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

a variable grouping based genetic algorithm for large scale integer programming to effectively reduce the dimensionality of search space this paper proposes a variable grouping based genetic algorithm vgga for large scale integer programming problems ips the vgga first groups ip s decision variables based on the optimal solution to the ip s continuous relaxation problem and then applies a standard genetic algorithm ga to the subproblem for each group of variables we compare the vgga with the standard ga and gas based on even variable grouping by applying them to solve randomly generated convex quadratic knapsack problems and integer knapsack problems numerical results suggest that the vgga is superior to the standard ga and gas based on even variable grouping both on computation time and solution quality

exact and approximation algorithms for the tactical fixed interval scheduling problem the tactical fixed interval scheduling problem tfisp is the problem of determining the minimum number of parallel nonidentical machines such that a feasible schedule exists for a given set of jobs in tfisp each job must belong to a specific job class and must be carried out in a prespecified time interval the problem is complicated by the restrictions that 1 each machine can handle only one job at a time 2 each machine can handle only jobs from a subset of the job classes and 3 preemption is not allowed in this paper we discuss the occurrence of tfisp in practice we analyze the computational complexity of tfisp and we present exact and approximation algorithms for solving tfisp the paper concludes with a computational study

symmetry breaking for δu 2δe u 0 on a disk with general boundary conditions we discuss the radially symmetric solutions and the symmetry breaking of the equation u 2 e u 0 in d and u b u n 0 on d where d is the unit disk in 2 0 and b is a constant we prove that for any b 0 such that there are exactly two radially symmetric solutions for 0 one for and none for b for where m is a positive integer there are b k 1 m such that the equation has symmetry breaking at k b on the lower branch of radially symmetric solutions

on the invariant measures for the two dimensional euler flow it is proven that the canonical gibbs measure associated with a gas of vortices of intensity converges in the limitn 0 n const to a gaussian measure which is invariant for the two dimensional euler equation

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

an efficient evolutionary algorithm for accurate polygonal approximation an optimization problem for polygonal approximation of 2 d shapes is investigated in this paper the optimization problem for a digital contour of n points with the approximating polygon of k vertices has a search space of c n k instances i e the number of ways of choosing k vertices out of n points a genetic algorithm based method has been proposed for determining the optimal polygons of digital curves and its performance is better than that of several existing methods for the polygonal approximation problems this paper proposes an efficient evolutionary algorithm eea with a novel orthogonal array crossover for obtaining the optimal solution to the polygonal approximation problem it is shown empirically that the proposed eea outperforms the existing genetic algorithm based method under the same cost conditions in terms of the quality of the best solution average solution variance of solutions and the convergence speed especially in solving large polygonal approximation problems

symmetry breaking for a class of semilinear elliptic problems abstract this document considers a nonlinear elliptic problem suppose this problem has a family of positive radial solutions parameterized by r i e ur x this paper studies the possibility of the existence of nonradial solutions of bifurcating from the radial solutions family answering a question posed by smoller and wasserman it is shown this happens if f satisfies suitable assumptions therefore the global structure of the nonradial solution set is investigated

approximation algorithms for fixed job schedule problems we consider two generalizations of the fixed job schedule problem obtained by imposing a bound on the spread time or on the working time of each processor these np hard problems studied by the authors in previous works can arise in bus driver scheduling we introduce several polynomial time approximation algorithms give efficient implementations for them and analyze their complexity and worst case performance the average behavior is also investigated through extensive computational experiments

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

exact and approximation algorithms for the operational fixed interval scheduling problem the operational fixed interval scheduling problem ofisp is characterized as the problem of scheduling a number of jobs each with a fixed starting time a fixed finishing time a priority index and a job class the objective is to find an assignment of jobs to machines with maximal total priority the problem is complicated by the restrictions that i each machine can handle only one job at a time ii each machine can handle only jobs from a prespecified subset of all possible job classes and iii preemption is not allowed it follows from the above that ofisp has both the character of a job scheduling problem and the character of an assignment problem in this paper we discuss the occurrence of the problem in practice and we present newly developed exact and approximation algorithms for solving ofisp finally some computational results are shown

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

on the nonlinear eigenvalue problem δu λe u 0 on etudie la structure de l ensemble 7b c des solutions du probleme aux valeurs propres non lineaire u exp u 0 sous des conditions de dirichlet dans un domaine borne simplement connexe

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

the fixed job schedule problem with working time constraints we consider a generalization of the fixed job schedule problem where a bound is imposed on the total working time of each processor it is shown that the problem is np hard but polynomially solvable in the preemptive case we introduce several lower bounds one is determined through definition of a special class of graphs for which the maximum clique problem is shown to be polynomial lower bounds and dominance criteria are exploited in a branch and bound algorithm for optimal solution of the problem the effectiveness of the algorithm is analyzed through computational experiments

filtering atmostnvalue with difference constraints application to the shift minimisation personnel task scheduling problem the problem of minimising the number of distinct values among a set of variables subject to difference constraints occurs in many real life contexts this is the case of the shift minimisation personnel task scheduling problem introduced by krishnamoorthy et al which is used as a case study all along this paper constraint programming enables to formulate this problem easily through several alldifferent constraints and a single atmostnvalue constraint however the independence of these constraints results in a poor lower bounding hence a difficulty to prove optimality this paper introduces a formalism to describe a family of propagators for atmostnvalue in particular we provide simple but significant improvement of the state of the art atmostnvalue propagator of bessiere et al to filter the conjunction of an atmostnvalue constraint and disequalities in addition we provide an original search strategy which relies on constraint reification extensive experiments show that our contribution significantly improves a straightforward model so that it competes with the best known approaches from operational research

an effective genetic algorithm approach to large scale mixed integer programming problems abstract to effectively reduce the search space of gas on large scale mip problems this paper proposed a new variable grouping method based on structure properties of a problem taking the capacity expansion and technology selection problem as a typical example this method groups problem s decision variables over time period and machine line based on this new variable grouping method we developed a variable grouping based genetic algorithm according to problem s structure properties vgga s we tested the performance of vgga s by applying it on the capacity expansion and technology selection problem numerical experiments suggested that vgga s outperforms the standard ga and variable grouping based gas without considering problem s structure properties both on computation time and solution quality although vgga s is proposed based on structure properties of a specific mip problem it is a general optimization algorithm and theoretically applicable to other large scale mip problems

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

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

randomized gravitational emulation search algorithm for symmetric traveling salesman problem abstract this paper presents a new heuristic method called randomized gravitational emulation search rges algorithm for solving symmetric traveling salesman problems stsp this algorithm is found upon introducing randomization concept along with the two of the four primary parameters velocity and gravity in physics through swapping in terms of groups by using random numbers in the existing local search algorithm gels in order to avoid local minima and thus can yield global minimum for stsp to validate the proposed method numerous simulations were conducted to compare the quality of solutions with other existing algorithms like genetic algorithm ga simulated annealing sa hill climbing hc etc using a range of stsp benchmark problems according to the results of the simulations the performance of rges is found significantly enhanced and provided optimal solutions in almost all test problems of sizes up to 76 also a comparative computational study of 11 chosen benchmark problems from tsplib library shows that this rges algorithm is an efficient tool of solving stsp and this heuristic is competitive with other heuristics too

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

particle swarm optimization combined with genetic operators for job shop scheduling problem with fuzzy processing time job shop scheduling problem is a np hard problem the processing time for each job is often imprecise in many real world applications and the imprecision in the data is critical for the scheduling procedures therefore job shop scheduling problem with fuzzy processing time is addressed in the paper the processing time is described by triangular fuzzy numbers the objective is to find a job sequence that minimizes the makespan and the uncertainty of the makespan by using an approach for ranking fuzzy numbers the particle swarm optimization pso is a randomized population based optimization method that was inspired by the flocking behavior of birds and human social interactions pso has been successfully applied to various real world applications but there is a little literature reported regarding application to scheduling problems as it was unsuitable for them in this paper pso is redefined and modified by introducing genetic operators such as crossover and mutation operator to update the particles we call this particle swarm optimization combined with genetic operators gpso this is successfully employed to solve the formulated problem ten benchmarks with fuzzy processing time are used to test gpso the feasibility as well as the efficiency of the proposed method is assessed in comparison with genetic algorithm ga

symmetry breaking for a class of semilinear elliptic problems we study positive solutions of the dirichlet problem for aji up a p 1 a 0 on the unit ball 0 we show that there exists a positive solution uo ao of this problem which satisfies in addition duo dn 0 on fi we prove also that at uo ao the symmetry breaks i e asymmetric solutions bifurcate from the positive radial solutions 1 introduction we consider here the symmetry breaking for semilinear el liptic problems of the type a f u x in bu u 0 on dbi where bi is the unit ball on r n 2 and rxr risac2 function we say that the symmetry breaks at a if the radial solution un ao is a nonradial bifurcation point i e if every neighborhood of u0 x0 contains solutions of 1 1 u a with u nonradial symmetry breaking problems for positive solutions of semilinear elliptic equa tions have been studied by smoller and wasserman 4 5 in fact they prove that for the function g c2 satisfying

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

a double loop hybrid algorithm for the traveling salesman problem with arbitrary neighbourhoods abstract this paper addresses the traveling salesman problem with arbitrary neighbourhoods which is an np hard problem in combinatorial optimization and is important in operations research and theoretical computer science existing methods based on neighbourhood predigestion and discretization are impractical in cases in which the neighbourhoods are arbitrary in this paper the neighbourhoods of the traveling salesman problem are generalized to arbitrarily connected regions in planar euclidean space a novel approach to solving this problem is proposed including a boundary based encoding scheme and a double loop hybrid algorithm based on particle swarm optimization and genetic algorithm in the hybrid algorithm linear descending inertia weight particle swarm optimization is adopted to search continuous visiting positions in the outer loop and the genetic algorithm is used to optimize the discrete visiting sequence in the inner loop the boundary based encoding scheme can reduce the search space significantly without degrading the solution quality and the hybrid algorithm can find a high quality solution in a reasonable time the computational results on both small and large instances demonstrate that the proposed approach can guarantee a high quality solution in a reasonable time compared with three other state of the art algorithms iterative algorithm branch and bound algorithm and upper and lower bound algorithm moreover the proposed approach works efficiently in a real world application that cannot be solved by existing algorithms

algorithms for preemptive scheduling of different classes of processors to do jobs with fixed times abstract we consider scheduling problems that involve processors of different classes and jobs with fixed start and finish times we allow preemption of jobs but impose restrictions on the assignment of processors to jobs we present polynomial time algorithms for finding the minimal cost preemptive solutions for two special cases in which the number of job classes is at most one more than the number of processor classes we then discuss a generalized version in which a processor can be assigned only to a subset of the jobs and a job can be done only by a subset of the processors we prove that for this case the problem of finding a preemptive feasible schedule with a given combination of processors can be solved in polynomial time by transforming it into a transportation network problem next we extend these results to cyclic scheduling problems further we discuss some transformations among these class scheduling problems and provide simpler proofs of np completeness for the nonpreemptive versions of the first two cases

a new genetic algorithm for solving nonconvex nonlinear programming problems abstract in nonlinear programming problems especially nonconvex problems attaining the global optimum is crucial to reach this purpose the current paper represents a new genetic algorithm for solving nonconvex nonlinear programming problems the new method is simpler and more intuitive than the existing models and finds the global optimum of the problem in a reasonable time the proposed technique to attain the global optimum of problem especially in large scale problems instead of increasing the size of population which usually conduces to the curse of dimensionality that is widespread in usual genetic algorithms decomposes the main problem into several sub problems but with lower size of population in each sub problem this decomposition has been formed in such a way that it could envelop the total solution space the proposed genetic algorithm which determines the sufficient sub problems could converge to global optimum of problem to be able to measure the proposed genetic algorithm several problems have been solved in different spectrums herein it has been shown that the proposed technique both in run time and in solution quality is preferred to the usual genetic algorithm in this domain nonlinear continuous programming problems

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

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

on the computational complexity of maximum class scheduling in this paper we consider several generalizations of the fixed job scheduling problem fsp which appear in a natural way in the aircraft maintenance process at an airport a number of jobs have to be carried out where the main attributes of a job are a fixed start time a fixed finish time a value representing the job s priority and a job class for carrying out these jobs a number of machines are available these machines can be split up into a number of disjoint machine classes for each combination of a job class and a machine class it is known whether or not it is allowed to assign a job in the job class to a machine in the machine class furthermore the jobs must be carried out in a non preemptive way and each machine can be carrying out at most one job at the same time within this setting one can ask for a feasible schedule for all jobs or if such a schedule does not exist for a feasible schedule for a subset of the jobs of maximum total value in this paper we present a complete classification of the computational complexity of two classes of combinatorial problems related this operational job scheduling problem

minimizing the number of machines for minimum length schedules in this paper we present a new objective function for scheduling on parallel machines minimizing the number of machines for schedules of minimum length we study its complexity and we prove the np completeness of this problem even if there is no precedences or for unitary execution times we propose several polynomial algorithms for various particular cases