet of pi n as well is introduced which applies to the dk dk c3 comb and c2 inequalities in addition a simple procedure for transforming any facet defining inequality of pn into an equivalent one which also defines a facet of pi n is given

a generalized insertion heuristic for the traveling salesman problem with time windows this article describes a generalized insertion heuristic for the traveling salesman problem with time windows in which the objective is the minimization of travel times the algorithm gradually builds a route by inserting at each step a vertex in its neighbourhood on the current route and performing a local reoptimization this is done while checking the feasibility of the remaining part of the route backtracking is sometimes necessary once a feasible route has been determined an attempt is made to improve it by applying a post optimization phase based on the successive removal and reinsertion of all vertices tests performed on 375 instances indicate that the proposed heuristic compares very well with alternative methods and very often produces optimal or near optimal solutions

mobility allowance shuttle transit mast services mip formulation and strengthening with logic constraints abstract we study a hybrid transportation system referred to as mobility allowance shuttle transit mast where vehicles may deviate from a fixed path consisting of a few mandatory checkpoints to serve demand distributed within a proper service area in this paper we propose a mixed integer programming mip formulation for the static scheduling problem of a mast type system since the problem is np hard we develop sets of logic cuts by using reasonable assumptions on passengers behavior the purpose of these constraints is to speed up the search for optimality by removing inefficient solutions from the original feasible region experiments show the effectiveness of the developed inequalities achieving a reduction up to 90 of the cpu solving time for some of the instances

minimum directed 1 subtree relaxation for score orienteering problem score orienteering is a variant of the orienteering game where competitors start at a specified control point visit as many control points as possible and thereby collect prizes and return to the starting point within a prescribed amount of time the competitor with the highest amount of prizes wins the game in this paper we propose the minimum directed 1 subtree problem as a new relaxation to this problem and develop two methods to improve its lower bound we describe a cut and dual simplex method and a lagrangian relaxation method and construct an algorithm that combines these two methods in an appropriate way computational experiments are carried out to analyse the behavior of the proposed relaxation with respect to the characteristics of the test problems especially for large scale instances we find that the proposed relaxation is superior to the ordinary assignment relaxation

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

a heuristic algorithm for the asymmetric capacitated vehicle routing problem abstract we consider the asymmetric capacitated vehicle routing problem acvrp a particular case of the standard asymmetric vehicle routing problem arising when only the vehicle capacity constraints are imposed acvrp is known to be np hard and finds practical applications e g in distribution and scheduling in this paper we describe the extension to acvrp of the two well known clarke wright and fisher jaikumar heuristic algorithms we also propose a new heuristic algorithm for acvrp that starting with an initial infeasible solution determines the final set of vehicle routes through an insertion procedure as well as intea route and inter route arc exchanges the initial infeasible solution is obtained by using the additive bounding procedures for acvrp described by fischetti toth and vigo in 1992 extensive computational results on several classes of randomly generated test problems involving up to 300 customers and on some real instances of distribution problems in urban areas are presented the results obtained show that the proposed approach favourably compares with previous algorithms from the literature

multiprogramming genetic algorithm for optimization problems with permutation property permutation property has been recognized as a common but challenging feature in combinatorial problems because of their complexity recent research has turned to genetic algorithms to address such problems although genetic algorithms have been proven to facilitate the entire space search but they lack in fine tuning capability for obtaining the global optimum therefore in this study a multiprogramming genetic algorithm mga was developed for permutation optimization both the global exploration through crossover operation and mutation operation among the population and the local exploitation through selection operation around chromosomes are integrated to mga in order to improve the performance of mga the authors establish some regulations replacement regulation local optimization regulation and global optimization regulation to help the evolvement of mga computational experiments are conducted on most of atsp instances available in the tsplib and on a set of larger asymmetric instances with known optimal solutions the comparisons show that the results obtained by our method compare favorably with those obtained by several other algorithms recently proposed for the atsp

performance characteristics of alternative genetic algorithmic approaches to the traveling salesman problem using path representation an empirical study abstract the potential of genetic algorithmic ga approaches for solving order based problems including the traveling salesman problem tsp is recognized in a number of recent studies by applying various gas these studies developed a set of unresolved ga design and configuration issues the purpose of this study is to resolve the conflicting ga design and configuration issues by 1 concentrating on the classical tsp and 2 developing implementing and testing a complete set of alternative ga configurations 144 gas are developed and evaluated by solvinh 5000 tsps a carefully designed statistical experimental plan accompanied by rigorous statistical analysis isolate the most promising configurations and identify their effect on solution time and quality although the emphasis is on the tsp the final results are applicable to other order based problems that use sequence encoding

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

an optimal algorithm for the traveling salesman problem with time windows this paper presents the development of new elimination tests which greatly enhance the performance of a relatively well established dynamic programming approach and its application to the minimization of the total traveling cost for the traveling salesman problem with time windows the tests take advantage of the time window constraints to significantly reduce the state space and the number of state transitions these reductions are performed both a priori and during the execution of the algorithm the approach does not experience problems stemming from increasing problem size wider or overlapping time windows or an increasing number of states nearly as rapidly as other methods our computational results indicate that the algorithm was successful in solving problems with up to 200 nodes and fairly wide time windows when the density of the nodes in the geographical region was kept constant as the problem size was increased the algorithm was capable of solving problems with up to 800 nodes for these problems the cpu time increased linearly with problem size these problem sizes are much larger than those of problems previously reported in the literature

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

cluster based branching for the asymmetric traveling salesman problem abstract this paper presents a new branching scheme for the asymmetric traveling salesman problem atsp based on clusters a cluster is defined as a node set with the characteristic that there exists an optimal solution in which the nodes in the node set are visited consecutively the paper considers identification of clusters implementation of a cluster based branching scheme and cluster based dominance tests the new approach is implemented in a branch and bound algorithm using a well known additive bounding procedure considerable savings in computing time are obtained compared to previously published assignment based branch and bound algorithms for the atsp

flatness of heavy chain systems in this paper the flatness m fliess j levine p martin and p rouchon internat j control 61 1995 pp 1327 1361 m fliess j levine p martin and p rouchon ieee trans automat control 44 1999 pp 922 937 of heavy chain systems i e trolleys carrying a fixed length heavy chain that may carry a load is addressed in the partial derivatives equations framework we parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem namely steering from one state to another state when considered as a finite set of small pendulums these systems were shown to be flat r m murray in proceedings of the ifac world congress san francisco ca 1996 pp 395 400 our study is an extension to the infinite dimensional case under small angle approximations these heavy chain systems are described by a one dimensional 1d partial differential wave equation dealing with this infinite dimensional description we show how to get the explicit parameterization of the chain trajectory using distributed and punctual advances and delays of its free end this parameterization results from symbolic computations replacing the time derivative by the laplace variable s yields a second order differential equation in the spatial variable where s is a parameter its fundamental solution is for each point considered along the chain an entire function of s of exponential type moreover for each we show that thanks to the liouville transformation this solution satisfies modulo explicitly computable exponentials of s the assumptions of the paley wiener theorem this solution is in fact the transfer function from the flat output the position of the free end of the system to the whole state of the system using an inverse laplace transform we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators

a new crossover approach for solving the multiple travelling salesmen problem using genetic algorithms abstract this paper proposes a new crossover operator called two part chromosome crossover tcx for solving the multiple travelling salesmen problem mtsp using a genetic algorithm ga for near optimal solutions we adopt the two part chromosome representation technique which has been proven to minimise the size of the problem search space nevertheless the existing crossover method for the two part chromosome representation has two limitations firstly it has extremely limited diversity in the second part of the chromosome which greatly restricts the search ability of the ga secondly the existing crossover approach tends to break useful building blocks in the first part of the chromosome which reduces the ga s effectiveness and solution quality therefore in order to improve the ga search performance with the two part chromosome representation we propose tcx to overcome these two limitations and improve solution quality moreover we evaluate and compare the proposed tcx with three different crossover methods for two mtsp objective functions namely minimising total travel distance and minimising longest tour the experimental results show that tcx can improve the solution quality of the ga compared to three existing crossover approaches

exact boundary controllability of 1 d parabolic and hyperbolic degenerate equations the boundary controllability of a class of one dimensional degenerate equations is studied in this paper the novelty of this work is that the control acts through the part of the boundary where degeneracy occurs first we consider a class of degenerate hyperbolic equations then we prove sharp observability estimates for these equations using nonharmonic fourier series the transmutation method yields a result for the corresponding class of degenerate parabolic equations

the tree representation for the pickup and delivery traveling salesman problem with lifo loading the feasible solutions of the traveling salesman problem with pickup and delivery tsppd are commonly represented by vertex lists however when the tsppd is required to follow a policy that loading and unloading operations must be performed in a last in first out lifo manner we show that its feasible solutions can be represented by trees consequently we develop a novel variable neighborhood search vns heuristic for the tsppd with last in first out loading tsppdl involving several search operators based on the tree data structure extensive experiments suggest that our vns heuristic is superior to the current best heuristics for the tsppdl in terms of solution quality while requiring no more computing time as the size of the problem increases

the shortest path problem with resource constraints with k 2 loop elimination and its application to the capacitated arc routing problem abstract in many branch and price algorithms the column generation subproblem consists of computing feasible constrained paths in the capacitated arc routing problem carp elementarity constraints concerning the edges to be serviced and additional constraints resulting from the branch and bound process together impose two types of loop elimination constraints to fulfill the former constraints it is common practice to rely on a relaxation where loops are allowed in a k loop elimination approach all loops of length k and smaller are forbidden following bode and irnich 2012 for solving the carp branching on followers and non followers is the only known approach to guarantee integer solutions within branch and price however it comes at the cost of additional task 2 loop elimination constraints in this paper we show that a combined k 2 loop elimination in the shortest path subproblem can be accomplished in a computationally efficient way overall the improved branch and price often allows the computation of tighter lower bounds and integer optimal solutions for several instances from standard benchmark sets

new formulations of the hop constrained minimum spanning tree problem via miller tucker zemlin constraints abstract given an undirected network with positive edge costs and a natural number p the hop constrained minimum spanning tree problem hmst is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops edges in this paper we develop new formulations for hmst the formulations are based on miller tucker zemlin mtz subtour elimination constraints mtz based liftings in the literature offered for hmst and a new set of topology enforcing constraints we also compare the proposed models with the mtz based models in the literature with respect to linear programming relaxation bounds and solution times the results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to mtz constraints some of which are based on well known strong liftings of desrochers and laporte 1991

genetic algorithms and traveling salesman problems abstract a genetic algorithm ga with an asexual reproduction plan through a generalized mutation for an evolutionary operator is developed that can be directly applied to a permutation of n numbers for an approximate global optimal solution of a traveling salesman problem tsp schema analysis of the algorithm shows that a sexual reproduction with the generalized mutation operator preserves the global convergence property of a genetic algorithm thus establishing the fundamental theorem of the ga for the algorithm avoiding an intermediate step of encoding through random keys to preserve crossover or permuting n and using fixing states for legal crossover are the chief benefits of the innovations reported in this paper the algorithm has been applied to a number of natural and artificial problems and the results are encouraging

a branch first cut second approach for locomotive assignment the problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules the force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines this complicates the solution process of the integer programming formulation and usually creates a large integrality gap furthermore the solution of the linearly relaxed problem is strongly fractional to obtain integer solutions we propose a novel branch and cut approach the core of the method consists of branching decisions that define on one branch the projection of the problem on a low dimensional subspace there the facets of the polyhedron describing a restricted constraint set can be easily derived we call this approach branch first cut second we first derive facets when at most two types of engines are used we then extend the branching rule to cases involving additional locomotive types we have conducted computational experiments using actual data from the canadian national railway company simulated test problems involving two or more locomotive types were solved over 1 2 and 3 day rolling horizons the cuts were successful in reducing the average integrality gap by 52 for the two type case and by 34 when more than 25 types were used furthermore the branch first cut second approach was instrumental in improving the best known solution for an almost 2 000 leg weekly problem involving 26 locomotive types it reduced the number of locomotives by 11 or 1 1 at an equivalent savings of 3 000 000 per unit additional tests on different weekly data produced almost identical results

new lower bounds for the symmetric travelling salesman problem in this paper new lower bounds for the symmetric travelling salesman problem are proposed and combined in additive bounding procedures efficient implementations of the algorithms are given in particular fast procedures for computing the linear programming reduced costs of the shortest spanning tree sst problem and for finding all ther sst of a given graph are presented computational results on randomly generated test problems are reported

an additive bounding procedure for the asymmetric travelling salesman problem in this paper new lower bounds for the asymmetric travelling salesman problem are presented based on spanning arborescences the new bounds are combined in an additive procedure whose theoretical performance is compared with that of the balas and christofides procedure 1981 both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems

dynamic programming strategies for the traveling salesman problem with time window and precedence constraints the traveling salesman problem with time window and precedence constraints tsp twpc is to find an hamiltonian tour of minimum cost in a graph g x a of n vertices starting at vertex 1 visiting each vertex i x during its time window and after having visited every vertex that must precede i and returning to vertex 1 the tsp twpc is known to be np hard and has applications in many sequencing and distribution problems in this paper we describe an exact algorithm to solve the problem that is based on dynamic programming and makes use of bounding functions to reduce the state space graph these functions are obtained by means of a new technique that is a generalization of the state space relaxation for dynamic programming introduced by christofides et al christofides n a mingozzi p toth 1981b state space relaxation for the computation of bounds to routing problems networks 11 145 164 computational results are given for randomly generated test problems

models for a steiner ring network design problem with revenues abstract consider a graph g v e where the set of nodes v is partitioned into two sets the set r of the nodes required to be in the ring and the set s of the nodes which may or may not be included in the ring besides the usual edge link costs revenue profits between each pair of nodes are also considered the objective is to find a ring including all the nodes in r plus some nodes in s such that the objective link costs minus revenues is minimized for a similar problem gendreau labbe and laporte 1995 have presented a formulation with a quadratic term in the objective function in this paper we follow a different strategy by pointing out that extra variables used in extended formulations for the asymmetric travelling salesman problem atsp can be used to model the revenues in a linearized way we adapt a recent formulation for the atsp proposed in gouveia and pires 1999 and discuss several classes of valid inequalities we also point out that the flow variables in a multicommodity flow model can also be used for modelling the revenues in a linearized way relationships between the linear programming relaxations of all the models presented in this paper are also discussed computational results will also be presented

a multiplier adjustment method for the generalized assignment problem we describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a lagrangian relaxation with the multipliers set by a heuristic adjustment method the algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms although comparison of running times on different machines is difficult the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem

sharp observability estimates for heat equations the goal of this article is to derive new estimates for the cost of observability of heat equations we have developed a new method allowing one to show that when the corresponding wave equation is observable the heat equation is also observable this method allows one to describe the explicit dependence of the observability constant on the geometry of the problem the domain in which the heat process evolves and the observation subdomain we show that our estimate is sharp in some cases particularly in one space dimension and in the multi dimensional radially symmetric case our result extends those in fattorini and russell arch rational mech anal 43 272 292 1971 to the multi dimensional setting and improves those available in the literature namely those by miller j differ equ 204 1 202 226 2004 siam j control optim 45 2 762 772 2006 atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 17 4 351 366 2006 and tenenbaum and tucsnak j differ equ 243 1 70 100 2007 our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation in contrast to the standard application of transmutation methods which uses a reverse representation of the heat solution in terms of the wave one we shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side we will also comment on the applications of our techniques to controllability properties of heat type equations

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs