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

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

an inverse boundary value problem for schrodinger operators with vector potentials we consider the schrodinger operator for a magnetic potential a and an electric potential q which are supported in a bounded domain in r n with n 3 we prove that knowledge of the dirichlet to neumann map associated to the schrodinger operator determines the magnetic field rot a and the electric potential 4 simultaneously provided rot a is small in the l topology

a time indexed formulation of non preemptive single machine scheduling problems we consider the formulation of non preemptive single machine scheduling problems using time indexed variables this approach leads to very large models but gives better lower bounds than other mixed integer programming formulations we derive a variety of valid inequalities and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities a cutting plane branch and bound algorithm based on these inequalities has been implemented computational experience on small problems with 20 30 jobs and various constraints and objective functions is presented

detection of mines by electric measurements in a homogeneous region omega of earth with conductivity 1 a mine is possibly buried occupying a region d and having conductivity ne 1 d is unknown but meas d geqq c 0 we let electric current enter omega at some point of partial omega and exit it at another point of partial omega and we measure the resulting potential at some open portion of partial omega it is proved that if the error in the measurements is small enough then one can assert whether or not a mine d is in fact buried in omega some results on the location of d are also given

stronger lagrangian bounds by use of slack variables applications to machine scheduling problems lagrangian relaxation is a powerful bounding technique that has been applied successfully to manynp hard combinatorial optimization problems the basic idea is to see annp hard problem as an easy to solve problem complicated by a number of nasty side constraints we show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds the trick is widely applicable but we focus on a broad class of machine scheduling problems for which it is particularly useful we provide promising computational results for three problems belonging to this class for which lagrangian bounds have appeared in the literature the single machine problem of minimizing total weighted completion time subject to precedence constraints the two machine flow shop problem of minimizing total completion time and the single machine problem of minimizing total weighted tardiness

uniqueness for the inverse boundary value problem with singular potentials in 2d in this paper we consider the inverse boundary value problem for the schrodinger equation with potential in l p class p 4 3 we show that the potential is uniquely determined by the boundary measurements

an exact algorithm for the precedence constrained single machine scheduling problem abstract this study proposes an efficient exact algorithm for the precedence constrained single machine scheduling problem to minimize total job completion cost where machine idle time is forbidden the proposed algorithm is based on the ssdp successive sublimation dynamic programming method and is an extension of the authors previous algorithms for the problem without precedence constraints in this method a lower bound is computed by solving a lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence constrained total weighted tardiness and total weighted earliness tardiness problems and most instances with 100 jobs of the former problem

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

inverse scattering for singular potentials in two dimensions we consider the schrodinger equation for a compactly supported potential having jump type singularities at a subdomain of r2 we prove that knowledge of the scattering amplitude at a fixed energy determines the location of the singularity as well as the jump across the curve of discontinuity this result follows from a similar result for the dirichlet to neumann map associated to the schrodinger equation for a compactly supported potential with the same type of singularities 1 introduction and statement of the results in this paper we consider the schrodinger equation for a compactly supported potential q having jump type singularities at a subdomain of k2 we prove that knowledge of the scattering amplitude at a fixed energy ao determines the location of the singularity as well as the jump across the curve of discontinuity this problem is reduced to the study of the dirichlet to neumann map for the schrodinger operator a q x in a bounded domain of r2 for the application considered here it is enough to consider the domain to be a ball containing the support of q we prove that in dimension two the dirichlet to neumann map for the schrodinger operator a q x determines uniquely the location of the singularity of q as well as its jump across the curve of discontinuity the scattering amplitude of a potential q l rn with compact support is defined via the outgoing eigenfunctions namely mx r 0 co s x there exists i x x co solution of

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

on uniqueness of recovery of a discontinuous conductivity coefficient on considere le probleme de retrouver le coefficient a de l equation elliptique div a u 0 dans un domaine avec la condition aux limites u sur quand u n est donnee pour tout regulier on demontre l unicite pour a a 0 b discontinu ou est la fonction indicateur d un ensemble ouvert a o est une fonction c 2 donnee et b est une fonction c 2 inconnue

recovery of singularities for formally determined inverse problems in this paper we considered several formally determined problems in two dimensions there are no global identifiability results for these problems however we can recover an important feature of these functions namely their singularities more precisely we prove that one can determine the location and strength of singularities of anl compactly supported potential by knowing the associated scattering amplitude at a fixed energy also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium

the two machine total completion time flow shop problem in this paper we study the np hard scheduling problem of minimizing total completion time in a two machine flow shop five known lower bounds are discussed and two new ones are presented a new dominance criterion is also proposed several versions of a branch and bound method are derived by applying both individually and combined these lower bounds a heuristic procedure is also presented that uses a constructive o n2 time method which computes a good starting solution together with a neighborhood search based on pairwise interchanges computational results show that the exact method can handle problems of up to 30 jobs in size within a reasonable amount of time and that the heuristic procedure has an average error of less than 0 5 from the optimal value and less than 2 7 from the lower bound

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

a lagrangean based branch and bound algorithm for single machine sequencing with precedence constraints to minimize total weighted completion time the single machine sequencing problem is considered in which there are precedence constraints on the jobs the objective is to minimize the sum of weighted completion times a lower bound is obtained by successively performing a lagrangean relaxation of appropriate constraints each lagrange multiplier is chosen to provide the maximum increment to the lower bound subject to retaining the nonnegativity of the coefficients of the variables when no further suitable constraints can be introduced into the lagrangean function the variables having zero cost coefficient are used to obtain a feasible sequence which provides an upper bound the gap between the lower and upper bound is reduced by removing some constraints from the lagrangean function and replacing them with others this lower bounding procedure is used in a branch and bound algorithm computational results indicate that the algorithm can satisfactorily solve problems with up to 100 jobs

uniqueness for formally determined inverse boundary value problems abstract in this paper we study the inverse conductivity problem in two dimensions this problem is reduced to a formally determined inverse boundary value problem for the schrodinger equation some recent generic uniqueness results for the inverse problems will be discussed and a new continuous dependence result presented

inverse problems in quasilinear anisotropic media we consider the question of whether one can determine the anisotropic quasilinear conductivity of a medium by making voltage and current measurements at the boundary we prove that all the known results for the linear case also hold in the quasilinear case

on the construction of a potential from cauchy data abstract we investigate the problem of recovering a potential q y in the differential equation u q y u 0 x y 0 1 0 1 u 0 y u 1 y u x 0 0 u x 1 f x u y x 1 g x the method of separation of variables reduces the recovery of q y to a nonstandard inverse sturm liouville problem an asymptotic formula is developed that suggests that under appropriate conditions on the cauchy pair f g q y is uniquely determined up to the mean moreover the recovery of q y is comparable to finding a function from its polynomial moments a reconstruction scheme is suggested and numerical examples are considered

single machine scheduling polyhedra with precedence constraints we consider nonpreemptive single machine scheduling subject to precedence constraints we define feasible schedules by the vector of the job completion times and study the structure of the convex hull of all feasible schedules called the scheduling polyhedron p we derive classes of valid inequalities for p and necessary and sufficient conditions under which they are facet inducing our main result is a complete description of the minimal linear system defining p when the precedence constraints are series parallel moreover this system consists of two classes of facet inducing inequalities associated with series and parallel compositions respectively if the precedence constraints are not series parallel we present another class of facet inducing inequalities for p associated with induced z subgraphs we also show that the convex hull of all preemptive feasible schedules is identical to p if and only if the precedence constraints form an out forest

generalized pencil of function method for extracting poles of an em system from its transient response a generalized pencil of function gpof method is developed for extracting the poles of an electromagnetic system from its transient response the gpof method needs the solution of a generalized eigenvalue problem to find the poles subspace decomposition is also used to optimize the performance of the gpof method the gpof method has advantages over the prony method in both computation and noise sensitivity and approaches the cramer rao bound when the signal to noise ratio snr is above threshold an application of the gpof method to a thin wire target is presented

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

transparent potentials at fixed energy in dimension two fixed energy dispersion relations for the fast decaying potentials for the two dimensional schrodinger equation delta v x psi e psi x in mathbb r 2 e e fixed 0 at a fixed positive energy with a fast decaying at infinity potentialv x dispersion relations on the scattering data are given under small norm assumption using these dispersion relations we give without a complete proof of sufficiency a characterization of scattering data for the potentials from the schwartz classs c 2 for the potentials with zero scattering amplitude at a fixed energyefixed transparent potentials we give a complete proof of this characterization as a consequence we construct a family parametrized by a function of one variable of two dimensional spherically symmetric real potentials from the schwartz classs transparent at a given energy for the two dimensional case without assumption that the potential is small we show that there are no nonzero real exponentially decreasing at infinity potentials transparent at a fixed energy for any dimension greater or equal to 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval we show that kdv type equations in dimension 2 1 related with the scattering problem the novikov veselov equations do not preserve in general these dispersion relations starting from the second one as a corollary these equations do not preserve in general the decay rate faster than x 3 for initial data from the schwartz class

minimizing total completion time in a two machine we consider the problem of minimizing total completion time in a two machine owshop we present a heuristic with worst case bound 2 where and denote the minimum and maximum processing time of all operations furthermore we analyze four special cases equal processing times on the first machine equal processing times on the second machine processing a job on the first machine takes time no less than its processing on the second machine and processing a job on the first machine takes time no more than its processing on the second machine we prove that the rst special case is np hard in the strong sense and present an o n log n approximation algorithm for it with worst case bound 4 3 we repeat the easy polynomial algorithms for the cases two and three and show that problem four is solvable in polynomial time as well

small and large tsp two polynomially solvable cases of the traveling salesman problem abstract the problem of minimizing and maximizing the objective function of the traveling salesman problem tsp is discussed in particular we consider two special cases of the tsp namely the small tsp and the large tsp for which both a minimum and a maximum of the objective function can be obtained in polynomial time the cost matrix c c ij corresponding to the small large tsp is defined by c ij min a i b j c ij max a i b j for each 1 j 1 n and n dimensional real vector a and ba the main purpose of this paper is to show the relationship between the subtour elimination algorithm of gilmore and gomory for the large tsp and gabovich s algorithm for the small tsp the recognition problem for these two types of matrices is solved by using deineko and filonenko s algorithm for recognizing permuted distribution matrices finally as an application a problem from machine scheduling theory is considered

limiting carleman weights and anisotropic inverse problems in this article we consider the anisotropic calderon problem and related inverse problems the approach is based on limiting carleman weights introduced in kenig et al ann math 165 567 591 2007 in the euclidean case we characterize those riemannian manifolds which admit limiting carleman weights and give a complex geometrical optics construction for a class of such manifolds this is used to prove uniqueness results for anisotropic inverse problems via the attenuated geodesic ray transform earlier results in dimension n 3 were restricted to real analytic metrics

inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in euclidean space we propose a new approach to solve the inverse boundary value problems in the euclidean space the idea consists of embedding the problem into hyperbolic manifolds and using their spectral properties as a by product one can discuss the reconstruction of local conformal deformation of the metric of hyperbolic manifolds from the spectral data at infinity we also propose a new spectral data observed from the cusp neighborhood at infinity

one processor scheduling with symmetric earliness and tardiness penalties we consider one processor scheduling problems having the following form tasks t1 t2 tn are given with each ti having a specified length li and a preferred starting time ai or equivalently a preferred completion time bi the tasks are to be scheduled nonpreemptively i e a task cannot be split on a single processor to begin as close to their preferred starting times as possible we examine two different cost measures for such schedules the sum of the absolute discrepancies from the preferred starting times and the maximum such discrepancy for the first of these we show that the problem of finding minimum cost schedules is np complete however we give an efficient algorithm that finds minimum cost schedules whenever the tasks either all have the same length or are required to be executed in a given fixed sequence for the second cost measure we give an efficient algorithm that finds minimum cost schedules in general with no constraints on the ordering or lengths of the tasks

the neumann to dirichlet map in two dimensions abstract for the two dimensional schrodinger equation in a bounded domain we prove uniqueness of the determination of potentials in w p 1 p 2 in the case where we apply all possible neumann data supported on an arbitrarily non empty open set of the boundary and observe the corresponding dirichlet data on an immediate consequence is that one can uniquely determine a conductivity in w p 3 with p 2 by measuring the voltage on an open subset of the boundary corresponding to a current supported in the same set

dual decomposition of a single machine scheduling problem we design a fast ascent direction algorithm for the lagrangian dual problem of the single machine scheduling problem of minimizing total weighted completion time subject to precedence constraints we show that designing such an algorithm is relatively simple if a scheduling problem is formulated in terms of the job completion times rather than as an 0 1 linear program also we show that upon termination of such an ascent direction algorithm we get a dual decomposition of the original problem which can be exploited to develop approximative and enumerative approaches for it computational results exhibit that in our application the ascent direction leads to good lagrangian lower and upper bounds

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

a hybrid fast hankel transform algorithm for electromagnetic modeling a hybrid fast hankel transform algorithm has been developed that uses several complementary features of two existing algorithms anderson s digital filtering or fast hankel transform fht algorithm and chave s quadrature and continued fraction algorithm a hybrid fht subprogram called hybfht written in standard fortran 77 provides a simple user interface to call either subalgorithm the hybrid approach is an attempt to combine the best features of the two subalgorithms in order to minimize the user s coding requirements and to provide fast execution and good accuracy for a large class of electromagnetic problems involving various related hankel transform sets with multiple arguments special cases of hankel transforms of double order and double argument are discussed where use of hybfht is shown to be advantageous for oscillatory kernel functions

an improved branch and bound algorithm for the two machine total completion time flow shop problem we consider the two machine total completion time flow shop scheduling problem this problem is known to be np hard in the strong sense an improved lagrangean relaxation approach is proposed two new dominance criteria are introduced to curtail the enumeration tree a branch and bound algorithm capable of solving to optimality medium size problem instances is presented

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

new lower and upper bounds for scheduling around a small common due date we consider the single machine problem of scheduling n jobs to minimize the sum of the deviations of the job completion times from a given small common due date for this np hard problem we develop a branch and bound algorithm based on lagrangian lower and upper bounds that are found in on log n time we identify conditions under which the bounds concur these conditions can be expected to be satisfied by many instances with n not too small in our experiments with processing times drawn from a uniform distribution the bounds concur for n 40 for the case where the bounds do not concur we present a refined lower bound that is obtained by solving a subset sum problem of small dimension to optimality we further develop a 4 3 approximation algorithm based upon the lagrangian upper bound

a uniqueness theorem for an inverse boundary value problem in two dimensions abstract we consider the inverse problem to determine the potential q entering the schrodinger equation u qu 0 in a bounded smooth domain in r 2 in terms of the boundary measurements dirichlet to neumann map when n 3 the uniqueness for the inverse problem was well established in this paper we prove the uniqueness in two dimensions under the assumption that the lp p 2 norm of q is small this result improves a uniqueness result of sylvester uhlmann under the assumption that the w1 norm of q is small

global uniqueness for a two dimensional semilinear elliptic inverse problem for a general class of nonlinear schrodinger equations au a x u 0 in a bounded planar domain 2 we show that the function a x u can be recovered from knowledge of the corresponding dirichlet to neumann map on the boundary dq

application of dcim on marine controlled source electromagnetic survey the discrete complex image method dcim is used to calculate the mixed potential electric field integral equation the dcim has been used broadly in high frequency domain and now is successfully introduced to the extremely low frequency situation of marine controlled source electromagnetic mcsem survey the integral path in high frequency situation does not hold valid for extremely low frequency situation the reason can be ascribed to the change of wave number formulas a new integral path suitable for the extremely low frequency situation is proposed the appropriate ranges of the parameters used in dcim are given the computation is stable at these considerably large ranges the results of a five layer model calculated with dcim are compared with those with the filter method and the direct integral method the computation time with dcim is comparable to the time with filter method the effects of water depth upper sediment thickness and oil layer thickness are studied with dcim the results show that electric field varies with the change of thickness the effects accord to the physical meanings of mcsem survey it is safe to conclude that the dcim with the new path is valid in mcsem survey

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs