on the exact separation of mixed integer knapsack cuts during the last decades much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets which we call knapsack cuts bixby et al the sharpest cut the impact of manfred padberg and his work mps siam series on optimization pp 309 326 2004 empirically observe that within the context of branch and cut algorithms to solve mixed integer programming problems the most important inequalities are knapsack cuts derived by the mixed integer rounding mir procedure in this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set the main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships the exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them using these benchmarks on miplib 3 0 and miplib 2003 instances we analyze the performance of mir inequalities our computations which are performed in exact arithmetic are surprising in the vast majority of the instances in which knapsack cuts yield bound improvements mir cuts alone achieve over 87 of the observed gain

the second pinching theorem for hypersurfaces with constant mean curvature in a sphere we generalize the second pinching theorem for minimal hypersurfaces in a sphere due to peng terng wei xu zhang and ding xin to the case of hypersurfaces with small constant mean curvature let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exist two positive constants gamma n and delta n depending only on n such that if h le gamma n and beta n h le s le beta n h delta n then s equiv beta n h and m is one of the following cases i s k big sqrt frac k n big times s n k big sqrt frac n k n big 1 le k le n 1 ii s 1 big frac 1 sqrt 1 mu 2 big times s n 1 big frac mu sqrt 1 mu 2 big here beta n h n frac n 3 2 n 1 h 2 frac n n 2 2 n 1 sqrt n 2h 4 4 n 1 h 2 and mu frac n h sqrt n 2h 2 4 n 1 2

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

cut generation through binarization for a mixed integer linear program where all integer variables are bounded we study a reformulation introduced by roy that maps general integer variables to a collection of binary variables we study theoretical properties and empirical strength of rank 2 simple split cuts of the reformulation we show that for a pure integer problem with two integer variables these cuts are sufficient to obtain the integer hull of the problem but that this does not generalize to problems in higher dimensions we also give an algorithm to compute an approximation of the rank 2 simple split cut closure we report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice free cut closure it gives a better bound than the split closure on 6 instances while it is weaker on 9 instances for an average gap closed 3 8 smaller than the one for the split closure

markov decision drift processes conditions for optimality obtained by discretization in a recent paper hordijk a f a van der duyn schouten 1983 discretization and weak convergence in markov decision drift processes math oper res 8 112 141 the authors gave sufficient conditions for the weak convergence of a sequence of discrete time markov decision drift processes to a related continuous time markov decision drift process the goal of this analysis was to obtain for the continuous time model conditions for optimality of a limit point of a sequence of discounted discrete time optimal policies however the general conditions in hordijk a f a van der duyn schouten 1983 discretization and weak convergence in markov decision drift processes math oper res 8 112 141 can only be applied to a very restrictive class of models to obtain more widely applicable conditions we are concerned in this paper not with the weak convergence of the stochastic processes induced by policies but rather with the convergence of the total expected discounted costs special attention is p

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

n dimensional area of minimal rotational hypersurfaces in spheres abstract let m r n denote the set of compact minimal rotational hypersurfaces in the n 1 dimensional sphere s n 1 for any compact hypersurface m s n 1 let us denote by m its n dimensional area in this paper we show that for any integer n 2 the set of numbers a n m m m r n is a discrete set we also show that if m m r 3 then either m s 3 or m m 0 3 or m m 0 3 where m 0 n s n 1 n 1 n s 1 1 n finally for 2 n 100 we show numerically evidence that if m m r n then either m s n or m m 0 n or m 2 m 0 n we strongly believe that for any m m r n either m s n or m m 0 n or m 2 m 0 n for all n

the gauss kronecker curvature of minimal hypersurfaces in four dimensional space forms let mathbb q 4 c be a four dimensional space form of constant curvature c in this paper we show that the infimum of the absolute value of the gauss kronecker curvature of a complete minimal hypersurface in mathbb q 4 c c leq 0 whose ricci curvature is bounded from below is equal to zero further we study the connected minimal hypersurfaces m 3 of a space form mathbb q 4 c with constant gauss kronecker curvature k for the case c 0 we prove by a local argument that if k is constant then k must be equal to zero we also present a classification of complete minimal hypersurfaces of mathbb q 4 c with k constant

generalized chvatal gomory closures for integer programs with bounds on variables integer programming problems that arise in practice often involve decision variables with one or two sided bounds in this paper we consider a generalization of chvatal gomory inequalities obtained by strengthening chvatal gomory inequalities using the bounds on the variables we prove that the closure of a rational polyhedron obtained after applying the generalized chvatal gomory inequalities is also a rational polyhedron this generalizes a result of dunkel and schulz on 0 1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded

intersection cuts with infinite split rank we consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables when this model only has two integer variables dey and louveaux characterized the intersection cuts that have infinite split rank we show that for any number of integer variables the split rank of an intersection cut generated from a rational lattice free polytope l is finite if and only if the integer points on the boundary of l satisfy a certain 2 hyperplane property the dey louveaux characterization is a consequence of this more general result

a new characterization of the clifford torus via scalar curvature pinching abstract let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exists an explicit positive constant n depending only on n such that if h n and n h s n h n 23 then s n h and m is one of the following cases i s k k n s n k n k n 1 k n 1 ii s 1 1 1 2 s n 1 1 2 here n h n n 3 2 n 1 h 2 n n 2 2 n 1 n 2 h 4 4 n 1 h 2 and n h n 2 h 2 4 n 1 2 for n 6 12 24 we give examples to show that the assumption h n cannot be removed

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

ancient solutions to mean curvature flow for isoparametric submanifolds mean curvature flow for isoparametric submanifolds in euclidean spaces and spheres was studied in liu and terng duke math j 147 1 157 179 2009 in this paper we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity we also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the chern s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres

gap theorems for minimal submanifolds of a hyperbolic space abstract this paper provides some gap theorems for complete immersed minimal submanifolds of dimension no less than five in a hyperbolic space namely we show that an n 5 dimensional complete immersed minimal submanifold m in a hyperbolic space is totally geodesic if the l 2 norm of a on geodesic balls centered at some point p m has less than quadratic growth and if either sup x m a 2 x is not too large or the l n norm of a on m is small here a is the second fundamental form of m

new modeling approaches for the design of local access transport area networks abstract this paper is concerned with the design of a local access and transport area lata network that arises in the context of providing a special access service by a local exchange facility the problem includes locating suitable hub facilities having adequate multiplexing capacity and routing the demand traffic from various clients end offices through their hubs or alternatively through a special point of presence node in order to minimize total costs these costs involve various traffic distribution and equipment component costs where the latter typically exhibit nonlinear economies of scale we develop a model for this problem and apply the reformulation linearization technique rlt to construct various enhanced tightened versions of the proposed model we also design a hybrid heuristic procedure that combines a limited run of an exact method with a lagrangian dual relaxation procedure extensive computational results are provided to study the relative tightness resulting from the various formulations and their effect on providing good quality feasible solutions

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

reverse split rank the reverse split rank of an integral polyhedron p p is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is p p already in mathbb r 3 r3 there exist polyhedra with infinite reverse split rank we give a geometric characterization of the integral polyhedra in mathbb r n rn with infinite reverse split rank

design and verify a new scheme for generating cutting planes a cutting plane procedure for integer programming ip problems usually involves invoking a black box procedure such as the gomory chvatal procedure to compute a cutting plane in this paper we describe an alternative paradigm of using the same cutting plane black box this involves two steps in the first step we design an inequality cx le d where c and d are integral independent of the cutting plane black box in the second step we verify that the designed inequality is a valid inequality by verifying that the set p cap x in mathbb r n mid cx ge d 1 cap mathbb z n is empty using cutting planes from the black box here p is the feasible region of the linear programming relaxation of the ip we refer to the closure of all cutting planes that can be verified to be valid using a specific cutting plane black box as the verification closure of the considered cutting plane black box this paper undertakes a systematic study of properties of verification closures of various cutting plane black box procedures

discretization and weak convergence in markov decision drift processes in this paper we deal with continuous time markov decision drift processes ctmdp which permit both controls affecting jump rates of the process and impulsive controls causing immediate transitions between two successive jump epochs the state of the process evolves according to a deterministic drift function given a ctmdp we construct a sequence of discrete time markov decision drift processes dtmdp with decreasing distance between the successive decision epochs sufficient conditions are provided under which the law of the ctmdp controlled by a fixed policy is the limit in the sense of weak convergence of probability measures of the laws of the approximating dtmdf s controlled by fixed discrete time policies the conditions concern both the parameters of the ctmdp and the relation between the discrete time and continuous time policies an application to a maintenance replacement model is given

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

convex hull of two quadratic or a conic quadratic and a quadratic inequality in this paper we consider an aggregation technique introduced by yildiran j math control inf 26 417 450 2009 to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality yildiran 2009 shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using linear matrix inequalities we show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality we also show that for sets defined by a strict conic quadratic and a strict quadratic inequality under one additional containment assumption these valid inequalities characterize the convex hull exactly we also show that under certain topological assumptions the results from the open setting can be extended to characterize the closed convex hull of sets defined with non strict conic and quadratic inequalities

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

a connection between cutting plane theory and the geometry of numbers in this paper we relate several questions about cutting planes to a fundamental problem in the geometry of numbers namely the closest vector problem using this connection we show that the dominance membership and validity problems are np complete for chvatal and split cuts

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

stability and ergodicity of piecewise deterministic markov processes the main goal of this paper is to establish some equivalence results on stability recurrence and ergodicity between a piecewise deterministic markov process pdmp x t and an embedded discrete time markov chain theta n generated by a markov kernel g that can be explicitly characterized in terms of the three local characteristics of the pdmp leading to tractable criterion results first we establish some important results characterizing theta n as a sampling of the pdmp x t and deriving a connection between the probability of the first return time to a set for the discrete time markov chains generated by g and the resolvent kernel r of the pdmp from these results we obtain equivalence results regarding irreducibility existence of sigma finite invariant measures and positive recurrence and positive harris recurrence between x t and theta n generalizing the results of f dufour and o l v costa siam j control optim 37 1999 pp 1483 1502 in several directions sufficient conditions in terms of a modified foster lyapunov criterion are also presented to ensure positive harris recurrence and ergodicity of the pdmp we illustrate the use of these conditions by showing the ergodicity of a capacity expansion model

average continuous control of piecewise deterministic markov processes this paper deals with the long run average continuous control problem of piecewise deterministic markov processes pdmps taking values in a general borel space and with compact action space depending on the state variable the control variable acts on the jump rate and transition measure of the pdmp and the running and boundary costs are assumed to be positive but not necessarily bounded our first main result is to obtain an optimality equation for the long run average cost in terms of a discrete time optimality equation related to the embedded markov chain given by the postjump location of the pdmp our second main result guarantees the existence of a feedback measurable selector for the discrete time optimality equation by establishing a connection between this equation and an integro differential equation our final main result is to obtain some sufficient conditions for the existence of a solution for a discrete time optimality inequality and an ordinary optimal feedback control for the long run average cost using the so called vanishing discount approach two examples are presented illustrating the possible applications of the results developed in the paper

maximal lattice free convex sets in linear subspaces we consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice free convex sets in an affine subspace we also show that these sets are polyhedra the latter result extends a theorem of lovasz characterizing maximal lattice free convex sets in rn

a new lift and project operator abstract in this paper we analyze the strength of split cuts in a lift and project framework we first observe that the lovasz schrijver and sherali adams lift and project operator hierarchies can be viewed as applying specific 0 1 split cuts to an appropriate extended formulation and demonstrate how to strengthen these hierarchies using additional split cuts more precisely we define a new operator that adds all 0 1 split cuts to the extended formulation for 0 1 mixed integer sets with k binary variables this new operator is guaranteed to obtain the integer hull in k 2 steps compared to k steps for the lovasz schrijver or the sherali adams operator we also present computational results on the stable set problem with our new operator

on chern s conjecture for minimal hypersurfaces and rigidity of self shrinkers using the generalized yau parameter method and the sylvester theory we verify that if m is a compact minimal hypersurface in sn 1 whose squared length of the second fundamental form satisfies 0 a 2 n n22 then a 2 n and m is a clifford torus moreover we prove that if m is a complete self shrinker with polynomial volume growth in rn 1 and if the squared length of the second fundamental form of m satisfies 0 a 2 1 121 then a 2 1 and m is a round sphere or a cylinder our results improve the rigidity theorems due to ding and xin 21 22

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

on the uncapacitated plant location problem i valid inequalities and facets the uncapacitated plant location problem is considered as a node packing problem for this problem several valid inequalities and facets are discussed necessary and sufficient conditions for trivial facets along with necessary conditions for nontrivial facets are derived in addition all of the facets for the case of three plants and three or more destinations are identified

a mobius rigidity of compact willmore hypersurfaces in sn 1 abstract let x m n s n 1 be an immersed hypersurface without umbilical point then one can define the mobius metric g the mobius second fundamental form b and the blaschke tensor a on the hypersurface m n which are invariant under the mobius transformation group of s n 1 a hypersurface is called a willmore hypersurface if it is the critical point of the volume functional of m n with respect to the mobius metric g in this paper we prove that if the hypersurface x is a compact willmore hypersurface without umbilical point then m n a b 2 n 1 n d m g 0 the equality holds if and only if the hypersurface m n is mobius equivalent to one of the willmore tori w n m s m n m n s n m m n s n 1 1 m n 1 where the tensor a a b 2 1 n t r a n 1 n g

stability of piecewise deterministic markov processes in this paper we study a form of stability for a general family of nondiffusion markov processes known in the literature as piecewise deterministic markov process pdmp by stability here we mean the existence of an invariant probability measure for the pdmp it is shown that the existence of such an invariant probability measure is equivalent to the existence of a sigma finite invariant measure for a markov kernel g linked to the resolvent operator u of the pdmp satisfying a boundedness condition or equivalently a radon nikod m derivative here we generalize existing results of the literature o costa j appl prob 27 1990 pp 60 73 m davis markov models and optimization chapman and hall 1993 since we do not require any additional assumptions to establish this equivalence moreover we give sufficient conditions to ensure the existence of such a sigma finite measure satisfying the boundedness condition they are mainly based on a modified foster lyapunov criterion for the case in which the markov chain generated by g is either recurrent or weak feller to emphasize the relevance of our results we study three examples and in particular we are able to generalize the results obtained by costa and davis on the capacity expansion model

ddvv type inequality for skew symmetric matrices and simons type inequality for riemannian submersions abstract in this paper we will first derive a ddvv type optimal inequality for real skew symmetric matrices then we apply it to establish a simons type integral inequality for riemannian submersions with totally geodesic fibres and yang mills horizontal distributions in this way we show phenomenons of duality between submanifold geometry and riemannian submersion particularly between second fundamental form of a submanifold and integrability tensor of a riemannian submersion

on the complexity of selecting disjunctions in integer programming the imposition of general disjunctions of the form pi x leq pi 0 vee pi x geq pi 0 1 where pi pi 0 are integer valued is a fundamental operation in both the branch and bound and cutting plane algorithms for solving mixed integer linear programs such disjunctions can be used for branching at each iteration of the branch and bound algorithm or to generate split inequalities for the cutting plane algorithm we first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction according to specific criteria described herein is mathcal np hard we further show that the problem remains mathcal np hard even for binary programs or when considering certain restricted classes of disjunctions we observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems which leads to a proof that the problem is mathcal np complete

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

closed willmore minimal hypersurfaces with constant scalar curvature in s5 1 are isoparametric abstract in this paper we will prove that any closed minimal willmore hypersurface m 4 of s 5 with constant scalar curvature must be isoparametric to be precise m 4 is either an equatorial 4 sphere a product of sphere s 2 2 2 s 2 2 2 or a cartan s minimal hypersurface in particular the value of the second fundamental form s can only be 0 4 12 this result strongly supports chern s conjecture

projected chvatal gomory cuts for mixed integer linear programs recent experiments by fischetti and lodi show that the first chvatal closure of a pure integer linear program ilp often gives a surprisingly tight approximation of the integer hull they optimize over the first chvatal closure by modeling the chvatal gomory cg separation problem as a mixed integer linear program milp which is then solved by a general purpose milp solver unfortunately this approach does not extend immediately to the gomory mixed integer gmi closure of an milp since the gmi separation problem involves the solution of a nonlinear mixed integer program or a parametric milp in this paper we introduce a projected version of the cg cuts and study their practical effectiveness for milp problems the idea is to project first the linear programming relaxation of the milp at hand onto the space of the integer variables and then to derive chvatal gomory cuts for the projected polyhedron though theoretically dominated by gmi cuts projected cg cuts have the advantage of producing a separation model very similar to the one introduced by fischetti and lodi which can typically be solved in a reasonable amount of computing time

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

on chernʼs problem for rigidity of minimal hypersurfaces in the spheres abstract for a compact minimal hypersurface m in s n 1 with the squared length of the second fundamental form s we confirm that there exists a positive constant n depending only on n such that if n s n n then s n i e m is a clifford minimal hypersurface in particular when n 6 the pinching constant n n 23

split rank of triangle and quadrilateral inequalities a simple relaxation consisting of two rows of a simplex tableau is a mixed integer set with two equations two free integer variables and nonnegative continuous variables recently andersen et al and cornuejols and margot showed that the facet defining inequalities of this set are either split cuts or intersection cuts obtained from lattice free triangles and quadrilaterals from an example given by cook et al it is known that one particular class of facet defining triangle inequality does not have finite split rank in this paper we show that all other facet defining triangle and quadrilateral inequalities have finite split rank

the compactness of a policy space in dynamic programming via an extension theorem for carathe odory functions the background for this paper is a dynamic programming model with a borel state space and compact action sets a new simple proof of the compactness of a space of measures corresponding to randomized stationary policies is given in a topology induced by caratheodory functions on the state action space for this purpose an extension theorem for caratheodory functions is proved which is a new version of the tietze s extension theorem for continuous functions

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

on the rank of disjunctive cuts let l be a family of lattice free polyhedra in rm containing the splits given a polyhedron p in rm n we characterize when a valid inequality for p zm rn can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l we also characterize the lattice free polyhedra m such that all the disjunctive cuts corresponding to m can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l for every polyhedron p our results imply interesting consequences related to split rank and to integral lattice free polyhedra that extend recent research findings

asymptotic control for a class of piecewise deterministic markov processes associated to temperate viruses we aim at characterizing the asymptotic behavior of value functions in the control of piecewise deterministic markov processes pdmp of switch type under nonexpansive assumptions for a particular class of processes inspired by temperate viruses we show that uniform limits of discounted problems as the discount decreases to zero and time averaged problems as the time horizon increases to infinity exist and coincide the arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics the approach strongly relies on viscosity techniques linear programming arguments and coupling via random measures associated to pdmp as an intermediate step in our approach we present the approximation of discounted value functions when using piecewise constant in time open loop policies

hermite approximation of a hyperbolic fokker planck optimality system to control a piecewise deterministic process abstractthe hermite spectral approximation of a hyperbolic fokker planck fp optimality system arising in the control of an unbounded piecewise deterministic process pdp is discussed to control the probability density function pdf corresponding to the pdp process an optimal control based on an fp strategy is considered the resulting optimality system consists of a hyperbolic system with opposite time orientation and an integral optimality condition equation a hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains it is proven that the proposed scheme satisfies the conservativity requirement of the pdfs the spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

generalized intersection cuts and a new cut generating paradigm intersection cuts are generated from a polyhedral cone and a convex set s whose interior contains no feasible integer point we generalize these cuts by replacing the cone with a more general polyhedron c the resulting generalized intersection cuts dominate the original ones this leads to a new cutting plane paradigm under which one generates and stores the intersection points of the extreme rays of c with the boundary of s rather than the cuts themselves these intersection points can then be used to generate in a non recursive fashion cuts that would require several recursive applications of some standard cut generating routine a procedure is also given for strengthening the coefficients of the integer constrained variables of a generalized intersection cut the new cutting plane paradigm yields a new characterization of the closure of intersection cuts and their strengthened variants this characterization is minimal in the sense that every one of the inequalities it uses defines a facet of the closure

scalar curvature of minimal hypersurfaces in a sphere abstract we improve the well known scalar curvature pinching theorem due to peng terng for n n 5 dimensional minimal hypersurfaces to the case of arbitrary n precisely if m is a closed and minimal hypersurface in a unit sphere s n 1 then there exists a positive constant n depending only on n such that if n s n n then s n i e m is a clifford torus s k k n s n k n k n k 1 2 n 1

a gap theorem of self shrinkers in this paper we study complete self shrinkers in euclidean space and prove that an n dimensional complete self shrinker with polynomial volume growth in euclidean space r n 1 is isometric to either r n s n n or r n m s m m 1 m n 1 if the squared norm s of the second fundamental form is constant and satisfies s 10 7

on convergence in mixed integer programming let p subseteq mathbb r m n be a rational polyhedron and let pi be the convex hull of p cap mathbb z m times mathbb r n we define the integral lattice free closure of p as the set obtained from p by adding all inequalities obtained from disjunctions associated with integral lattice free polyhedra in mathbb r m we show that the integral lattice free closure of p is again a polyhedron and that repeatedly taking the integral lattice free closure of p gives pi after a finite number of iterations such results can be seen as a mixed integer analogue of theorems by chvatal and schrijver for the pure integer case one ingredient of our proof is an extension of a result by owen and mehrotra in fact we prove that for each rational polyhedron p the split closures of p yield in the limit the set pi

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

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