on the singular set for lipschitzian critical points of polyconvex functionals abstract partial regularity is proved for lipschitzian critical points of polyconvex functionals provided d u l is small enough in particular the singular set for a lipschitzian critical point has hausdorff dimension strictly less than n when d u l is small enough model problems treated include u 2 det u 2 where u r 2 r 2 and u 2 u s ad u s det u s where u r 3 r 3 with s 2 moreover it is shown that the singular set of a lipschitzian global minimizer has hausdorff dimension strictly less than n

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

an interactive reference point approach for multiobjective mixed integer programming using branch and bound abstract we propose an interactive reference point approach for multiple objective mixed integer linear programming problems that exploits the use of branch and bound techniques for solving the scalarizing programs at each dialogue phase the decision maker must specify a criterion reference point or just choose an objective function he she wants to improve in respect to the previous efficient nondominated solution in the latter case a directional search is performed adjusting automatically the reference point used at each stage tchebycheff mixed integer scalarizing programs are successively solved by branch and bound postoptimality techniques have been developed enabling the algorithm to profit from previous computations to solve the next scalarizing programs the previous branch and bound tree is used as a starting point and operations of simplification and branching are then performed to obtain a new efficient solution computational results have shown that this approach is effective for carrying out directional or local searches for efficient solutions

efficient computation of the search region in multi objective optimization multi objective optimization procedures usually proceed by iteratively producing new solutions for this purpose a key issue is to determine and efficiently update the search region which corresponds to the part of the objective space where new nondominated points could lie in this paper we elaborate a specific neighborhood structure among local upper bounds thanks to this structure the update of the search region with respect to a new point can be performed more efficiently compared to existing approaches moreover the neighborhood structure provides new insight into the search region and the location of nondominated points

a reduction dynamic programming algorithm for the bi objective integer knapsack problem abstract this paper presents a backward state reduction dynamic programming algorithm for generating the exact pareto frontier for the bi objective integer knapsack problem the algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept first an approximate core is obtained by eliminating dominated items second the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted sum functions associated with the efficient extreme solutions of the linear relaxation of the multi objective integer knapsack problem third the items are classified according to the values of their upper bounds items with zero upper bounds can be eliminated finally the remaining items are used to form a mixed network with different upper bounds the numerical results obtained from different types of bi objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm

multiobjective solution of the uncapacitated plant location problem abstract in this paper we consider the discrete multiobjective uncapacitated plant location problem we present an exact and an approximate approach to obtain the set of non dominated solutions the two approaches resort to dynamic programming to generate in an efficient way the non dominated solution sets the solution methods that solve the problems associated with the generated states are based on the decomposition of the problem on two nested subproblems we define lower and upper bound sets that lead to elimination tests that have shown to have a high performance computational experiments on a set of test problems show the good performance of the proposal

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

applications of universal context modeling to lossless compression of gray scale images inspired by theoretical results on universal modeling a general framework for sequential modeling of gray scale images is proposed and applied to lossless compression the model is based on stochastic complexity considerations and is implemented with a tree structure it is efficiently estimated by a modification of the universal algorithm context several variants of the algorithm are described the sequential lossless compression schemes obtained when the context modeler is used with an arithmetic coder are tested with a representative set of gray scale images the compression ratios are compared with those obtained with state of the art algorithms available in the literature with the results of the comparison consistently favoring the proposed approach

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

an interactive procedure for multiple objective integer linear programming problems abstract there are many practical problems where the objective functions and the constraints can be represented by linear functions but the decision variables are restricted to only integer values our objective is to present an interactive procedure for the multiple objective integer linear programming problem that is fast does not place too many demands on the decision maker and can be implemented without requiring any special software we illustrate the proposed procedure with a numerical example and report some computational experience

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

a method for finding the set of non dominated vectors for multiple objective integer linear programs abstract an algorithm for enumerating all non dominated vectors of multiple objective integer linear programs is presented using a straightforward theoretical approach the problem is solved using a sequence of progressively more constrained integer linear programs generating a new solution at each step the algorithm can also give subsets of efficient solutions that can be useful for designing interactive procedures for large real life problems

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

an algorithm for the multiple objective integer linear programming problem abstract a technique is presented for solving the multiple objective integer linear programming problem the technique can be used to identify some or all efficient solutions while the technique is applicable with any integer programming algorithm it is well suited for implementation using integer postoptimality techniques such an implementation based on balas additive algorithm is described for problems with zero one variables

on a biobjective search problem in a line formulations and algorithms abstract in this article we describe three formulations of a multiobjective combinatorial optimization problem as well as several complexity results and structural properties of these formulations a multiobjective dynamic programming algorithm is proposed for each of the three formulations based on our theoretical and computational results we argue that a clever definition of the recursion allowing for strong dominance criteria is crucial in the design of a multiobjective dynamic programming algorithm

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

zero one programming with multiple criteria abstract the study of multiple criteria problems is not new but during the last seventeen years more and more study has been done in this area the study of 0 1 multiple criteria problems started much later than the study of continuous multiple criteria problems some of the earliest works on 0 1 problems with multiple criteria was first presented in 1973 quite a few solution methods have been developed since then especially during the last seven years in this paper most of these methods and some applications will be treated

a branch and bound algorithm for mixed zero one multiple objective linear programming abstract in this paper a branch and bound algorithm for the generation of the efficient set in mixed zero one multiple objective linear programming problems is presented the algorithm is developed as to take account of the multiple objectives in the node fathoming procedure in order to extend the algorithm s applicability to large sized problems from real life an interactive procedure is introduced which systematically reduces the number of efficient points and thus saves considerable computational effort without losing essential information the algorithm is tested in randomly generated problems along with a case study conceming the power generation sector

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

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

multi objective branch and bound abstract branch and bound is a well known generic method for computing an optimal solution of a single objective optimization problem based on the idea divide to conquer it consists in an implicit enumeration principle viewed as a tree search although the branch and bound was first suggested by land and doig 1960 the first complete algorithm introduced as a multi objective branch and bound that we identified was proposed by kiziltan and yucaoglu 1983 rather few multi objective branch and bound algorithms have been proposed this situation is not surprising as the contributions on the extensions of the components of branch and bound for multi objective optimization are recent for example the concept of bound sets which extends the classic notion of bounds has been mentioned by villarreal and karwan 1981 but it was only developed for the first time in 2001 by ehrgott and gandibleux and fully defined in 2007 this paper describes a state of the art of multi objective branch and bound which reviews concepts components and published algorithms it mainly focuses on the contributions belonging to the class of optimization problems who has received the most of attention in this context from 1983 until 2015 the linear optimization problems with zero one variables and mixed 0 1 continuous variables only papers aiming to compute a complete set of efficient solutions are discussed

multicriteria integer programming a hybrid dynamic programming recursive approach dynamic programming recursive equations are used to develop a procedure to obtain the set of efficient solutions to the multicriteria integer linear programming problem an alternate method is produced by combining this procedure with branch and bound rules computational results are reported

the lavrentiev gap phenomenon in nonlinear elasticity the main objective of this paper is to present examples of the lavrentiev phenomenon within the framework of two dimensional nonlinear elasticity loosely speaking this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings we provide a physically natural stored energy density and reasonable though nontraditional boundary conditions such that the energy functional exhibits the lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations the stored energy density w that we produce is smooth materially homogeneous frame indifferent isotropic and polyconvex furthermore the corresponding minimization problem is such that existence of a continuous minimizer follows from known results the basis for our examples is a convex integrand w 0 for which the euler lagrange equations have a very special form we show that the functional associated with this w 0 exhibits the lavrentiev phenomenon for certain problems by making a perturbation to w 0 we create the stored energy density w described in the previous paragraph with other perturbations to the integrand w 0 and modifications of the boundary conditions we are able to produce additional examples of the lavrentiev phenomenon finally we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon

an algorithm for multiobjective zero one linear programming a branch and bound algorithm is presented which is based on the extension of implicit enumeration techniques to multiobjective zero one linear programming and which appears to be computationally quite efficient domination tests aiming at identifying paths of the enumeration tree that lead to dominated solutions as high up the tree as possible are developed some computational results are also given

interactive relaxation method for a broad class of integer and continuous nonlinear multiple criteria problems an interactive decomposition algorithm for solving a broad class of multiple criteria mc problems is presented in this paper on the one hand the method is designed to solve a broad class of mc problems including complex ones i e problems with nonlinear objective functions and nonlinear constraints with continuous and or integer valued decision variables and with an unknown underlying nonlinear preference function on the other hand the method is intended to reduce the assessment burden on the decision maker dm by simplifying and facilitating the task of making preference assessments and tradeoffs the method is efficient from a computational standpoint since the only program that needs to be solved at each iteration is linear in the decision variables also the algorithm is capable of producing lower and upper bounds lb and ub on the optimal objective value of the mc problem at each iteration although the dm s preference function is not assumed to be known explicitly lb and ub can be used to terminate the search prematurely at a satisfactory good enough solution and consequently to reduce the computational time as well as the number of preference assessments that the dm must make finally all the tradeoffs that the dm is required to make are of the form of ordinal paired comparisons most of which are particularly simple since they involve changes in only two criteria at a time also all the preference assessments are made in the context of feasible solutions only

an interactive dynamic programming approach to multicriteria discrete programming abstract several interactive schemes for solving multicriteria discrete programming problems are developed under a dynamic programming framework it is assumed that the decision maker s preference structure satisfies the conditions of transitivity monotonicity and nonsatiation hybrid procedures are also structured by including branch and bound ideas into the recursions initial computational results are offered

an interactive algorithm for solving multiple objective integer linear programming problems abstract we propose a reference direction based interactive algorithm to solve multiple objective integer linear programming moilp problems at each iteration of the solution procedure the algorithm finds weak nondominated solutions to the relaxed moilp problem only at certain iterations if the dm so desires an additional mixed integer programming problem is solved to find an integer weak nondominated solution which is close to the current continuous weak nondominated solution to the relaxed moilp problem in the proposed algorithm dm has to provide only the reference point at each iteration no special software is required to implement the proposed algorithm the algorithm is illustrated with an example

a review of interactive methods for multiobjective integer and mixed integer programming this paper makes a review of interactive methods devoted to multiobjective integer and mixed integer programming moip momip problems the basic concepts concerning the characterization of the non dominated solution set are first introduced followed by a remark about non interactive methods vs interactive methods then we focus on interactive moip momip methods including their characterization according to the type of preference information required from the decision maker the computing process used to determine non dominated solutions and the interactive protocol used to communicate with the decision maker we try to draw out some contrasts and similarities of the different types of methods

the regularity of critical points of polyconvex functionals in this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 we construct a smooth strongly polyconvex eq2 and lipschitzian weak solutions eq3 to the corresponding euler lagrange system which are nowhere c1 moreover we show that f can be chosen in such a way that these irregular weak solutions are weak local minimisers

accelerating the optimization of requantization codebook for vector quantization requantization is an important technique for variable rate data compression and communication and optimal design for requantization is key to guaranteeing the minimization of the requantization noise recently an iterative optimization algorithm for designing the requantization codebook was proposed we develop further results regarding the acceleration of the algorithm and a better understanding of the problem of requantization in particular our result shows that the quantization and requantization processes can be decoupled into two nearly independent sequential stages and that requantization does not need the original training vectors all these features pave the way toward real time variable rate data compression and transmission over heterogeneous networks simulation results support the theory and the advantages of our method

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

requantization for transcoding of mpeg 2 intraframes an investigation on requantization for transcoding of video signals is carried out specifically mpeg 2 compatible discrete cosine transform dct intraframe coding is addressed the aim of this work is twofold first to provide a theoretical analysis of the transcoding problem and second to derive quantization methods for efficient transcoding based on the results of the analysis the mean squared error mse cost function is proposed for designing a quantizer with minimum distortion resulting in up to 1 3 db gain compared with the quantizer used in the mpeg 2 reference coder tm5 however the mse quantizer leads in general to a larger bit rate and may therefore only be applied locally to blocks of sensitive image content a better rate distortion performance can be provided by the maximum a posteriori map cost function in critical cases the map quantizer gives a 0 4 db larger signal to noise ratio snr at the same bit rate compared with the tm5 quantizer the results are not limited to mpeg 2 and can be adapted to other coding schemes such as h 263 or jpeg

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

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

lipschitz regularity for inner variational equations we obtain lipschitz regularity results for a fairly general class of nonlinear first order pdes these equations arise from the inner variation of certain energy integrals even in the simplest model case of the dirichlet energy the inner stationary solutions need not be differ entiable everywhere the lipschitz continuity is the best possible but the proofs even in the dirichlet case turn out to relay on topological arguments the appeal to the inner stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity specifically neo hookian type problems

an algorithm for the bi criterion integer programming problem abstract we specify a variation of the weighting method for multi criterion optimization which determines nondominated solutions to the bi criterion integer programming problem the technique makes use of imposed constraints based on nondominated points for the bi criterion case we develop an algorithm which finds all nondominated points by solving a sequence of single criterion integer programming problems we present computational results for the linear 0 1 case and discuss the extension of our algorithm to the general multi criterion integer programming problem

a novel approach for fast codebook re quantization in this paper we present a fast codebook re quantization algorithm fcra using codewords of a codebook being re quantized as the training vectors to generate the re quantized codebook our method is different from the available approach which uses the original training set to generate a re quantized codebook compared to the traditional approach our method can reduce the computing time dramatically since the number of codewords of a codebook being re quantized is usually much smaller than the number of original training vectors our method first classifies codewords of a re quantized codebook into static and active groups this approach uses the information of codeword displacements between successive partitions to reject impossible candidates in the partition process of codebook re quantization by implementing a fast search algorithm used for vector quantization encoding mfaupi in the partition step of fcra the computational complexity of codebook re quantization can be further reduced significantly using mfaupi the computing time of fcra can be reduced by a factor of 1 55 3 78 compared with the available approach oarc optimization algorithm for re quantization codebook our proposed method can reduce the codebook re quantization time by a factor of about 8005 using a training set of six real images this reduction factor is increased when the re quantized codebook size and or training set size are increased it is noted that our proposed algorithm can generate the same re quantized codebook as that produced by the oarc

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

quasiconvexity and partial regularity in the calculus of variations we prove partial regularity of minimizers of certain functionals in the calculus of variations under the principal assumption that the integrands be uniformly strictly quasiconvex this is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate sobolev spaces examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix

an overview of techniques for solving multiobjective mathematical programs multiobjective mathematical programming has been one of the fastest growing areas of or ms during the last 15 years this paper presents some reasons for the rapidly growing increase in interest in multiobjective mathematical programming a discussion of the advantages and disadvantages of the three general approaches articulation of the decision maker s preference structure over the multiple objectives prior to during or after the optimization towards multiobjective mathematical programming a nontechnical overview of many of the specific solution techniques for multiobjective mathematical programming and a discussion of important areas for further research the overview concentrates on those techniques which require an articulation of the decision maker s preference structure either during or after the optimization since these are the areas where most of the recent research has been conducted it differs from previous overviews in that in addition to the timing of the elicited preference informa

an algorithm for optimizing a linear function over an integer efficient set optimizing a linear function over the efficient set of a multiobjective integer linear programming moilp problem is a topic of unquestionable practical as well as mathematical interest within the field of multiple criteria decision making as known those problems are particularly difficult to deal with due to the discrete nature of the efficient set which is not explicitly known nor a suitable implicit description is available in this work an exact algorithm is presented to optimize a linear function over the efficient set of a moilp the approach here proposed defines a sequence of progressively more constrained single objective integer problems that successively eliminates undesirable points from further consideration the algorithm has been coded in c sharp using cplex solver and computational experiments have been undertaken in order to analyze performance properties of the algorithm over different problem instances randomly generated

dynamic programming algorithms for the bi objective integer knapsack problem abstract this paper presents two new dynamic programming dp algorithms to find the exact pareto frontier for the bi objective integer knapsack problem first a property of the traditional dp algorithm for the multi objective integer knapsack problem is identified the first algorithm is developed by directly using the property the second algorithm is a hybrid dp approach using the concept of the bound sets the property is used in conjunction with the bound sets next the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set it means that the upper bound set is underestimated then an extended upper bound set is proposed on the basis of the set of linear relaxation solutions the efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set the numerical results obtained from different types of bi objective instances show the effectiveness of the proposed approach

a class of practical interactive branch and bound algorithms for multicriteria integer programming abstract a practical interactive solution approach to multicriteria integer programming problems is developed the problem is solved by a branch and bound method that employs the zionts and wallenius procedure 23 for solving the multicriteria linear programming problem the development of algorithms for multicriteria decision problems itself is a multicriteria problem which involves the simultaneous minimization of the number of questions asked of the decision maker and the solution time two branch and bound algorithms that follow different search strategies to meet different levels of these criteria have been developed further two families of hybrid algorithms that incorporate a combination of the strategies of the two algorithms have also been developed strategies for the exploration of the decision maker s preference structure are discussed computational experience with the algorithms is presented the class of algorithms represents a collection of viable solution strategies applicable to a variety of decision making styles