finite dimensional vertex algebra modules over fixed point commutative subalgebras abstract let a be a connected commutative c algebra with derivation d g a finite linear automorphism group of a which preserves d and r a g the fixed point subalgebra of a under the action of g we show that if a is generated by a single element as an r algebra and is a galois extension over r in the sense of m auslander and o goldman then every finite dimensional indecomposable vertex algebra r module has a structure of twisted vertex algebra a module

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

finite dimensional vertex algebra modules over fixed point differential subfields abstract let k be a differential field over c with derivation d g a finite linear automorphism group over k which preserves d and k g the fixed point subfield of k under the action of g we show that every finite dimensional vertex algebra k g module is contained in some twisted vertex algebra k module

a note on the selection of benders cuts a new cut selection criterion for benders cuts is proposed and computationally analyzed the results show that the new criterion is more robust and often considerably faster than the standard ones

a branch and cut approach to the cardinality constrained circuit problem the cardinality constrained circuit problem cccp is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges the cccp is np hard we present classes of facet inducing inequalities for the convex hull of feasible circuits and a branch and cut solution approach using these inequalities

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

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

the arithmetic of function fields 2 the cyclotomic theory contents i the ciassical rheoy qf cutx es orer a finite jeld 2 the cfclotomic construction for number jelds and function jields 2 1 number fields 2 2 function fields 3 complex arld p adic zeta functions 3 1 the complex theory 3 2 the p adic theory 4 classical iwasabila rheory 4 1 kummer s criterion 4 2 the theoq of the tower 5 two variable zera functions 5 1 the theory at infinity 5 2 special values 6 the theory for futzction fields 6 1 the basic set up 6 2 the kummer criterion 6 3 the theory of the tower 7 l nirs 7 i number fields 7 2 function fields references

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

harmonic analysis and the flow of a drinfeld module abstract let c be the carlitz module associated to a fr u associated to c there are two types of flows a naive flow coming from the additive group and a twisted flow coming from the theory of v adic measures v e spec a the study of these flows leads us to the introduction of a new type of fourier transform for continuous characteristic p valued functions on zp this fourier transform is an invertible operator from continuous functions to continuous functions we establish analogs of classical results such as parseval s theorem and poisson summation the transforms in this paper should have applications in the study of the special functions i e characteristic p valued zeta and gamma functions that appear in the arithmetic of function fields indeed the techniques introduced here may eventually lead to the construction of such objects as theta series

the classical umbral calculus and the flow of a drinfeld module the notion of a flow appears in many areas of mathematics and physics for example it is one of the fundamental notions in studying ordinary differential equations classically a flow on a set x is a group action of the additive group of the set of real numbers r on the set x one of the classical theorems in analysis the classical taylor theorem can be expressed in terms of a flow more precisely let d dt be the formal derivative acting on the polynomial ring r t for each x r let e r t r t be the linear operator defined by

finite dimensional modules for the polynomial ring in one variable as a vertex algebra abstract a commutative associative algebra a over c with a derivation is one of the simplest examples of a vertex algebra however the differences between the modules for vertex algebra a and the modules for associative algebra a are not well understood in this paper i give the classification of finite dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over c as a vertex algebra

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

reduce and split cuts improving the performance of mixed integer gomory cuts mixed integer gomory cuts have become an integral part of state of the art software for solving mixed integer linear programming problems therefore improvements in the performance of these cutting planes can be of great practical value in this paper we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed integer gomory cuts this is motivated by the fact that in a mixed integer gomory cut the coefficient of an integer variable lies between 0 and 1 whereas for a continuous variable there is no upper bound the heuristic tries to reduce the coefficients of the continuous variables we call the resulting cuts reduce and split cuts we found that on several test problems reduce and split cuts can substantially enhance the performance of a branch and bound algorithm

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

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

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

parallel branch cut and price for large scale discrete optimization in discrete optimization most exact solution approaches are based on branch and bound which is conceptually easy to parallelize in its simplest forms more sophisticated variants such as the so called branch cut and price algorithms are more difficult to parallelize because of the need to share large amounts of knowledge discovered during the search process in the first part of the paper we survey the issues involved in parallelizing such algorithms we then review the implementation of symphony and coin bcp two existing frameworks for implementing parallel branch cut and price these frameworks have limited scalability but are effective on small numbers of processors finally we briefly describe our next generation framework which improves scalability and further abstracts many of the notions inherent in parallel bcp making it possible to implement and parallelize more general classes of algorithms

clique relaxations in social network analysis the maximum k plex problem this paper introduces and studies the maximum k plex problem which arises in social network analysis and has wider applicability in several important areas employing graph based data mining after establishing np completeness of the decision version of the problem on arbitrary graphs an integer programming formulation is presented followed by a polyhedral study to identify combinatorial valid inequalities and facets a branch and cut algorithm is implemented and tested on proposed benchmark instances an algorithmic approach is developed exploiting the graph theoretic properties of a k plex that is effective in solving the problem to optimality on very large sparse graphs such as the power law graphs frequently encountered in the applications of interest

a cutting plane approach to mixed 0 1 stochastic integer programs abstract we consider a mixed 0 1 integer programming problem with dual block angular structure arising in two stage stochastic programming a relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable the convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how lift and project cuts can be generated for one subproblem and made valid for different outcomes

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

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

partial convexification cuts for 0 1 mixed integer programs abstract in this research we propose a new cut generation scheme based on constructing a partial convex hull representation for a given 0 1 mixed integer programming problem by using the reformulation linearization technique rlt we derive a separation problem that projects the extended space of the rlt formulation into the original space in order to generate a cut that deletes a current fractional solution naturally the success of such a partial convexification based cutting plane scheme depends on the process used to tradeoff the strength of the cut derived and the effort expended accordingly we investigate several variable selection rules for performing this convexification along with restricted versions of the accompanying separation problems so as to be able to derive strong cuts within a reasonable effort we also develop a strengthening procedure that enhances the generated cut by considering the binariness of the remaining unselected 0 1 variables finally we present some promising computational results that provide insights into implementing the proposed cutting plane methodology

optimizing over the split closure the polyhedron defined by all the split cuts obtainable directly i e without iterated cut generation from the lp relaxation p of a mixed integer program mip is termed the elementary or rank 1 split closure of p this paper deals with the problem of optimizing over the elementary split closure this is accomplished by repeatedly solving the following separation problem given a fractional point say x find a rank 1 split cut violated by x or show that none exists following caprara and letchford 17 we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program pmilp with a single parameter in the objective function and the right hand side we develop an algorithmic framework to deal with the resulting pmilp by creating and maintaining a dynamically updated grid of parameter values and use the corresponding mixed integer programs to generate rank 1 split cuts our approach was implemented in the coin or framework using cplex 9 0 as a general purpose mip solver we report our computational results on well known benchmark instances from miplib 3 0 and several classes of structured integer and mixed integer problems our computational results show that rank 1 split cuts close more than 98 of the duality gap on 15 out of 41 mixed integer instances from miplib 3 0 more than 75 of the duality gap can be closed on an additional 10 instances the average gap closed over all 41 instances is 72 78 in the pure integer case rank 1 split cuts close more than 75 of the duality gap on 13 out of 24 instances from miplib 3 0 on average rank 1 split cuts close about 72 of the duality gap on these 24 instances we also report results on several classes of structured problems capacitated versions of warehouse location single source facility location p median fixed charge network flow multi commodity network design with splittable and unsplittable flows and lot sizing the fraction of the integrality gap closed varies for these problem classes between 100 and 67 we also gathered statistics on the average coefficient size absolute value of the disjunctions generated they turn out to be surprisingly small