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

multiobjective programming in optimization of interval objective functions a generalized approach abstract we generalize known concepts those introduced by ishihuchi and tanaka 1 and rommerlfanger et al 2 of the solution of the linear programming problem with interval coefficients in the objective function based on preference relations between intervals we unify all the discussed concepts as well as the corresponding solution methods into one general framework

multistate components assignment problem with optimal network reliability subject to assignment budget abstract the typical assignment problem for finding the optimal assignment of a set of components to a set of locations in a system has been widely studied in practical applications however this problem mainly focuses on maximizing the total profit or minimizing the total cost without considering component s failure in practice each component should be multistate due to failure partially failure or maintenance that is each component has several capacities with a probability distribution and may fail when a set of multistate components is assigned to a system the system can be treated as a stochastic flow network the network reliability is the probability that d units of homogenous commodity can be transmitted through the network successfully the multistate components assignment problem to maximize the network reliability is never discussed therefore this paper focuses on solving this problem under an assignment budget constraint in which each component has an assignment cost the network reliability under a components assignment can be evaluated in terms of minimal paths and state space decomposition subsequently an optimization method based on genetic algorithm is proposed the experimental results show that the proposed algorithm can be executed in a reasonable time

a revised bound improvement sequence algorithm abstract in this paper we present a generalization and a computational improvement of the bound improvement sequence algorithm the main computational burden of this algorithm consists in determining whether there exists a feasible point on the objective hyperplane when the algorithm encounters a fixed point by generalizing the algorithm such that the objective function and constraints are treated alike the number of fixed points that are required can be reduced the computational results that we report allow us to conclude that the number of fixed points can generally be reduced for loosely constrained problems for this class of problems the new algorithm appears to be more efficient than a standard mip code such as fmps

zero one integer programs with few constraints lower bounding theory abstract the zero one integer programming problem and its special case the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems we present several methods for computing lower bounds on the optimal solution of the zero one integer programming problem they include lagrangean surrogate and composite relaxations new heuristic procedures are suggested for determining good surrogate multipliers based on theoretical results and extensive computational testing it is shown that for zero one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used lagrangean and linear programming relaxations these results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero one integer programming problems

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

on the norm of a hilbert s type linear operator and applications in this paper the norm of a hilbert s type linear operator t l r l r r 1 r p q is given as applications a new operator inequality and the equivalent forms with the norm are obtained and particularly some new extended hilbert s type inequalities and the equivalent forms with the best constant factors are established

exact methods for the knapsack problem and its generalizations abstract a unified approach and a summary of the most important results concerned with exact methods for solving the binary knapsack problem and its generalizations are given we stress the importance of dual methods for solving linear programming relaxations of the considered problems two ways of generalization of the knapsack problem are described if the special ordered sets are added then the multiple choice knapsack problem is obtained if the constraints have the nested structure then we get the nested knapsack problem also the multiple choice nested knapsack problem is discussed

nonlinear 0 1 programming i linearization techniques any real valued nonlinear function in 0 1 variables can be rewritten as a multilinear function we discuss classes of lower and upper bounding linear expressions for multilinear functions in 0 1 variables for any multilinear inequality in 0 1 variables we define an equivalent family of linear inequalities this family contains the well known system of generalized covering inequalities as well as other linear equivalents of the multilinear inequality that are more compact i e of smaller cardinality in a companion paper 7 we discuss dominance relations between various linear equivalents of a multilinear inequality and describe a class of algorithms for multilinear 0 1 programming based on these results

a bilinear inequality we establish an inequality for symmetric bilinear forms involving both the norm and the inner product of vectors we use the inequality to convert known inequalities in real hilbert spaces such as classical hilbert s inequality to sharper inequalities

type ii blow up mechanisms in a semilinear heat equation with lepin exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant in the present article we prove constructively the existence of type ii blow up solutions for p 1 6 n 10 so called lepin exponent their blow up mechanisms are strongly influenced on space dimensions for lower dimensions the mechanisms are determined by dirac type approximations of nonlinear term and on the other hand dominated by classical taylor approximation for high dimensions on the threshold dimension the both approximations are balanced consequently the blow up exhibits a new blow up profile classification of radial blow up solutions in terms of blow up rates is also presented

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

on the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity this paper is concerned with a cauchy problem mbox p left begin array ll u t delta u u p quad mbox in bf r n times 0 infty u x 0 lambda varphi x quad mbox in bf r n end array right where p p ast equiv n 2 n 2 lambda 0 and varphi is a nonnegative radially symmetric function in c 1 bf r n with compact support denote the solution of p by u lambda let p ast infty if 3 leq n leq 10 and p ast 1 6 n 10 if n geq 11 we show that if p ast p p ast then there is lambda varphi 0 such that i if lambda lambda varphi then u lambda exists globally in time in the classical sense and u lambda t converges to zero locally uniformly in bf r n as t to infty ii if lambda lambda varphi then u lambda blows upincompletely in finite time iii if lambda lambda varphi then u lambda blows upcompletely in finite time

decay of heat semigroups in l and applications to nonlinear parabolic problems in unbounded domains we characterize all domains of rn such that the heat semigroup decays in l l or l l1 as t namely we prove that this property is equivalent to the poincare inequality and that it is also equivalent to the solvability of u f in l for all f l in particular under mild regularity assumptions on these properties are equivalent to the geometric condition that has finite inradius next we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains first we consider the quenching problem for singular parabolic equations we prove that the solution in quenches in finite time no matter how small the nonlinearity is if and only if does not fulfill the poincare inequality second for the semilinear heat equation with a power nonlinearity we prove roughly speaking that the trivial solution is stable in l or in l1 if and only if has finite inradius

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

the generalized assignment problem valid inequalities and facets three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem general properties of the facet defining inequalities are discussed and for a special case the convex hull is completely characterized in addition we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint

morrey spaces and classification of global solutions for a supercritical semilinear heat equation in rn abstract we prove the boundedness of global classical solutions for the semilinear heat equation u t u u p 1 u in the whole space r n with n 3 and supercritical power p n 2 n 2 this is proved without any radial symmetry or sign assumptions unlike in all the previously known results for the cauchy problem and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter examples moreover we show that any global classical solution has to decay in time faster than t 1 p 1 which is also optimal and in contrast with the subcritical case the proof relies on nontrivial modifications of techniques developed by chou du and zheng 8 and by blatt and struwe 3 5 for the case of convex bounded domains they are based on weighted energy estimates of giga kohn type combined with an analysis of the equation in a suitable morrey space this method actually works for any convex bounded or unbounded smooth domain but at the same time captures some of the specific behaviors associated with the case of the whole space r n as a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies and we show that the so called borderline global weak solutions blow up in finite time and then become classical again and decay as t all these results confirm the key role played by the morrey space m 2 4 p 1 in the understanding of the structure of the set of global solutions for p p s

global unbounded solutions of the fujita equation in the intermediate range we consider the fujita equation u t vardelta u u p on mathbb r n with n ge 3 we prove the existence of global unbounded solutions for p s p p jl where p s n 2 n 2 and begin aligned p jl left begin array l quad l frac n 2 2 4n 8 sqrt n 1 n 2 n 10 text if n 10 infty text if n le 10 end array right end aligned previously it was known that global unbounded solutions exist for p ge p jl whereas for p p s there are no such solutions

genetic algorithm for asymmetric traveling salesman problem with imprecise travel times this paper presents a variant of the asymmetric traveling salesman problem atsp in which the traveling time between each pair of cities is represented by an interval of values wherein the actual travel time is expected to lie instead of a fixed deterministic value as in the classical atsp here the atsp with interval objective is formulated using the usual interval arithmetic to solve the interval atsp i atsp a genetic algorithm with interval valued fitness function is proposed for this purpose the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used the proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators to analyze the performance and effectiveness of the proposed algorithm and different genetic operators computational studies of the proposed algorithm on some randomly generated test problems are reported

nonlinear 0 1 programming ii dominance relations and algorithms a nonlinear 0 1 program can be restated as a multilinear 0 1 program which in turn is known to be equivalent to a linear 0 1 program with generalized covering g c inequalities in a companion paper 6 we have defined a family of linear inequalities that contains more compact smaller cardinality linearizations of a multilinear 0 1 program than the one based on the g c inequalities in this paper we analyze the dominance relations between inequalities of the above family in particular we give a criterion that can be checked in linear time for deciding whether a g c inequality can be strengthened by extending the cover from which it was derived we then describe a class of algorithms based on these results and discuss our computational experience we conclude that the g c inequalities can be strengthened most of the time to an extent that increases with problem density in particular the algorithm using the strengthening procedure outperforms the one using only g c inequalities whenever the number of nonlinear terms per constraint exceeds about 12 15 and the difference in their performance grows with the number of such terms

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

classification of type i and type ii behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume p ps n 2n 2 and that the initial data is bounded possibly sign changing our first goal is to establish various characterizations of type i and type ii blow ups among many other things we show that the following conditions are equivalent a the blow up is of type ii b the rescaled solution w y s converges to either y or y as s where denotes the singular stationary solution c u x t x tends to 1 as x 0 where t is the blow up time our second goal is to study continuation beyond blow up among other things we show that if a blow up is of type i and incomplete then its limit l1 continuation becomes smooth immediately after blow up and that type i blow up implies type i regularization that is t t 1 p 1 u t l is bounded as t t we also give various criteria for complete and incomplete blow ups

scaling in nonlinear parabolic equations the cauchy problem for parabolic equations with quadratic nonlinearity is studied we investigate the existence of global in time solutions and their large time behavior assuming some scaling property of the equation as well as of the norm of the banach space in which the solutions are constructed some applications of these abstract results are indicated

on a new generalization of hardy hilbert s inequality and its applications abstract this paper deals with a generalization of the hardy hilbert inequality with best constant factor which involves the function as an application we obtain a new equivalent form of the hardy hilbert inequality

an optimal column generation with ranking algorithm for very large scale set partitioning problems in traffic assignment abstract we show in this paper how branch and bound and column generation techniques can be conbined very efficiently to solve to optimality some very large scale set partitioning problems with special structure such as the matrix decomposition problem arising in the context of satellite communication system optimization the main contribution of our approach is the use of column generation during the tree search procedure combined with a ranking procedure which ensures that the exact optimal integer solution is obtained extensive computational experiments are reported for the matrix decomposition problem encountered in the search for optimal schedules in satellite switching systems showing the effectiveness of this approach

boundedness of global solutions of nonlinear diffusion equations it is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t cc and other choices for which the solutions blow up in finite time if we are interested in other types of behavior we may consider the possibility p u t u exists globally

threshold and generic type i behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume that n 3 and p ps n 2n 2 our first goal is to analyze a threshold behavior for solutions with initial data u0 v where v c h1 and v 0 v 0 it is known that there exists 0 such that the solution converges to 0 as t if 0 with j j 1 2 k the blow up is complete and of type i with a flat local profile our method is based on a combination of the zero number principle and energy estimates in the second part of the paper we employ the very same idea to show that the constant solution attains the smallest rescaled energy among all non zero stationary solutions of the rescaled equation using this result we derive a sharp criterion for no blow up

generalized assignment with nonlinear capacity interaction the utility of the generalized assignment problem gap is well known in regard to resource allocation problems arising in the areas of production planning scheduling and facility location this paper introduces an important generalization of the gap which is called the 0 1 generalized assignment problem with nonlinear capacity constraints nlgap and which allows for capacity interaction among tasks assigned to the same agent for example nlgap can be used to model the hierarchical production planning problem involving the assignment of product families to production facilities in the case where process changeover gives rise to nonlinear capacity interaction among product families assigned to be produced at the same facility other applications of nlgap occur readily throughout the areas of production planning and scheduling we define a branch and bound algorithm for the nlgap this algorithm draws upon the underlying structure of the problem by combining in a novel manner recent advances in both nonlinear 0 1 programming and bounding techniques for the gap a heuristic for obtaining approximate solutions to nlgap is also defined we discuss computational experience with both the exact algorithm and the heuristic the computational results demonstrate that the heuristic is extremely effective in obtaining near optimal solutions and that the branch and bound algorithm can solve to optimality nlgaps with 5 agents 20 tasks and over 1000 nonlinear terms per constraint

stochastic binary problems with simple penalties for capacity constraints violations this paper studies stochastic programs with first stage binary variables and capacity constraints using simple penalties for capacities violations in particular we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly mathcal n mathcal p hard in general we provide pseudo polynomial algorithms for three special cases of the problem constant weights and capacity uniformly distributed subset sum with gaussian weights and strictly positively distributed random capacity and subset sum with constant weights and arbitrary random capacity we then turn to a branch and cut algorithm based on the outer approximation of the objective function we provide computational results for the stochastic knapsack problem i with gaussian weights and constant capacity and ii with constant weights and capacity uniformly distributed on randomly generated instances inspired by computational results for the knapsack problem

improved lagrangean decomposition an application to the generalized assignment problem abstract recently two new ways of obtaining improved lagrangean bounds have been suggested lagrangean decomposition and bound improving sequences in this work we will obtain a lagrangean approach combining the two ideas mentioned above theoretical results are provided about the sharpness of the bounds obtained by the combined approach for the general case as well as an application to the generalized assignment problem computational experience is reported

zero one integer programs with few constraints efficient branch and bound algorithms abstract the 0 1 integer programming problem and its special case the 0 1 knapsack problem are frequently encountered in modeling various design and decision making processes this paper is a follow up paper to 4 and deals with the development of an effective solution procedure for 0 1 integer programs with few constraints detailed computational experiments are carried out and different separation branching and bounding rules are compared using an experimental branch and bound code an efficient branch and bound procedure is developed tested and compared with previously developed optimal algorithms it is suggested that this procedure may also be used as a heuristic method for large problems by early termination of the tree search this scheme is tested and found to be very effective

blow up and global existence for the porous medium equation with reaction on a class of cartan hadamard manifolds abstract we consider the porous medium equation with power type reaction terms u p on negatively curved riemannian manifolds and solutions corresponding to bounded nonnegative and compactly supported data if p m small data give rise to global in time solutions while solutions associated to large data blow up in finite time if p m large data blow up at worst in infinite time and under the stronger restriction p 1 1 m 2 all data give rise to solutions existing globally in time whereas solutions corresponding to large data blow up in infinite time the results are in several aspects significantly different from the euclidean ones as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation

boundedness of global solutions of a supercritical parabolic equation we study radially symmetric classical solutions of the dirichlet problem for a heat equation with a supercritical nonlinear source we give a sufficient condition under which blow up in infinite time cannot occur this condition involves only the growth rate of the source term at infinity we do not need the homogeneity property which played a key role in previous proofs of similar results we also establish the blow up rate for a class of solutions which blow up in finite time

asymptotically self similar global solutions of a general semilinear heat equation we consider the general nonlinear heat equation partial t u delta u a u p 1 1 u g u u 0 varphi on 0 infty times i r n where a in i r p 1 1 2 n and g satisfies certain growth conditions we prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation we also prove that some of those global solutions are asymptotic for large time to self similar solutions of the single power nonlinear heat equation i e with g equiv 0

an application oriented guide for designing lagrangean dual ascent algorithms abstract lagrangean relaxation is an effective method for providing bounds in integer programming an efficient solution of the lagrangean dual often requires a specialized algorithm that must be tailored for each application model lagrangean dual ascent is a generic name that describes a broad class of algorithms lagrangean dual ascent is often preferred to subgradient optimization in solving a lagrangean dual since an ascent procedure guarantees monotonic bound improvement also if embedded within a branch and bound scheme lagrangean dual ascent may adjust only a few multipliers at each node thus conserving computer storage and enhancing processing efficiency this paper describes a framework to construct a lagrangean dual ascent procedure and adapts it for the generalized assignment problem and the constrained arborescence problem

a survey of algorithms for the generalized assignment problem abstract this paper surveys algorithms for the well known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded all approaches seem to be based on branch and bound with bound supplied through heuristics and through relaxations of the primal problem formulation from the survey one can select building blocks for the design of one s own tailor made algorithm the survey also reveals that although just about every mathematical programming technique was tried on this problem there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared as well as a shortage of effective heuristics

on a hilbert type linear series operator and its applications this paper deals with a hilbert type linear series operator and its norm several generalizations of the hilbert type inequalities whose kernels are symmetric and homogeneous of the first order are presented included are applications of some new extended hilbert type inequalities with the best possible constant factors and the equivalent forms are established also the reverse forms and some particular forms are obtained as special cases of the results of this paper

generating experimental data for the generalized assignment problem the generalized assignment problem gap is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine subject to capacity restrictions on the machines we propose a new stochastic model for the gap a tight condition on this stochastic model under which the gap is feasible with probability one when the number of jobs goes to infinity is derived this new stochastic model enables us to analyze the adequacy of most of the random generators given for the gap in the literature we demonstrate that the random generators commonly used to test solution procedures for the gap tend to create easier problem instances when the number of machines m increases we describe a greedy heuristic for the gap and use it to illustrate the results from the paper

an application of genetic algorithm in a marketing oriented inventory model with interval valued inventory costs and three component demand rate dependent on displayed stock level the objective of this research is to investigate an inventory model for a single item with imprecise inventory costs by considering the impact of marketing strategies such as pricing and advertising on three component demand rate this rate is dependent on selling price frequency of advertisement and displayed stock level dsl in a show room shop here the impreciseness of inventory costs like carrying cost purchase cost ordering cost and advertisement cost has been represented by interval valued numbers analyzing the relative size of the storage capacity of the show room shop and the stock level dependency parameters of demand different scenarios with sub scenarios of each have been mentioned then for each sub scenario the model has been formulated as a constrained optimization problem with interval objective to solve these problems an advanced genetic algorithm ga for mixed integer non linear programming has been developed with interval valued fitness function in this developed ga the order relations of interval valued numbers have been used with respect to pessimistic decision maker s point of view this approach has been used in the ranked based selection process for selecting better chromosomes individuals for the next generation and also for finding the best chromosomes individuals in each generation finally the model has been illustrated with some numerical examples and the performance of the ga has been tested by computing appropriate statistical measures and the computing time

a hilbert type integral inequality in the whole plane related to the hypergeometric function and the beta function abstract a new hilbert type integral inequality in the whole plane with the non homogeneous kernel and parameters is given the constant factor related to the hypergeometric function and the beta function is proved to be the best possible as applications equivalent forms the reverses some particular examples two kinds of hardy type inequalities and operator expressions are considered

technical note an improved dual based algorithm for the generalized assignment problem the generalized assignment problem gap determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent subject to an agent s capacity existing solution algorithms have not solved problems with more than 100 decision variables this paper designs an optimization algorithm for the gap that effectively solves problems with up to 500 variables compared with existing procedures this algorithm requires fewer enumeration nodes and shorter running times improved performance stems from an enhanced lagrangian dual ascent procedure that solves a lagrangian dual at each enumeration node adding a surrogate constraint to the lagrangian relaxed model and an elaborate branch and bound scheme an empirical investigation of various problem structures not considered in existing literature is also presented

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

elitist genetic algorithm for assignment problem with imprecise goal the objective of this research paper is to solve a generalized assignment problem with imprecise cost s time s instead of precise one by elitist genetic algorithm ga here the impreciseness of cost s time s has been represented by interval valued numbers as interval valued numbers are the best representation than others like random variable representation with a known probability distribution and fuzzy representation to solve these types of problems an elitist ga has been developed with interval valued fitness function in this developed ga the existing ideas about the order relations of interval valued numbers have been modified from the point of view of two types of decision making viz optimistic decision making and pessimistic decision making this modified approach has been used in the selection process for selecting better chromosomes individuals for the next generation and in finding the best as well as the worst chromosomes individuals in each generation here two new crossover schemes and two new mutation schemes have been introduced in order to maintain the feasibility with crossover operations a repair algorithm has been suggested extensive comparative computational studies based on different parameters of our developed algorithm on one illustrative example have also been reported

relaxation heuristics for a generalized assignment problem abstract we propose relaxation heuristics for the problem of maximum profit assignment of n tasks to m agents n m such that each task is assigned to only one agent subject to capacity constraints on the agents using lagrangian or surrogate relaxation the heuristics perform a subgradient search obtaining feasible solutions relaxation considers a vector of multipliers for the capacity constraints the resolution of the lagrangian is then immediate for the surrogate the resulting problem is a multiple choice knapsack that is again relaxed for continuous values of the variables and solved in polynomial time relaxation multipliers are used with an improved heuristic of martello and toth or a new constructive heuristic to find good feasible solutions six heuristics are tested with problems of the literature and random generated problems best results are less than 0 5 from the optimal with reasonable computational times for an at 386 computer it seems promising even for problems with correlated coefficients

logic cuts for multilevel generalized assignment problems abstract in the multilevel generalized assignment problem mgap agents can perform tasks at more than one efficiency level important manufacturing problems such as lot sizing can be easily formulated as mgaps however the large number of variables in the related 0 1 integer program makes it hard to find optimal solutions to these problems even when using powerful commercial optimization packages the mgap includes a set of knapsack constraints one per agent that can be useful for generating simple logical constraints or logic cuts we exploit the fact that logic cuts can be generated in linear time and can be easily added to the model before solving it with classical branch and bound methodology we generate all contiguous 1 cuts for every knapsack in large mgap s problems and report the effect of adding these cuts in our experimental results

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

a new lagrangian relaxation approach to the generalized assignment problem abstract in this paper we present a new lagrangian relaxation approach to the generalized assignment problem gap the new approach is based on a reformulation of gap into an equivalent problem which is then relaxed by traditional lagrangian relaxation techniques the reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints by relaxing the coupling constraints we get subproblems where both types of constraint structures present in the gap are active the lower bound from the lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional lagrangian relaxation approaches an important feature is also the possibility of generating upper bounds quite easily the new approach has been tested on 10 problems of size m 4 n 25 i e 100 0 1 variables and the results have been compared with the traditional approaches

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

algorithms for the generalized quadratic assignment problem combining lagrangean decomposition and the reformulation linearization technique in this paper we propose two exact algorithms for the gqap generalized quadratic assignment problem in this problem given m facilities and n locations the facility space requirements the location available space the facility installation costs the flows between facilities and the distance costs between locations one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized this problem generalizes the well known quadratic assignment problem qap both exact algorithms combine a previously proposed branch and bound scheme with a new lagrangean relaxation procedure over a known rlt reformulation linearization technique formulation we also apply transformational lower bounding techniques to improve the performance of the new procedure we report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time six of these instances were open

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

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs