on the relationship between combinatorial and lp based lower bounds for np hard scheduling problems enumerative approaches to solving optimization problems such as branch and bound require a subroutine that produces a lower bound on the value of the optimal solution in the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear programming lp relaxation of the problem or the solution to a combinatorial relaxation in this paper we investigate from a theoretical perspective the relationship between several lp based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled we establish a number of facts about the relationship between these different sorts of lower bounds including the equivalence of certain lp based lower bounds for these problems to combinatorial lower bounds used in successful branch and bound algorithms as a result we obtain the first worst case analysis of the quality of the lower bounds delivered by these combinatorial relaxations

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

multidimensional lower bounds for the eigenvalues of stokes and dirichlet laplacian operators this article is concerned with the improvements of certain eigenvalue inequalities of stokes operator and dirichlet laplacian related to the berezin li yau type inequalities the formulas proved extend the earlier works of melas a lower bound for sums of eigenvalues of the laplacian proc am math soc 131 2 631 636 2002 10 1090 s0002 9939 02 06834 x on dirichlet laplacian and of ilyin lower bounds for the spectrum of the laplace and stokes operators discrete contin dyn syst 28 1 131 146 2010 10 3934 dcds 2010 28 131 on stokes operator for any dimension d 2 and they are asymptotically sharp as are the earlier inequalities of berezin li yau melas and ilyin

on the equivalence of the max min transportation lower bound and the time indexed lower bound for single machine scheduling problems new observations are made about two lower bound schemes for single machine min sum scheduling problems we find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program we show the equivalence of this strongest bound and the bound provided by the lp relaxation of the time indexed integer programming formulation these observations lead to a new lower bound scheme that yields fast approximation of the time indexed bound several techniques are developed to facilitate the effective use of the new lower bound in branch and bound numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from or library results obtained with our new method are spectacular we are able to solve all 125 open problems to optimality

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

hybrid optimization methods for time dependent sequencing problems abstract in this paper we introduce novel optimization methods for sequencing problems in which the setup times between a pair of tasks depend on the relative position of the tasks in the ordering our proposed methods rely on a hybrid approach where a constraint programming model is enhanced with two distinct relaxations one discrete relaxation based on multivalued decision diagrams and one continuous relaxation based on linear programming both relaxations are used to generate bounds and enhance constraint propagation experiments conducted on three variants of the time dependent traveling salesman problem indicate that our techniques substantially outperform general purpose methods such as mixed integer linear programming and constraint programming models

eigenvalue inequalities for klein gordon operators we consider the pseudodifferential operators hm associated by the prescriptions of quantum mechanics to the klein gordon hamiltonian p 2 m2 when restricted to a bounded open domain rd when the mass m is 0 the operator h0 coincides with the generator of the cauchy stochastic process with a killing condition on the operator h0 is sometimes called the fractional laplacian with power 12 cf r banuelos t kulczycki eigenvalue gaps for the cauchy process and a poincare inequality j funct anal 211 2 2004 355 423 r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 234 2006 199 225 e giere the fractional laplacian in applications http www eckhard giere de math publications review pdf we prove several universal inequalities for the eigenvalues 0 1 2 of hm and their means k 1k l 1k l among the inequalities proved are k cst k 1 d for an explicit optimal semiclassical constant depending only on the dimension d for any dimension d 2 and any k k 1 d 1d 1 k furthermore when d 2 and k 2j k j d21 d d 1 kj 1d finally we present some analogous estimates allowing for an operator including an external potential energy field i e hm v x for v x in certain function classes

points de petite hauteur sur les courbes modulaires x0 n let n be a square free integer prime to 6 let the imbeding of x 0 n in its jacobian relative to the point we show that the set is finite and that is infinite this explicit form of the bogomolov conjecture is obtained by an estimation of the self intersection of the dualizing sheaf in the sense of arakelov theory of modular curves this result is obtained by estimating several quantities attached to the arakelov metric on x 0 n starting with petersson s trace formula

dirichlet and neumann eigenvalue problems on domains in euclidean spaces abstract we obtain here some inequalities for the eigenvalues of dirichlet and neumann value problems for general classes of operators or system of operators acting in l 2 or l 2 c m r d d 1

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

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

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

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

two dimensional berezin li yau inequalities with a correction term we improve the berezin li yau inequality in dimension two by adding a positive correction term to its right hand side it is also shown that the asymptotical behaviour of the correction term is almost optimal this improves a previous result by melas 11

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

refined eigenvalue bounds on the dirichlet fractional laplacian the purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the dirichlet fractional laplacian operator 2 restricted to a bounded domain d with d 2 1 2 and d 3 0 2 our main result yields a sharper lower bound in the sense of weyl asymptotics for the berezin li yau type inequality improving the previous result s yildirim yolcu and t yolcu discrete contin dyn syst 35 5 2209 2225 2015 furthermore we give a result improving the bounds for analogous elliptic operators y c kim e print arxiv 1406 5407

attractors for navier stokes equations in domains with finite measure university of wales college of cardiff school of mathematics senghennydd road carditt cf2 4yh u k and hydrometeorological research centre bolshevistskaya 9 l 3 moscow 123376 russia received 28 june 1993 received in revisedform 24 march 1995 received for publication i

estimates on the eigenvalues of the clamped plate problem on domains in euclidean spaces the purpose of this article is two fold first we obtain some certain bounds for the sums of positive and negative powers of the eigenvalues of the clamped plate problem of the dirichlet bi laplacian operator 2 d restricted to a bounded domain d rd with d 2 second we establish lower bounds for the sums of eigenvalues of 2 d sharper than the bounds recently obtained by cheng and wei a lower bound for eigenvalues of a clamped plate problem calculus var partial differ equ 42 3 4 579 590 2011 10 1007 s00526 011 0399 6 all these estimates are sharp in the sense of weyl asymptotics

the relation of time indexed formulations of single machine scheduling problems to the node packing problem the relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules schedules in which not all jobs have to be started the equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established the main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5 hole inequalities of the convex hull of the set of feasible node packings in this graph respectively

a polyhedral approach to single machine scheduling problems we report new results for a time indexed formulation of nonpreemptive single machine scheduling problems we give complete characterizations of all facet inducing inequalities with integral coefficients and right hand side 1 or 2 for the convex hull of the set of feasible partial schedules i e schedules in which not all jobs have to be started furthermore we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope which is the convex hull of the set of feasible complete schedules i e schedules in which all jobs have to be started to obtain insight in the effectiveness of these classes of facet inducing inequalities we develop a branch and cut algorithm based on them we evaluate its performance on the strongly np hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates