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

spectral properties of a matrix of redheffer abstract define n n matrices d n d ij and c n c ij by d ij 1 if i j 0 otherwise and c n 0 1 1 1 t 1 0 0 0 let a n d n c n the matrix is of number theoretic significance because det a n m n is mertens function we give a simplified derivation of the characteristic polynomial of a n give precise asymptotic estimates for its two large eigenvalues and use rouche s theorem to significantly improve a previous bound on its small eigenvalues finally we present some attractive conjectures about these small eigenvalues based on numerical evidence

solving linear programming relaxations associated with lagrangean relaxations by fenchel cutting planes abstract for nearly 30 years lagrangean relaxation has been extensively used in integer programming and combinatorial optimization today it is one of the most used techniques for obtaining strong bounds although it generally provides no linear programming relaxation our main objective in this paper is to show how to obtain linear programming relaxations with a value improving that of the lagrangean dual problem we first show how the recently introduced fenchel cutting planes solve the convexified problem associated with every lagrangean relaxation moreover we show that fenchel cutting planes are much more effective than lagrangean cutting planes and that they have very good computational properties for the 0 1 problems treated

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

separation algorithms for 0 1 knapsack polytopes valid inequalities for 0 1 knapsack polytopes often prove useful when tackling hard 0 1 linear programming problems to generate such inequalities one needs separation algorithms for them i e routines for detecting when they are violated we present new exact and heuristic separation algorithms for several classes of inequalities namely lifted cover extended cover weight and lifted pack inequalities moreover we show how to improve a recent separation algorithm for the 0 1 knapsack polytope itself extensive computational results on miplib and or library instances show the strengths and limitations of the inequalities and algorithms considered

wave breaking analysis for the fornberg whitham equation abstract in this paper the wave breaking of the fornberg whitham equation is studied we first investigate the local well posedness and show a blow up scenario by local in time a priori estimates next we present a sufficient condition on the initial data to ensure wave breaking moreover the estimates of life span and blow up rate are obtained

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

on uniqueness properties of solutions of the k generalized kdv equations in this paper we study uniqueness properties of solutions of the so called k generalized korteweg de vries equations our goal is to obtain sufficient conditions on the behavior of the difference u1 u2 of two solutions u1 u2 of 1 1 at two different times t0 0 and t1 1 which guarantee that u1 u2

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

computing deep facet defining disjunctive cuts for mixed integer programming the problem of separation is to find an affine hyperplane or cut that lies between the origin o and a given closed convex set q in a euclidean space we study cuts which are deep in a well defined geometrical sense and facet defining the cases when the deepest cut is decomposable as a combination of facet defining cuts are characterized using the reverse polar set when q is a disjunctive polyhedron a description of the reverse polar linked to the so called cut generating linear program of lift and project techniques is given a successive projections algorithm onto the reverse polar is proposed that computes the decomposition of the deepest cut into facet defining cuts illustrative numerical experiments show how these cuts compare with the deepest cut and with the most violated cut

global analyticity for a generalized camassa holm equation and decay of the radius of spatial analyticity abstract global analytic solution in both the time and the space variables is proved for the cauchy problem of a generalized ch equation which contains as its members two integrable equations namely the camassa holm and the novikov equations the main assumptions are that the initial datum u 0 x is analytic on the line it has uniform radius of analyticity r 0 0 and is such that the mckean quantity m 0 x 1 x 2 u 0 x does not change sign furthermore an explicit lower bound on the radius of space analyticity at later times is obtained which is of the form l 3 exp l 1 exp l 2 t where l 1 l 2 and l 3 are appropriate positive constants

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

on the spectral radius of a 0 1 matrix related to merten s function abstract define n n matrices dn dij and cn cij by dij 1 if i j 0 otherwise and cn 0 1 1 1 t 1 0 0 0 let an dn cn we use the directed graph of an in to obtain the characteristic polynomial of an then we show that all but log2n 1 of the eigenvalues of an are equal to 1 and that an is asymptotically equal to n as n

a dominant negative eigenvalue of a matrix of redheffer abstract define the matrix an as the sum of dn dij and cn where dij 1 when i j and zero when i j and where cn 0 1 1 1 t 1 0 0 0 we use the gersgorin disc theorem and previous results regarding the spectral radiusand the characteristic polynomial of an to prove the existence of a real eigenvalue which is asymptotically equal to n as n we also show that all eigenvalues except the spectral radius and eigenvalue of order n are for large n contained in the circle z c z 1 n log n where an is the spectral radius

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

a cut and solve based algorithm for the single source capacitated facility location problem abstract in this paper we present a cut and solve cs based exact algorithm for the single source capacitated facility location problem sscflp at each level of cs s branching tree it has only two nodes corresponding to the sparse problem sp and the dense problem dp respectively the sp whose solution space is relatively small with the values of some variables fixed to zero is solved to optimality by using a commercial mip solver and its solution if it exists provides an upper bound to the sscflp meanwhile the resolution of the lp of dp provides a lower bound for the sscflp a cutting plane method which combines the lifted cover inequalities and fenchel cutting planes to separate the 0 1 knapsack polytopes is applied to strengthen the lower bound of sscflp and that of dp these lower bounds are further tightened with a partial integrality strategy numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the cs approach in searching an optimal solution in a reasonable time computational results on large sized instances are also presented

breakdown of the camassa holm equation the camassa holm equation is written in its eulerian form as displaystyle begin array rcl text ch quad frac partial v partial t v frac partial v partial x frac partial p partial x 0 end array with pressure p g v 2 frac 1 2 v 2 here g 1 d 2 1 frac 1 2 e vert x y vert it is easy to see that if at time t 0 v is odd with v x 0 for x 0 and v 0 0 then the slope s t v t 0 satisfies s bullet frac 1 2 s 2 and so is driven down to infty at some time t leq 2 s 0 in short the flow may break down this is the steepening lemma of camassa and holm 2 nothing worse happens v t x itself cannot jump like the usual kind of shock actually it is not the obvious v but rather the auxiliary function m v v that controls this m is a caricature of the vorticity that controls the behavior of honest euler in dimensions 2 and 3 see bertozzi and majda 1 for a full account

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

resolving degeneracy in combinatorial linear programs steepest edge steepest ascent and parametric ascent while variants of the steepest edge pivot rule are commonly used in linear programming codes they are not known to have the theoretically attractive property of avoiding an infinite sequence of pivots at points of degeneracy an example is presented demonstrating that the steepest edge pivot rule can fail to terminate finitely it is then shown that a natural extension of the steepest edge pivot rule based on steepest ascent is guaranteed to determine a direction of ascent or a proof that no such direction exists after a finite number of pivots although without modification the extension may not terminate with an ascent direction corresponding to an edge finally it is demonstrated that a computationally more efficient pivot rule proposed by magnanti and orlin has similar theoretical properties to steepest ascent with probability 1independent of the linear program being solved unlike alternative methods such as primal lexicographic rules and bland s rule the algorithms described here have the advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy thus making them attractive in combinatorial settings where the linear program is represented implicitly

on the cauchy problem for the periodic b family of equations and of the non uniform continuity of degasperis procesi equation abstract we consider the problem of the existence of the global solutions and formation of singularities for a b family of equations which includes the camassa holm and degasperis procesi equation we also consider the problem of the uniformly continuity of degasperis procesi equation

steepest edge simplex algorithms for linear programming we present several new steepest edge simplex algorithms for solving linear programming problems including variants of both the primal and the dual simplex method these algorithms differ depending upon the space in which the problem is viewed as residing and include variants in which this space varies dynamically we present computational results comparing steepest edge simplex algorithms and approximate versions of them against simplex algorithms that use standard pivoting rules on truly large scale realworld linear programs with as many as tens of thousands of rows and columns these results demonstrate unambiguously the superiority of steepest edge pivot selection criteria to other pivot selection criteria in the simplex method

decay properties of solutions to a 4 parameter family of wave equations in this paper persistence properties of solutions are investigated for a 4 parameter family k abc equation of evolution equations having k 1 degree nonlinearities and containing as its integrable members the camassa holm the degasperis procesi novikov and fokas olver rosenau qiao equations these properties will imply that strong solutions of the k abc equation will decay at infinity in the spatial variable provided that the initial data does furthermore it is shown that the equation exhibits unique continuation for appropriate values of the parameters k a b and c

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the jordan 1 structure of a matrix of redheffer abstract let 1 1 0 0 and 1 1 1 be complex vectors of length n and let d n d ij with d ij 1 if i j and 0 otherwise be an n n complex matrix then the sizes of the jordan blocks of the matrix t 1 d n corresponding to the eigenvalue 1 are log 2 n 3 1 log 2 n 5 1 log 2 n n 1 where n is the greatest odd integer less than or equal to n

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

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

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

persistence properties and unique continuation for a generalized camassa holm equation in this paper persistence properties of solutions are investigated for a generalized camassa holm equation g k bch having k 1 degree nonlinearities and containing as its integrable members the camassa holm the degasperis procesi and the novikov equations the persistence properties will imply that strong solutions of the g k bch equation will decay at infinity in the spatial variable provided that the initial data does furthermore it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k finally existence of global solutions is established when b k 1

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

cutting planes for mixed integer knapsack polyhedra an algorithm for generating cutting planes for mixed integer knapsack polyhedra is described the algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper computational results are presented 1998 the mathematical programming society inc published by elsevier science b v

local and global lifted cover inequalities for the 0 1 multidimensional knapsack problem abstract the 0 1 multidimensional knapsack problem 0 1 mkp is a well known and strongly np hard combinatorial optimization problem with many applications up to now the majority of upper bounding techniques for the 0 1 mkp have been based on lagrangian or surrogate relaxation we show that good upper bounds can be obtained by a cutting plane method based on lifted cover inequalities lcis as well as using traditional lcis we use some new global lcis which take the whole constraint matrix into account

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation