some model theory of polish structures we introduce a notion of polish structure and in doing so provide a setting which allows the application of ideas and techniques from model theory descriptive set theory topology and the theory of profinite groups we define a topological notion of independence in polish structures and prove that it has some nice properties using this notion we prove counterparts of some basic results from geometric stability theory in the context of small polish structures then we prove some structural theorems about compact groups regarded as polish structures each small nm stable compact g group is solvable by finite each small compact g group of finite n m rank is nilpotent by finite examples of small polish structures and groups are also given

the multi handler knapsack problem under uncertainty the multi handler knapsack problem under uncertainty is a new stochastic knapsack problem where given a set of items characterized by volume and random profit and a set of potential handlers we want to find a subset of items which maximizes the expected total profit the item profit is given by the sum of a deterministic profit plus a stochastic profit due to the random handling costs of the handlers on the contrary of other stochastic problems in the literature the probability distribution of the stochastic profit is unknown by using the asymptotic theory of extreme values a deterministic approximation for the stochastic problem is derived the accuracy of such a deterministic approximation is tested against the two stage with fixed recourse formulation of the problem very promising results are obtained on a large set of instances in negligible computing time

the shortest path problem with two objective functions abstract we present methods to find the shortest path in a network where each path is associated with two objectives we describe how to obtain the nondominated paths and the extreme nondominated paths and compare the expected complexity of the methods an improvement in efficiency can be obtained when quasiconcave or quasiconvex utility functions are assumed in the first case we describe how to find the optimal extreme nondominated path and bounds for the optimal path value then the optimal path can be located by calculating the k th shortest path in the second case we suggest a branch and bound method to solve the problem

approximation of pareto optima in multiple objective shortest path problems we study methods for approximating the set of pareto optimal paths in multiple objective shortest path problems known generalizations of standard shortest path methods will compute this set but can suffer from rapidly increasing computational and storage demands as problem size increases in an effort to avoid such difficulties we develop approximation methods that can estimate the pareto optima to any required degree of accuracy the approximation methods are fully polynomial that is they operate in time and space bounded by a polynomial in problem size and accuracy of approximation the greater the accuracy the more time required to reach a solution we show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of np complete single objective problems

system optimal routing of traffic flows with user constraints in networks with congestion the design of route guidance systems faces a well known dilemma the approach that theoretically yields the system optimal traffic pattern may discriminate against some users in favor of others proposed alternate models however do not directly address the system perspective and may result in inferior performance we propose a novel model and corresponding algorithms to resolve this dilemma we present computational results on real world instances and compare the new approach with the well established traffic assignment model the essence of this study is that system optimal routing of traffic flow with explicit integration of user constraints leads to a better performance than the user equilibrium while simultaneously guaranteeing superior fairness compared to the pure system optimum

adaptivity in the stochastic blackjack knapsack problem we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size in every stage of insertion if the subset of the items inserted thus far is feasible then it has a total value that equals to the sum of values of all items of this subset otherwise if the subset violates the constraint then its value equals to zero the goal is to compute a policy for insertion of the items that maximizes the expected total value of items placed in the knapsack we consider both non adaptive policies that designate a priori a fixed subset of items to insert and adaptive policies that can make dynamic decisions based on the instantiated sizes of items placed in the knapsack thus far our work characterizes the benefit of adaptivity for this purpose we use a measure called the adaptivity gap the supremum over instances of the ratio between the expected value obtained by an optimal adaptive policy and the expected value obtained by an optimal non adaptive policy first we show a tight bound of 32 on the adaptivity gap for the case of inputs consisting of only two items then we present a non adaptive policy with expected value that is at least 2 1 2 2 1 11 66 times the expected value of the optimal adaptive policy thus the adaptivity gap in this model is at most 11 66 additionally this non adaptive policy is computed in polynomial time finally we consider a special case of the model where all sizes are distributed according to bernoulli distribution with different parameters for this special case we improve our result and bound the adaptivity gap by 8

a class of variance constrained problems a lagrange multiplier solution procedure is developed for a class of variance constrained problems the method based on the separation of the variance and an algorithm for determining the optimal value of the mean relies on the availability of a method of solving a modified problem in which the variance operation is replaced by the generalized variance operator it is indicated that the procedure can be used in conjunction with dynamic programming to solve a variety of variance constrained problems which allow the consideration of adaptive strategies the procedure is demonstrated by a simple numerical example

reference points and approximation algorithms in multicriteria discrete optimization abstract mathematical research on multicriteria optimization problems predominantly revolves around the set of pareto optimal solutions in practice on the other hand methods that output a single solution are more widespread reference point methods are a successful example of this approach and are widely used in real world multicriteria optimization a reference point solution is the solution closest to a given reference point in the objective space we study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets in particular we establish that in terms of computational complexity computing approximate reference point solutions is polynomially equivalent to approximating the pareto set complementing these results we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization in particular we lift the link between dynamic programming and fptas as well as certain lp rounding techniques the latter applies e g to set cover and several machine scheduling problems

preference structure representation using convex cones in multicriteria integer programming a new efficient system of representing the decision maker s preference structure in solving multicriteria integer programming problems is developed the problem is solved by an interactive branch and bound method that employs the procedure of zionts and wallenius zionts s j wallenius 1983 an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions management sci 295 for multicriteria linear programming the decision maker s underlying utility function is assumed to be pseudoconcave and his pairwise comparisons of decision alternatives are used to determine his preference structure in terms of certain convex cones in the objective function space and constraints on the weights on the objectives in the weight space the two forms of preference structure representation are interrelated and their underlying theory is developed the primary objective of a representation scheme is exactness and in this respect it is shown that the constraints on the weights are not adequate for representing nonlinear utility functions on the other hand the convex cones exactly represent any quasiconcave utility function and clearly avoid the approximations and inaccuracies in other utility assessment systems accordingly an efficient ordered representation using convex cones is developed an algorithmic framework for multicriteria integer programming that integrates the representation using convex cones with the branch and bound solution procedure is developed computational experience with bicriteria problems having up to 80 variables and 40 constraints is presented

small approximate pareto sets for biobjective shortest paths and other problems we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy epsilon the pareto curve of a multiobjective optimization problem we show that for a broad class of biobjective problems containing many important widely studied problems such as shortest paths spanning tree matching and many others we can compute in polynomial time an epsilon pareto set that contains at most twice as many solutions as the minimum set furthermore we show that the factor of 2 is tight for these problems i e it is np hard to do better we present upper and lower bounds for three or more objectives as well as for the dual problem of computing a specified number k of solutions which provide a good approximation to the pareto curve

the median tour and maximal covering tour problems formulations and heuristics abstract in this paper the authors introduce two bicriterion routing problems the median tour problem mtp and the maximal covering tour problem mctp in both problems the tour must visit only p of the n nodes on the network in addition both problems have as one of their objectives the minimization of total tour length the second objective in both problems maximizes access to the tour for the nodes not directly on it in the mtp the access objective is the minimization of the total travel distance necessary for demand at the nodes to reach their nearest stop on the tour in the mctp the access objective is to maximize the total demand within some prespecified maximal travel distance from a tour stop it is shown that the mctp may be viewed as a special case of the mtp consequently solution procedures developed for the mtp may also be employed to solve the mctp unfortunately both problems are np hard in addition the number of efficient solutions may grow exponentially with the number of nodes in the particular problem instance consequently a heuristic procedure is developed to generate a good approximation of the efficient frontier not only does this approximation provide the decision marker with insight into the magnitudes of the inherent tradeoffs between the two objectives but it also provides him or her with a set of feasible options to consider computational experience on a realistically scaled problem is provided applications of the problems include the design of mobile service delivery systems e g healthcare delivery in rural areas of developing countries bi modal transportation systems e g overnight mail delivery and distributed computer networks among others

decision making based on approximate and smoothed pareto curves we consider bicriteria optimization problems and investigate the relationship between two standard approaches to solving them i computing the pareto curve and ii the so called decision maker s approach in which both criteria are combined into a single usually nonlinear objective function previous work by papadimitriou and yannakakis showed how to efficiently approximate the pareto curve for problems like shortest path spanning tree and perfect matching we wish to determine for which classes of combined objective functions the approximate pareto curve also yields an approximate solution to the decision maker s problem we show that an fptas for the pareto curve also gives an fptas for the decision maker s problem if the combined objective function is growth bounded like a quasi polynomial function if the objective function however shows exponential growth then the decision maker s problem is np hard to approximate within any polynomial factor in order to bypass these limitations of approximate decision making we turn our attention to pareto curves in the probabilistic framework of smoothed analysis we show that in a smoothed model we can efficiently generate the complete and exact pareto curve with a small failure probability if there exists an algorithm for generating the pareto curve whose worst case running time is pseudopolynomial this way we can solve the decision maker s problem w r t any non decreasing objective function for randomly perturbed instances of e g shortest path spanning tree and perfect matching

approximation schemes for the restricted shortest path problem this note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint the main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value whose ratio is polynomially bounded are known in spite of this difficulty one of the algorithms presented here is strongly polynomial applications to other problems are also discussed

locally finite profinite rings abstract we investigate the structure of locally finite profinite rings we classify jacobson semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields and we prove that the jacobson radical of any locally finite profinite ring is nil of finite nilexponent our results apply to the context of small compact g rings where we also obtain a description of possible actions of g on the underlying ring

the multiobjective discrete optimization problem a weighted min max two stage optimization approach and a bicriteria algorithm we study the multiple objective discrete optimization modo problem and propose two stage optimization problems as subproblems to be solved to obtain efficient solutions the mathematical structure of the first level subproblem has similarities to both tchebycheff type of approaches and a generalization of the lexicographic max ordering problem that are applicable to multiple objective optimization we present some results that enable us to develop an algorithm to solve the bicriteria discrete optimization problem for the entire efficient set we also propose a modification of the algorithm that generates a sample of efficient solutions that satisfies a prespecified quality guarantee we apply the algorithm to solve the bicriteria knapsack problem our computational results on this particular problem demonstrate that our algorithm performs significantly better than an equivalent tchebycheff counterpart moreover the computational behavior of the sampling version is quite promising

a multicriteria pareto optimal path algorithm abstract in this paper we present an algorithm for finding in a multicriteria network pareto optimal paths one each for each efficient objective vector validation of the algorithm and two illustrative examples are included computational experience is also reported this algorithm is a generalization of an earlier one for the bicriterion case

optimal paths in bi attribute networks with fractional cost functions abstract an important routing problem is to determine an optimal path through a multi attribute network which minimizes a cost function of path attributes in this paper we study an optimal path problem in a bi attribute network where the cost function for path evaluation is fractional the problem can be equivalently formulated as the bi attribute rational path problem which is known to be np complete we develop an exact approach to find an optimal simple path through the network when arc attributes are non negative the approach uses some path preference structures and elimination techniques to discard from further consideration those partial paths that cannot be parts of an optimal path our extensive computational results demonstrate that the proposed method can find optimal paths for large networks in very attractive times

enumeration and interactive selection of efficient paths in a multiple criteria graph for scheduling an earth observing satellite we consider a satellite following orbits around the earth in order to take shots corresponding to images requested by various customers the daily operations of such a satellite consist of defining a feasible and satisfactory shot sequence this problem involves both combinatorial and multiple criteria difficulties indeed the number of feasible shot sequences grows significantly with the number of images asked for and the evaluation of a shot sequence is based on several conflicting criteria we propose to formulate this problem as the selection of a multiple criteria path in a graph without circuit our approach for solving this problem involves two stages generation of efficient paths and selection of a satisfactory path using a multiple criteria interactive procedure

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

an efficient time and space k point to point shortest simple paths algorithm abstract we address the problem identifying the k best point to point simple paths connecting a given pair of nodes in a directed network with arbitrary lengths the main result in this paper is the proof that a path tree containing the kth point to point shortest simple path can be obtained by using one of the previous k 1 path trees containing each one of the previous k 1 best point to point shortest simple paths the proof implies that at most n single source shortest path computations re optimizations in a network with non negative length arcs are made in each iteration of the method in the optimistic case this strategy only needs o m time to compute the best neighbor associated with a path tree that is the second shortest simple path given a shortest simple path the algorithm runs in o k nf n m cmax time and uses o k m space to determine the k point to point shortest simple paths in a directed network with n nodes m arcs and maximum absolute length cmax here o f n m cmax is the best time needed to determine the shortest simple paths connecting a source node with any other non source node in a network with non negative length arcs we improve required space in yen s algorithm by a multiplicative factor of o n2 for each best solution moreover our algorithm runs in the optimistic case in o kf n m cmax time this affirmation is confirmed by an experimental study where o k shortest paths are used to determine the k point to point shortest simple paths in two versions of our algorithm

the minmax regret robust shortest path problem in a finite multi scenario model the robust shortest path problem is a network optimization problem that can be defined to deal with uncertainty of costs associated with the arcs of a network two models have been considered for the robust shortest path problem interval data and discrete data sets this work addresses the robust shortest path problem with a minmax regret objective function on a finite multi scenario model this problem consists in finding an optimal path in the sense that it has the minimum maximum deviation from the shortest one over all scenarios with this goal some properties of the problem and of its optimal solutions are derived these results allow to introduce three approaches a labeling algorithm an algorithm based on ranking loopless paths and a hybrid algorithm which ranks loopless paths in a suitable way to apply the early elimination of useless solutions the algorithms are tested on random networks and compared with a previous method for the same problem the obtained computational results are reported and discussed they show that the labeling and the hybrid approaches outperform the others

approximating the stochastic knapsack problem the benefit of adaptivity we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size our goal is to compute a solution policy that maximizes the expected value of items successfully placed in the knapsack where the final overflowing item contributes no value we consider both nonadaptive policies that designate a priori a fixed sequence of items to insert and adaptive policies that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far an important facet of our work lies in characterizing the benefit of adaptivity for this purpose we advocate the use of a measure called the adaptivity gap the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy we bound the adaptivity

integrating partial optimization with scatter search for solving bi criteria 0 1 knapsack problems abstract this paper deals with the problem of inaccuracy of the solutions generated by metaheuristic approaches for combinatorial optimization bi criteria 0 1 knapsack problems a hybrid approach which combines systematic and heuristic searches is proposed to reduce that inaccuracy in the context of a scatter search method the components of this method are used to determine regions in the decision space to be systematically searched comparisons with small and medium size instances solved by exact methods are presented large size instances are also considered and the quality of the approximation is evaluated by taking into account the proximity to the upper frontier devised by the linear relaxation and the diversity of the solutions comparisons with other two well known metaheuristics are also performed the results show the effectiveness of the proposed approach for both small medium and large size instances

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

an interactive branch and bound algorithm for multiple criteria optimization we present a new interactive algorithm for multiple criteria optimization the algorithm is of the branch and bound type and differs from previous interactive algorithms in several ways first the field of application is wider because it applies to two important classes of multiple criteria decision problems those for which the feasible set is convex and those for which the feasible set is discrete secondly the algorithm does not require a great deal from the decision maker he is merely required to indicate his preference between two vectors whenever the algorithm so demands

the bicriterion semi obnoxious location bsl problem solved by an ϵ approximation abstract locating an obnoxious undesirable facility is often modeled by the maximin or maxisum problem but the obnoxious facility is often placed unrealistically far away from the demand points nodes resulting in prohibitively high transportation cost time one solution is to model the problem as a semi obnoxious location problem here we model the problem as a bicriterion problem not in advance determining the importance of the obnoxious objective compared to the cost time objective we consider this model for both the planar and the network case the two problems are solved by an approximation algorithm and the models are briefly compared by means of a real life example

computation of ideal and nadir values and implications for their use in mcdm methods abstract in this paper we investigate the problem of finding the nadir point for multicriteria optimization problems mop the nadir point is characterized by the componentwise maximal values of efficient points for mop it can be easily computed in the bicriteria case however in general this problem is very difficult we review some existing methods and heuristics and also discuss some new ones we propose a general method to compute nadir values based on theoretical results on pareto optimal solutions of subproblems with fewer criteria the general scheme is valid for any number of criteria and practical for the case of three objectives we also investigate the use of the nadir point for compromise programming when the goal is to be as far away as possible from the worst outcomes we prove some results about weak pareto optimality of the resulting solutions and present examples which show the limitations of this idea and therefore limitations of multicriteria methods using this type of problems

an empirical investigation of some bicriterion shortest path algorithms abstract several applications involve finding shortest paths in bicriterion networks those in which each arc possesses two characteristics such as time and cost in these circumstances it is appropriate to find efficient pareto optimal paths between various points in the network that is paths are sought for which no alternative path has a smaller value for the first characteristic without a larger value for the second in this paper several implementations of a general label correcting algorithm are discussed for bicriterion networks these implementations are then compared on networks of varying size and having varying degrees of correlation between the two criteria the empirical findings shed some light on the ability to solve bicriterion problems in practice and on those factors that significantly influence the computational effort

transversality of cr mappings we prove here new results about transversality and related geometric properties of a holomorphic formal or cr mapping sending one generic submanifold of c into another one of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type for the case of hypersurfaces transversality in this context was proved by baouendi and the second author in 1990 the general case of generic manifolds of higher codimension which we treat in this paper had remained an open problem since then applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism

shortest path problems with partial information models and algorithms for detecting dominance abstract in this work we focus on partial information models for the well known shortest path problem where we consider multiple instances of values for the parameters that determine the cost of each arc this allows coping with the uncertainty about the future the imprecision of data the arbitrariness of some options the evolving values of the decision makers dms and or the multiplicity of dms in group decision making this paper proves some results and presents detailed algorithms to identify the set of non dominated paths a concept from decision theory under partial information we first address problems with a finite set of instances then problems with a general eventually not discrete set of instances and finally we study a particular case of the latter which complies with a condition that may hold in some situations to deal with these partial information problems we propose a new use for existing multicriteria algorithms based on the ranking of shortest paths

discrete dynamic programming with outcomes in random variable structures abstract multiobjective approach is the common way of generalization single criterion dynamic programming models another way is to consider partially ordered criteria structures that approach is rather rare the aim of the paper is to present such a model generalization of bellman s principle of optimality is employed to create a forward procedure to find the set of all maximal elements as this set is usual large the second problem under consideration is to find its subsets to reduce the number of solutions presented to decision maker we propose to apply a family of narrowing relations that approach is similar to scalarization in multiobjective programming ordered structures of random variables based on mean variance stochastic dominance and inverse stochastic dominance are considered numerical illustration is given at the end of the paper

an enhanced k sp algorithm with pruning strategies to solve the constrained shortest path problem a new exact algorithm for the constrained shortest path problem is presented extensive experiments comparing the behavior of the new scheme and the recent best method from the literature are provided the proposed approach solves the constrained shortest path problem in usa road networks containing millions of nodes and arcs we propose an exact method to solve the constrained shortest path csp problem the new approach takes advantage of the binary partition strategy of a recent k shortest paths k sp algorithm that allows posing numerous additional path constraints in the model without extra difficulty we adapt and incorporate several pruning strategies from the literature on the csp problem in this scheme and obtain a robust and scalable algorithm the method is compared with the current best algorithm to solve the csp problem and produces significant speedups in a wide range of network configurations an example is given where the csp problem is solved in road networks containing a million nodes and arcs

the most critical path in a pert network a heuristic approach abstract the classical pert approach uses a deterministic critical path to estimate the probability of completing a project by a given deadline in general such a path is not the most critical path and does not provide the lowest probability of completion time this paper studies the most critical path problem and formulates it as an optimal path problem through a deterministic network with a two attribute fractional objective function a heuristic approach is developed for the optimal path problem which gives a solution to the most critical path problem the illustrative examples and computational results demonstrate that the proposed procedure provides critical activities and estimates for the probabilities of completion time that are more accurate than those resulting from the classical pert approach further our computational experiments show that the proposed heuristic performs well in identifying the most critical paths

on the bicriterion minimal cost minimal label spanning tree problem we address a bicriterion spanning tree problem relevant in some application fields such as telecommunication networks or transportation networks each edge is assigned with a cost value and a label such as a color the first criterion intends to minimize the total cost of the spanning tree the summation of its edge costs while the second intends to get the solution with a minimal number of different labels since these criteria in general are conflicting criteria we developed an algorithm to generate the set of non dominated spanning trees computational experiments are presented and results discussed

on a multicriteria shortest path problem abstract multicriteria shortest path problems have not been treated intensively in the specialized literature despite their potential applications in fact a single objective function may not be sufficient to characterize a practical problem completely for instance in a road network several parameters as time cost distance etc can be assigned to each arc clearly the shortest path may be too expensive to be used nevertheless the decision maker must be able to choose some solution possibly not the best for all the criteria in this paper we present two algorithms for this problem one of them is an immediate generalization of the multiple labelling scheme algorithm of hansen for the bicriteria case based on this algorithm it is proved that any pair of nondominated paths can be connected by nondominated paths this result is the support of an algorithm that can be viewed as a variant of the simplex method used in continuous linear multiobjective programming a small example is presented for both algorithms

small profinite structures we propose a model theoretic framework for investigating profinite structures we prove that in many cases small profinite structures interpret infinite groups this corresponds to results of hrushovski and peterzil on interpreting groups in locally modular stable and o minimal structures

a parametric approach to solving bicriterion shortest path problems abstract in this paper a new algorithm is developed to solve bicriterion shortest path problems bsp this algorithm first relaxes the integrality conditions and solves a simple bicriterion network problem the bicriterion network problem is solved parametrically exploiting properties associated with adjacent basis trees those pareto optimal paths not obtained by solving the lp relaxation are obtained using a label correcting procedure computational results comparing the parametric approach to the label setting approach and the k th shortest path approach are also presented they indicate that the parametric approach is orders of magnitude faster than the k th shortest path approach for most problems tested for problems with a positive correlation between the two cost coefficients the parametric approach is seen to be significantly faster than the label setting approach

the interactive analysis of the multicriteria shortest path problem by the reference point method abstract the multicriteria shortest path problem is considered the paper presents the interactive method of analyzing this problem by the reference point approach the reference point guided labeling algorithm was developed this algorithm finds the pareto optimal shortest path which is best attuned to the specified preferences

a general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into one in this paper we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function examples of such problems include those in which the objective function is a product or ratio of two linear functions parallel machine scheduling problems with the makespan objective robust versions of weighted multiobjective optimization problems and assortment optimization problems with logit choice models the main idea behind our approximation schemes is the construction of an approximate pareto optimal frontier of the functions that constitute the given objective using this idea we give the first fully polynomial time approximation schemes for the max min resource allocation problem with a fixed number of agents combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions or the product of a fixed number of linear functions and assor

technical note the adaptive knapsack problem with stochastic rewards given a set of items with associated deterministic weights and random rewards the adaptive stochastic knapsack problem adaptive skp maximizes the probability of reaching a predetermined target reward level when items are inserted sequentially into a capacitated knapsack before the reward of each item is realized this model arises in resource allocation problems that permit or require sequential allocation decisions in a probabilistic setting one particular application is in obsolescence inventory management in this paper the adaptive skp is formulated as a dynamic programming dp problem for discrete random rewards the paper also presents a heuristic that mixes adaptive and static policies to overcome the curse of dimensionality in the dp the proposed heuristic is extended to problems with normally distributed random rewards the heuristic can solve large problems quickly and its solution always outperforms a static policy the numerical study indicates that a near optimal solution can be obtained by using an algorithm with limited look ahead capabilities

multilevel dynamic programming for general multiple linear quadratic control in discrete time systems abstract this paper extends the reach of dynamic programming to general multiple linear quadratic control problems in discrete time systems where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel dynamic programming solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem at the lower level the formulated auxiliary lagrangian problem is of a parametric linear quadratic structure and is solved by dynamic programming the weighting vector in the auxiliary lagrangian problem is adjusted by the upper level iteratively this two level process repeats until an optimal condition is satisfied

on general multiple linear quadratic control problems general multiple linear quadratic control problems are studied where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem the auxiliary lagrangian problem formulated at the lower level is of a parametric linear quadratic structure in which the weighting vector is adjusted by the upper level until a derived optimal condition is met

maximizing classes of two parameter objectives over matroids let m n be a matroid suppose that each element i in n is associated with an ordered pair of rational numbers ai bi for each subset s in define a s i s ai and b s i s bi let g be a real convex function defined on r2 consider the problem of maximizing g a s b s over all bases s of m we present a polynomial algorithm for this problem when g is a general polynomial this algorithm is strongly polynomial when the degree of g is at most cubic using the latter result we apply appropriate transformations to obtain strongly polynomial algorithms for some cases when g is not polynomial in particular we find in strongly polynomial time the minimal cost reliability ratio spanning tree of an undirected graph

on the k shortest path trees problem abstract we address the problem of finding the k best path trees connecting a source node with any other non source node in a directed network with arbitrary lengths the main result in this paper is the proof that the k th shortest path tree is adjacent to at least one of the previous k 1 shortest path trees consequently we design an o f n m c max km time and o k m space algorithm to determine the k shortest path trees in a directed network with n nodes m arcs and maximum absolute length c max where o f n m c max is the best time needed to solve the shortest simple paths connecting a source node with any other non source node

network reduction for the acyclic constrained shortest path problem abstract this paper presents a procedure for reducing networks used with the constrained shortest path problem the network considered herein is acyclic with each arc having two attributes time and length with a time limit on the shortest length path the procedure traverses the network twice and depends on establishing time labels for the nodes similar to those established in shortest path algorithms of the label setting type these labels are used to identify the arcs and nodes that render a path infeasible the computational requirements of the procedure are modest which should enhance the application of the constrained shortest path problem

risk criteria in a stochastic knapsack problem we consider an investor who wants to allocate funds among several projects each project is expected to yield a certain reward and the objective is that a total reward will achieve a certain given amount called the target this problem is relatively easy to solve when rewards are deterministic but may be hard in a more realistic setting when the rewards are stochastic and the investor wants to maximize the probability of attaining the target we show that by combining dynamic programming with a search procedure the stochastic version of the problem can be solved relatively fast when rewards are normally distributed the procedure is also useful for other risk criteria which involve both the mean and the variance of the total reward

a bicriterion shortest path algorithm abstract among the network models one of the more popular is the so called shortest path problem this model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network often a single objective function is not sufficient to completely characterize some real world problems for instance in a road network two parameters as cost and time can be assigned to each arc clearly the fastest path may be too costly nevertheless the decision maker must choose one solution possibly not the best for both criteria in this paper we present an algorithm for this problem with this algorithm a special set of paths the set of pareto optimal paths is determined one objective for any pareto optimal path can not be improved without worsening the other one

a general decomposition approach for multi criteria decision trees in this paper we address the dynamic and multi criteria decision making problems under uncertainty generally represented by multi criteria decision trees decision making consists of choosing at each period a decision that maximizes the decision maker outcomes these outcomes should often be measured against a set of heterogeneous and conflicting criteria generating the set of non dominated solutions is a common approach considered in the literature to solve the multi criteria decision trees but it becomes very challenging for large problems we propose a new approach to solve multi criteria decision trees without generating the set of all non dominated solutions which should reduce the computation time and the cardinality of the solution set in particular the proposed approach combines the advantages of decomposition with the application of multi criteria decision aid mcda methods at each decision node a generalization of the bellman s principle of decomposition to the multi criteria context is put forward a decomposition theorem is therefore proposed under the sufficient conditions stated by the theorem the principle of decomposition will generate the set of best compromise strategies seven mcda methods are then characterized lexicographic weighted sum multi attribute value theory topsis electre iii and promethee ii against the conditions of the theorem of decomposition and against other properties neutrality anonymous fidelity dominance independency in order to confirm or infirm their applicability with the proposed decomposition principle moreover the relationship between independency and temporal consistence is discussed as well as the effects of incomparableness rank reversals and use of thresholds two conjectures resulted from this characterization

an exact e constraint method for bi objective combinatorial optimization problems application to the traveling salesman problem with profits abstract this paper describes an exact constraint method for bi objective combinatorial optimization problems with integer objective values this method tackles multi objective optimization problems by solving a series of single objective subproblems where all but one objectives are transformed into constraints we show in this paper that the pareto front of bi objective problems can be efficiently generated with the constraint method furthermore we describe heuristics based on information gathered from previous subproblems that significantly speed up the method this approach is used to find the exact pareto front of the traveling salesman problem with profits a variant of the traveling salesman problem in which a profit or prize value is associated with each vertex the goal here is to visit a subset of vertices while addressing two conflicting objectives maximize the collected prize and minimize the travel costs we report the first exact results for this problem on instances derived from classical vehicle routing and traveling salesman problem instances with up to 150 vertices results on approximations of the pareto front obtained from a variant of our exact algorithm are also reported

semi infinite relaxations for the dynamic knapsack problem with stochastic item sizes we consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it after every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items while an unsuccessful insertion terminates the process we propose a new semi infinite relaxation based on an affine value function approximation and show that an existing pseudo polynomial relaxation corresponds to a nonparametric value function approximation we compare both theoretically to other relaxations from the literature and also perform a computational study our main empirical conclusion is that our new relaxation provides tight bounds over a variety of different instances and becomes tighter as the number of items increases

relaxation analysis for the dynamic knapsack problem with stochastic item sizes we consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it after every successful insertion the decision maker ca

generalized dynamic programming for multicriteria optimization abstract applications of dynamic programming dp to multicriteria sequential decision problems involving the optimization of a multicriteria preference function have been rare this is due to the ease with which dp s monotonicity assumption can be violated in such situations generalized dp avoids the potential pitfalls created by this absence of monotonicity thereby guaranteeing optimality the methodology is applied to a prototypical multicriteria dp problem namely a multicriteria version of the shortest path problem

iterative parametric minimax method for a class of composite optimization problems this paper considers a class of composite optimization problems that are often difficult to solve directly due to large dimension nonlinearity nonseparability and or nonconvexity of the problem an iterative parametric minimax method is proposed in which the original optimization problem is embedded into a weighted minimax formulation the resulting auxiliary parametric optimization problems at the lower level often have simple structures that are readily tackled by efficient solution strategies such as the decomposition scheme in dynamic programming and in the primal dual method the analytical expression of the partial derivatives of systems performance indices with respect to the weighting vector in the parametric minimax formulation is derived the gradient method can be thus adopted at the upper level to adjust the value of the weighting vector the solution of the weighted minimax formulation converges to the optimal solution of the original problem in a multilevel iteration process an application of the proposed iterative parametric minimax method is demonstrated in constrained reliability optimization problems

experiments with dynamic programming algorithms for nonseparable problems abstract in this paper we report on numerical experiments with dynamic programming algorithms designed for solving problems with nonseparable objective functions of the c programming type the experiments were conducted with the view to evaluate the efficiency of c programming s exclusionary rules when they are incorporated in the conventional parametric lagrangian search method two classical nonseparable problems are considered 1 the variance minimization problem and 2 the nonlinear fractional knapsack problem the results indicate that the incorporation of c programming s exclusionary rules in the lagrangian procedure significantly improves the efficiency of the line search procedure

typical properties of winners and losers 0 2ex in discrete optimization we present a probabilistic analysis of a large class of combinatorial optimization problems containing all binary optimization problems defined by linear constraints and a linear objective function over 0 1 n our analysis is based on a semirandom input model that preserves the combinatorial structure of the underlying optimization problem by parameterizing which input numbers are of a stochastic and which are of an adversarial nature this input model covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with gaussian and other kinds of perturbation models as a special case in fact we can exactly characterize the smoothed complexity of binary optimization problems in terms of their worst case complexity a binary optimization problem has polynomial smoothed complexity if and only if it admits a possibly randomized algorithm with pseudo polynomial worst case complexity our analysis is centered around structural properties of binary optimization problems called winner loser and feasibility gap we show that if the coefficients of the objective function are stochastic then the gap between the best and second best solution is likely to be of order omega 1 n furthermore we show that if the coefficients of the constraints are stochastic then the slack of the optimal solution with respect to this constraint is typically of order omega 1 n 2 we exploit these properties in an adaptive rounding scheme that increases the accuracy of calculation until the optimal solution is found the strength of our techniques is illustrated by applications to various npc hard optimization problems from mathematical programming network design and scheduling for which we obtain the first algorithms with polynomial smoothed average case complexity

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

theory and applications of generalized dynamic programming an overview abstract extensions of dynamic programming dp into generalized preference structures such as exist in multicriteria optimization have invariably assumed sufficient structure to ensure the validity of bellman s principle of optimality or the presence of monotonicity hence ensuring the validity of the functional equations of dp often however the only reasonable formulation of problem in terms of the size of the resulting state space is one for which neither monotonicity nor the strong principle of optimality is satisfied generalized dp based on a weaker principle of optimality is a generalization of dp that produces optimal solutions even in the absence of monotonicity it is shown both to subsume conventional dp and to provide an alternative means of implementing mitten s preference order dp this paper provides an overview of generalized dp and its applications to date summarizing our computational experiences in the process

on a special class of bicriterion path problems abstract in this paper we are concerned with a special class of bicriterion path problems for the considered class of bicriterion problems at least one of the objectives is type maxmin for the other one besides an objective type maxmin also a minsum and a minratio type objective are considered two algorithms are presented for the considered class of bicriterion path problems the first one only determines the minimal complete set of nondominated paths and the second one determines the entire set of nondominated paths both algorithms can be used for any type of bicriterion path problems since one of the objectives is type maxmin and an algorithm exists to determine the best path for the other objective computational statistics for the three types of considered bicriterion path problems are also presented

on finding dissimilar pareto optimal paths the aim of the present paper is to provide a methodology for finding a set of alternative paths between an origin and a destination site on which routing one or a set of dangerous goods finding a set of paths allows one to equally distribute the total risk among the population exposed the concept of equity of risk is here related to the concept of determining spatially dissimilar paths we divide our approach into two phases in the first phase we find a set of pareto optimal paths between an origin and a destination by implementing a multicriteria shortest path algorithm in the second one for each path previously found and by using a geographical information system we construct a buffer zone approximating the impact area of a material being released after an accident based on these buffer zones a dissimilarity index between every pair of paths can be derived in order to find the most spatially different routes we then compare our method with an iterative penalty method and discuss computational results based both on a real application and on test problems

a comparison of heuristic best first algorithms for bicriterion shortest path problems abstract a variety of algorithms have been proposed to solve the bicriterion shortest path problem this article analyzes and compares the performance of three best first label setting algorithms that accept heuristic information to improve efficiency these are namoa moa and tung chew s algorithm tc a set of experiments explores the impact of heuristic information in search efficiency and the relative performance of the algorithms the analysis reveals that namoa is the best option for difficult problems its time performance can benefit considerably from heuristic information though not in all cases the performance of tc is similar but somewhat worse however the time performance of moa is found to degrade considerably with the use of heuristic information in most cases explanations are provided for these phenomena

an algorithm for the ranking of shortest paths an efficient computational implementation of a path deletion k shortest paths algorithm and a new algorithm for the same problem are presented in a path deletion k shortest paths algorithm a sequence g1 g2 gk of networks is is defined such that g1 is given network and its k th shortes path is trivially determined from the shortest path in gk in essence as soon as the shortest path in gk is determined it is excluded from gk in such a way that no new paths are formed and no more paths are deleted so far each gk two procedures are executed a shortest path algorithm and a path deletion algorithm in the presented computational implementation all the information resulting from the determination of the k th shortest path is carried throughout gk 1 gk 2 gk the new algorithm not only keeps this characteristics but also avoids the last k 1 executions of a shortest path algorithm which results in a suprising and very substantial reduction in the execution time in fact for randomly generated networks with 104 nodes and 105 arcs once the shortest the new algorithm computes the next 100 shortest paths in times of the order of 10 1 seconds to illustrate the efficiency of this algorithm comparative computational experiments are reported

small profinite groups and rings abstract a small profinite ring has an open nil ideal of finite nil exponent we also discuss various conjectures about small profinite rings and give examples illustrating them

an interactive procedure using domination cones for bicriterion shortest path problems abstract in this paper an interactive procedure is developed for the bicriterion shortest path problem it is assumed that the decision maker s inherant utility function is quasi concave and non increasing and that the network consists of non negative integer valued arc lengths the proposed procedure uses the concept of domination cones which it develops from pairwise comparisons of alternatives these domination cones are used to fathom a large number of pareto optimal solutions extensive computational testing was performed on large grid networks simulating the decision maker s response using polynomial utility functions the results indicate that our proposed procedure is able to converge to the optimal solution in a reasonably small number of pairwise comparisons even for those problems with a large number of pareto optimal solutions

exact algorithms for the stochastic shortest path problem with a decreasing deadline utility function in this paper the stochastic shortest path problem of determining a path that maximizes the expected utility is considered the nature of the utility function used to evaluate paths is of a decreasing deadline type the principal contribution of this paper is the development of exact algorithms that use two types of pruning techniques that are incorporated in labeling procedures one type of pruning makes use of the concept of local preference relations while the other type makes use of relaxations specifically two algorithms are developed each containing the same preference relation but two different relaxations our extensive computational testing indicate that both these algorithms are able to solve even large size problems quickly more importantly even for large problems both the algorithms solved them by enumerating a very small percentage of all paths

the static stochastic knapsack problem with normally distributed item sizes this paper develops exact and heuristic algorithms for a stochastic knapsack problem where items with random sizes may be assigned to a knapsack an item s value is given by the realization of the product of a random unit revenue and the random item size when the realization of the sum of selected item sizes exceeds the knapsack capacity a penalty cost is incurred for each unit of overflow while our model allows for a salvage value for each unit of capacity that remains unused we seek to maximize the expected net profit resulting from the assignment of items to the knapsack although the capacity is fixed in our core model we show that problems with random capacity as well as problems in which capacity is a decision variable subject to unit costs fall within this class of problems as well we focus on the case where item sizes are independent and normally distributed random variables and provide an exact solution method for a continuous relaxation of the problem we show that an optimal solution to this relaxation exists containing no more than two fractionally selected items and develop a customized branch and bound algorithm for obtaining an optimal binary solution in addition we present an efficient heuristic solution method based on our algorithm for solving the relaxation and empirically show that it provides high quality solutions

extension of dynamic programming to nonseparable dynamic optimization problems abstract the use of dynamic programming is extended to a general nonseparable class where multiobjective optimization is used as a separation strategy the original nonseparable dynamic optimization problem is first embedded into a separable albeit multiobjective optimization problem where multiobjective dynamic programming using the envelope approach is used as a solution scheme under certain conditions the optimal solution of the original nonseparable problem is proven to be attained by a noninferior solution

products of finite abelian groups as profinite groups abstract we investigate products of finite abelian groups of bounded exponent as profinite structures in the sense of newelski in such groups we describe orbits under the action of the standard structural group of automorphisms then we conclude that such groups are small m normal and m stable let x be a product of countably many finite abelian groups we also investigate the influence of modifications of the standard structural group of x on its smallness m normality and m stability

the dynamic and stochastic knapsack problem the dynamic and stochastic knapsack problem dskp is defined as follows items arrive according to a poisson process in time each item has a demand size for a limited resource the knapsack and an associated reward the resource requirements and rewards are jointly distributed according to a known probability distribution and become known at the time of the item s arrival items can be either accepted or rejected if an item is accepted the item s reward is received and if an item is rejected a penalty is paid the problem can be stopped at any time at which time a terminal value is received which may depend on the amount of resource remaining given the waiting cost and the time horizon of the problem the objective is to determine the optim al policy that maximizes the expected value rewards minus costs accumulated assuming that all items have equal sizes but random rewards optimal solutions are derived for a variety of cost structures and time horizons and recursive algorithms for computing them are developed optimal closed form solutions are obtained for special cases the dskp has applications in freight transportation in scheduling of batch processors in selling of assets and in selection of investment projects

hierarchical control for large scale systems with general multiple linear quadratic structure abstract a class of large scale linear control systems where the overall objective function is a nonlinear function of multiple quadratic performance indices is investigated in this paper this type of large scale control problem with a general multiple linear quadratic structure is nonseparable in the sense of conventional hierarchical control hierarchical control is extended in this paper to large scale nonseparable control problems where multiobjective optimization is used as a separation strategy the large scale general multiple linear quadratic control problem is embedded under certain conditions into a family of the weighted lagrangian formulation the weighted lagrangian formulation is separable with respect to subsystems and can be effectively solved using the interaction prediction approach at the two lower levels in the proposed three level solution structure at the third level the weighting vector for the weighted lagrangian formulation is adjusted iteratively to search the optimal weighting vector with which the optimal control of the original large scale nonseparable control problem is attained one feature of this hierarchical control scheme is a derived analytical solution for the large scale general multiple linear quadratic control

the tricriterion shortest path problem with at least two bottleneck objective functions abstract the focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type for example minmax or maxmin the third objective function may be of the same kind or we may consider for example minsum or maxprod let p n be the complexity of a classical single objective algorithm responsible for this third function where n is the number of nodes and m be the number of arcs of the graph an o m 2 p n algorithm is presented that can generate the minimal complete set of pareto optimal solutions finding the maximal complete set is also possible optimality proofs are given and extensions for several special cases are presented computational experience for a set of randomly generated problems is reported

the dynamic and stochastic knapsack problem with deadlines in this paper a dynamic and stochastic model of the well known knapsack problem is developed and analyzed the problem is motivated by a wide variety of real world applications objects of random weight and reward arrive according to a stochastic process in time the weights and rewards associated with the objects are distributed according to a known probability distribution each object can either be accepted to be loaded into the knapsack of known weight capacity or be rejected the objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward the optimal decision rules are derived and are shown to exhibit surprising behavior in some cases it is also shown that if the distribution of the weights is concave then the decision rules behave according to intuition

a new algorithm for the discrete fuzzy shortest path problem in a network in a network the arc lengths may represent time or cost in practical situations it is reasonable to assume that each arc length is a discrete fuzzy set we called it the discrete fuzzy shortest path problem there were several methods reported to solve this kind of problem in the open literature in these methods they can obtain either the fuzzy shortest length or the shortest path in this paper we propose a new algorithm that can obtain both of them the discrete fuzzy shortest length method is proposed to find the fuzzy shortest length and the fuzzy similarity measure is utilized to get the shortest path an illustrative example is given to demonstrate our proposed algorithm

generalized dynamic programming for stochastic combinatorial optimization in stochastic versions of combinatorial optimization problems the objective is to maximize or minimize a function of random variables for many problems of this type conventionally applied dynamic programming dp may fail to generate an optimal solution due to the potential violation of the monotonicity assumption of dp we develop a generalization of dp that guarantees optimality even in the absence of monotonicity we illustrate the methodology on a version of the stochastic traveling salesman problem for which a previously proposed dp algorithm e kao is potentially suboptimal due to the violation of monotonicity m sniedovich using generalized dp we are able to modify the algorithm to guarantee optimality