maximal lattice free polyhedra finiteness and an explicit description in dimension three a convex set with nonempty interior is maximal lattice free if it is inclusion maximal with respect to the property of not containing integer points in its interior maximal lattice free convex sets are known to be polyhedra the precision of a rational polyhedron p in rd is the smallest natural number s such that sp is an integral polyhedron in this paper we show that up to affine mappings preserving zd the number of maximal lattice free rational polyhedra of a given precision s is finite furthermore we present the complete list of all maximal lattice free integral polyhedra in dimension three our results are motivated by recent research on cutting plane theory in mixed integer linear optimization

a strong dual for conic mixed integer programs mixed integer conic programming is a generalization of mixed integer linear programming in this paper we present an extension of the duality theory for mixed integer linear programming see m guzelsoy and t k ralphs int j oper res taichung 4 2007 pp 118 137 g l nemhauser and l a wolsey integer and combinatorial optimization wiley interscience new york 1988 to the case of mixed integer conic programming in particular we construct a subadditive dual for mixed integer conic programming problems under a simple condition on the primal problem we show that strong duality holds

aggregation based cutting planes for packing and covering integer programs in this paper we study the strength of chvatal gomory cg cuts and more generally aggregation cuts for packing and covering integer programs ips aggregation cuts are obtained as follows given an ip formulation we first generate a single implied inequality using aggregation of the original constraints then obtain the integer hull of the set defined by this single inequality with variable bounds and finally use the inequalities describing the integer hull as cutting planes our first main result is to show that for packing and covering ips the cg and aggregation closures can be 2 approximated by simply generating the respective closures for each of the original formulation constraints without using any aggregations on the other hand we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general ips the proof of the above stated results for the case of covering ips with bounds require the development of some new structural results which may be of independent interest finally we examine the strength of cuts based on k different aggregation inequalities simultaneously the so called multi row cuts and show that every packing or covering ip with a large integrality gap also has a large k aggregation closure rank in particular this rank is always at least of the order of the logarithm of the integrality gap

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

some properties of convex hulls of integer points contained in general convex sets in this paper we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it we first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed this leads to useful results for special classes of convex sets such as pointed cones strictly convex sets and sets containing integer points in their interior we then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron this result generalizes the well known result due to meyer math program 7 223 225 1974 under a simple technical assumption we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron

a geometric perspective on lifting recently it has been shown that minimal inequalities for a continuous relaxation of mixed integer linear programs are associated with maximal lattice free convex sets in this paper we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice free convex sets in a higher dimensional space we apply this approach to several examples in particular we identify cases in which the lifting is unique

the triangle closure is a polyhedron recently cutting planes derived from maximal lattice free convex sets have been studied intensively by the integer programming community an important question in this research area has been to decide whether the closures associated with certain families of lattice free sets are polyhedra for a long time the only result known was the celebrated theorem of cook kannan and schrijver who showed that the split closure is a polyhedron although some fairly general results were obtained by andersen et al math oper res 35 1 233 256 2010 and averkov discret optimiz 9 4 209 215 2012 some basic questions have remained unresolved for example maximal lattice free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron in this paper we show that when the number of integer variables m 2 the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data the techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet defining due to cornuejols and margot math program 120 429 456 2009 and obtain polynomial complexity results about the mixed integer hull

on sublinear inequalities for mixed integer conic programs this paper studies mathcal k k sublinear inequalities a class of inequalities with strong relations to mathcal k k minimal inequalities for disjunctive conic sets we establish a stronger result on the sufficiency of mathcal k k sublinear inequalities that is we show that when mathcal k k is the nonnegative orthant or the second order cone mathcal k k sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set when mathcal k k is the nonnegative orthant mathcal k k sublinear inequalities are tightly connected to functions that generate cuts so called cut generating functions in particular we introduce the concept of relaxed cut generating functions and show that each mathbb r n r n sublinear inequality is generated by one of these we then relate the relaxed cut generating functions to the usual ones studied in the literature recently under a structural assumption cornuejols wolsey and yildiz established the sufficiency of cut generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant we provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of mathbb r n r n sublinear inequalities and their connection with relaxed cut generating functions

characterizations of mixed binary convex quadratic representable sets representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems in this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems we provide a complete characterization of the sets that can be obtained as the projection of such feasible regions in order to obtain this result we first provide a complete characterization of these sets in the special cases where 1 the feasible region is bounded 2 only binary extended variables are present and 3 only continuous variables are present

the structure of the infinite models in integer programming the infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions in this paper we study the relationships between these two descriptions our results have implications for finite dimensional corner polyhedra one consequence is that nonnegative continuous functions suffice to describe finite dimensional corner polyhedra with rational data we also discover new facts about corner polyhedra with non rational data

unique minimal liftings for simplicial polytopes for a minimal inequality derived from a maximal lattice free simplicial polytope in rn we investigate the region where minimal liftings are uniquely defined and we characterize when this region covers rn we then use this characterization to show that a minimal inequality derived from a maximal lattice free simplex in rn with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points

tighter bounds on the error probability of fixed convolutional codes we introduce new upper bounds for the bit and burst error probabilities when a convolutional code is used for transmission over a binary input additive white gaussian noise awgn channel and the decoder performs maximum likelihood decoding the bounds are tighter than existing bounds and nontrivial also for signal to noise ratios below the computational cutoff rate

constrained infinite group relaxations of mips recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized in this paper we consider a stronger relaxation of general mixed integer sets by allowing constraints such as bounds on the free integer variables in the continuous group relaxation we generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

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

valid inequalities based on simple mixed integer sets in this paper we use facets of simple mixed integer sets with three variables to derive a parametric family of valid inequalities for general mixed integer sets we call these inequalities two step mir inequalities as they can be derived by applying the simple mixed integer rounding mir principle of wolsey 1998 twice the two step mir inequalities define facets of the master cyclic group polyhedron of gomory 1969 in addition they dominate the strong fractional cuts of letchford and lodi 2002

on the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs we compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints two unrestricted integer variables and any number of nonnegative continuous variables in particular we prove that the closure of type 2 triangle resp type 3 triangle quadrilateral inequalities are all within a factor of 1 5 1 5 of the integer hull and provide examples showing that the approximation factor is not less than 1 125 1 125 there is no fixed approximation ratio for split or type 1 triangle inequalities however

decomposition algorithms with parametric gomory cuts for two stage stochastic integer programs we consider a class of two stage stochastic integer programs with binary variables in the rst stage and general integer variables in the second stage we develop decomposition algorithms akin to the l shaped or benders methods by utilizing gomory cuts to obtain iteratively tighter approximations of the second stage integer programs we show that the proposed methodology is exible in that it allows several modes of implementation all of which lead to nitely convergent algorithms we illustrate our algorithms using examples from the literature we report computational results using the stochastic server location problem instances which suggest that our decomposition based approach scales better with increases in the number of scenarios than a state of the art solver which was used to solve the deterministic equivalent formulation

facets of two dimensional infinite group problems in this paper we lay the foundation for the study of the two dimensional mixed integer infinite group problem 2dmiigp we introduce tools to determine if a given continuous and piecewise linear function over the two dimensional infinite group is subadditive and to determine whether it defines a facet of 2dmiigp we then present two different constructions that yield the first known families of facet defining inequalities for 2dmiigp the first construction uses valid inequalities of the one dimensional integer infinite group problem 1diigp as building blocks for creating inequalities for the two dimensional integer infinite group problem 2diigp we prove that this construction yields all continuous piecewise linear facets of the two dimensional group problem that have exactly two gradients the second construction we present has three gradients and yields facet defining inequalities of 2dmiigp whose continuous coefficients are not dominated by those of facets of the one dimensional mixed integer infinite group problem 1dmiigp

a counterexample to a conjecture of gomory and johnson in mathematical programming 2003 gomory and johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions in this paper we give an example showing that the gomory johnson conjecture is false

on the augmented lagrangian dual for integer programming we consider the augmented lagrangian dual for integer programming and provide a primal characterization of the resulting bound as a corollary we obtain proof that the augmented lagrangian is a strong dual for integer programming we are able to show that the penalty parameter applied to the augmented lagrangian term may be placed at a fixed large value and still obtain strong duality for pure integer programs

two dimensional lattice free cuts and asymmetric disjunctions for mixed integer polyhedra in this paper we study the relationship between 2d lattice free cuts the family of cuts obtained by taking two row relaxations of a mixed integer program mip and applying intersection cuts based on maximal lattice free sets in r 2 and various types of disjunctions recently li and richard 2007 studied disjunctive cuts obtained from t branch split disjunctions of mixed integer sets these cu ts generalize split cuts balas 2009 initiated the study of cuts for the two row continuous group relaxation obtained from 2 branch split disjunctions we study these cuts and call them cross cuts for the two row continuous group relaxation and for general mips we also consider cuts obtained from asymmetric 2 branch disjunctions which we call crooked cross cuts for the two row continuous group relaxation we show that unimodular cross cuts the coefficients of the two split inequalities fo rm a unimodular matrix are equivalent to the cuts obtained from maximal lattice free sets other than type 3 triangles we also prove that all 2d lattice free cuts and their s free extensions ar e crooked cross cuts for general mixed integer sets we show that crooked cross cuts can be generated from a structured three row relaxation finally we show that for the corner relaxation of an mip every crooked cross cut is a 2d lattice free cut

moderate deviations in channel coding we consider block codes whose rate converges to the channel capacity with increasing blocklength at a certain speed and examine the best possible decay of the probability of error for discrete memoryless channels we prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes

on minimal valid inequalities for mixed integer conic programs we study disjunctive conic sets involving a general regular closed convex full dimensional and pointed cone x1d4a6 such as the nonnegative orthant the lorentz cone or the positive semidefinite cone in a unified framework we introduce x1d4a6 minimal inequalities and show that under mild assumptions these inequalities together with the trivial cone implied inequalities are sufficient to describe the convex hull we focus on the properties of x1d4a6 minimal inequalities by establishing algebraic necessary conditions for an inequality to be x1d4a6 minimal this characterization leads to a broader algebraically defined class of x1d4a6 sublinear inequalities we demonstrate a close connection between x1d4a6 sublinear inequalities and the support functions of convex sets with a particular structure this connection results in practical ways of verifying x1d4a6 sublinearity and or x1d4a6 minimality of inequalities our study generalizes some of the results from the mixed integer linear case it is well known that the minimal inequalities for

cut generating functions for integer variables for an integer linear program gomory s corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation in this paper we do not relax these nonnegativity constraints we generalize a classical result of gomory and johnson characterizing minimal cut generating functions in terms of subadditivity symmetry and periodicity our result is based on the notion of generalized symmetry condition we also prove a 2 slope theorem for extreme cut generating functions in our setting in the spirit of the 2 slope theorem of gomory and johnson

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

discrete lossy gray wyner revisited second order asymptotics large and moderate deviations in this paper we revisit the discrete lossy gray wyner problem in particular we derive its optimal second order coding rate region its error exponent reliability function and its moderate deviations constant under mild conditions on the source to obtain the second order asymptotics we extend some ideas from watanabe s work in particular we leverage the properties of an appropriate generalization of the conditional distortion tilted information density which was first introduced by kostina and verdu the converse part uses a perturbation argument by gu and effros in their strong converse proof of the discrete gray wyner problem the achievability part uses two novel elements 1 a generalization of various type covering lemmas and 2 the uniform continuity of the conditional rate distortion function in both the source joint distribution and the distortion level to obtain the error exponent for the achievability part we use the same generalized type covering lemma and for the converse we use the strong converse together with a change of measure technique finally to obtain the moderate deviations constant we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities

an algorithm for the separation of two row cuts we consider the question of finding deep cuts from a model with two rows of the type p i x s in mathbb z 2 times mathbb r n x f rs to do that we show how to reduce the complexity of setting up the polar of mathop mathrm conv p i from a quadratic number of integer hull computations to a linear number of integer hull computations furthermore we present an algorithm that avoids computing all integer hulls a polynomial running time is not guaranteed but computational results show that the algorithm runs quickly in practice

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 k 1 slope theorem for the k dimensional infinite group relaxation we prove that any minimal valid function for the k dimensional innite group relaxation that is piecewise linear with at most k 1 slopes and does not factor through a linear map with non trivial kernel is extreme this generalizes a theorem of gomory and johnson for k 1 and cornu ejols and molinaro for k 2

he s homotopy perturbation method a strongly promising method for solving non linear systems of the mixed volterra fredholm integral equations in this paper he s homotopy perturbation method is applied to solve non linear systems of mixed volterra fredholm integral equations two examples are presented to illustrate the ability of the method also comparisons are made between the adomian decomposition method and the homotopy perturbation method the results reveal that he s homotopy perturbation method is very effective and simple and in these examples leads to the exact solutions

operations that preserve the covering property of the lifting region we contribute to the theory for minimal liftings of cut generating functions in particular we give three operations that preserve the so called covering property of certain structured cut generating functions this has the consequence of vastly expanding the set of undominated cut generating functions which can be used computationally compared to known examples from the literature the results of this paper are significant generalizations of previous results from the literature on such operations and use completely different proof techniques which we feel are more suitable for attacking future research questions in this area

refinement of the sphere packing bound asymmetric channels we provide a refinement of the sphere packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the subexponential factor in front of the exponent the order of our subexponential factor is n 0 5 1 e r for any 0 where r is the left derivative of the sphere packing exponent at rate r and n is the blocklength

master corner polyhedron vertices abstract we focus on the vertices of the master corner polyhedron mcp a fundamental object in the theory of integer linear programming we introduce two combinatorial operations that transform vertices to their neighbors this implies that each mcp can be defined by the initial vertices regarding these operations we call them support vertices we prove that the class of support vertices of all mcps over a group is invariant under automorphisms of this group and describe mcp vertex bases among other results we characterize its irreducible points establish relations between a vertex and the nontrivial facets that pass through it and prove that this polyhedron is of diameter 2

second order asymptotics of sequential hypothesis testing we consider the classical sequential binary hypothesis testing problem in which there are two hypotheses governed respectively by distributions p 0 and p 1 and we would like to decide which hypothesis is true using a sequential test it is known from the work of wald and wolfowitz that as the expectation of the length of the test grows the optimal type i and type ii error exponents approach the relative entropies d p 1 p 0 and d p 0 p 1 we refine this result by considering the optimal backoff or second order asymptotics from the corner point of the achievable exponent region d p 1 p 0 d p 0 p 1 under two different constraints on the length of the test or the sample size first we consider a probabilistic constraint in which the probability that the length of test exceeds a prescribed integer n is less than a certain threshold 0 second the expectation of the sample size is bounded by n in both cases and under mild conditions the second order asymptotics is characterized exactly numerical examples are provided to illustrate our results

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

unique lifting of integer variables in minimal inequalities this paper contributes to the theory of cutting planes for mixed integer linear programs milps minimal valid inequalities are well understood for a relaxation of an milp in tableau form where all the nonbasic variables are continuous they are derived using the gauge function of maximal lattice free convex sets in this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities we characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting this result first appeared in the proceedings of ipco 2010 the answer is a nonconvex region that can be obtained as a finite union of convex polyhedra we then establish a necessary and sufficient condition for the uniqueness of the lifting function

on the error probability of signals in additive white gaussian noise a new upper bound to the probability of error in detecting one of m equally probable signals in additive white gaussian noise is presented this bound is easy to calculate can be applied to any signal set it is always better than the union and minimum distance bounds examples demonstrate the use of the bound

on the relationship between standard intersection cuts lift and project cuts and generalized intersection cuts we examine the connections between the classes of cuts in the title we show that lift and project l p cuts from a given disjunction are equivalent to generalized intersection cuts from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction while l p cuts from split disjunctions are known to be equivalent to standard intersection cuts sics from the strip obtained by complementing the terms of the split we show that l p cuts from more general disjunctions may not be equivalent to any sic in particular we give easily verifiable necessary and sufficient conditions for a l p cut from a given disjunction d to be equivalent to a sic from the polyhedral counterpart of d irregular l p cuts i e those that violate these conditions have interesting properties for instance unlike the regular ones they may cut off part of the corner polyhedron associated with the lp solution from which they are derived furthermore they are not exceptional their frequency exceeds that of regular cuts a numerical example illustrates some of the above properties

a tight upper bound for the third order asymptotics for most discrete memoryless channels this paper shows that the logarithm of the e error capacity average error probability for n uses of a discrete memoryless channel dmc is upper bounded by the normal approximation plus a third order term that does not exceed 1 2 logn o 1 if the e dispersion of the channel is positive this matches a lower bound by y polyanskiy 2010 for dmcs with positive reverse dispersion if the e dispersion vanishes the logarithm of the e error capacity is upper bounded by n times the capacity plus a constant term except for a small class of dmcs and e 1 2

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

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

lifting properties of maximal lattice free polyhedra we study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice free polyhedra we prove a basic invariance property of unique minimal liftings for general maximal lattice free polyhedra this generalizes a previous result by basu et al math oper res 37 2 346 355 2012 for simplicial maximal lattice free polytopes thus completely settling this fundamental question about lifting for maximal lattice free polyhedra we further give a very general iterative construction to get maximal lattice free polyhedra with the unique lifting property in arbitrary dimensions this single construction not only obtains all previously known polyhedra with the unique lifting property but goes further and vastly expands the known list of such polyhedra finally we extend characterizations from basu et al 2012 about lifting with respect to maximal lattice free simplices to more general polytopes these nontrivial generalizations rely on a number of results from discrete geometry including the venkov alexandrov mcmullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces

strong large deviation and local limit theorems abstract an event is loosely called a large deviation event if the dominant term in the probability of that event goes to zero exponentially most papers give asymptotic expressions only to the algorithm of the probability of a large deviation event this paper obtains strong deviation results which give asymptotic expressions to the actual probability of a large deviation event based on an arbitrary sequence of random variables under some conditions on the moment generating functions the proof of these results depends on local limit theorems which also are proved in this paper by imposing some conditions on the characteristic functions a local limit theorem states that the pseudo densities of random variables converge which is stronger than the convergence of these random variables in distribution

mismatched decoding error exponents second order rates and saddlepoint approximations this paper considers the problem of channel coding with a given possibly suboptimal maximum metric decoding rule a cost constrained random coding ensemble with multiple auxiliary costs is introduced and is shown to achieve error exponents and second order coding rates matching those of constant composition random coding while being directly applicable to channels with infinite or continuous alphabets the number of auxiliary costs required to match the error exponents and second order rates of constant composition coding is studied and is shown to be at most two for independent identically distributed random coding asymptotic estimates of two well known non asymptotic bounds are given using saddlepoint approximations each expression is shown to characterize the asymptotic behavior of the corresponding random coding bound at both fixed and varying rates thus unifying the regimes characterized by error exponents second order rates and moderate deviations for fixed rates novel exact asymptotics expressions are obtained to within a multiplicative 1 o 1 term using numerical examples it is shown that the saddlepoint approximations are highly accurate even at short block lengths

minimal cut generating functions are nearly extreme we study continuous strongly minimal cut generating functions for the model where all variables are integer we consider both the original gomory johnson setting as well as a recent extension by yildiz and cornuejols math oper res 41 1381 1403 2016 we show that for any continuous minimal or strongly minimal cut generating function there exists an extreme cut generating function that approximates the strongly minimal function as closely as desired in other words the extreme functions are dense in the set of continuous strongly minimal functions

cut generating functions and s free sets we consider the separation problem for sets x that are pre images of a given set s by a linear mapping classical examples occur in integer programming as well as in other optimization problems such as complementarity one would like to generate valid inequalities that cut off some point not lying in x without reference to the linear mapping to this aim we introduce a concept cut generating functions cgf and we develop a formal theory for them largely based on convex analysis they are intimately related to s free sets and we study this relation disclosing several definitions for minimal cgf s and maximal s free sets our work unifies and puts into perspective a number of existing works on s free sets in particular we show how cgf s recover the celebrated gomory cuts

importance sampling for coded modulation error probability estimation this paper proposes an efficient simulation method based on importance sampling to estimate the random coding error probability of coded modulation the technique is valid for complex valued modulations over gaussian channels channels with memory and naturally extends to fading channels the simulation method is built on two nested importance samplers to respectively estimate the pairwise error probability and generate the channel input and output the effect of the respective number of samples on the overall bias and variance of the estimate of the error probability is characterized for a memoryless channel the estimator is shown to be consistent and with a small variance growing with the square root of the code length rather than the exponential growth of a standard monte carlo estimator

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

an operational approach to information leakage given two random variables x and y an operational approach is undertaken to quantify the leakage of information from x to y the resulting measure mathcal l left x to y right is called maximal leakage and is defined as the multiplicative increase upon observing y of the probability of correctly guessing a randomized function of x maximized over all such randomized functions a closed form expression for mathcal l left x to y right is given for discrete x and y and it is subsequently generalized to handle a large class of random variables the resulting properties are shown to be consistent with an axiomatic view of a leakage measure and the definition is shown to be robust to variations in the setup moreover a variant of the shannon cipher system is studied in which performance of an encryption scheme is measured using maximal leakage a single letter characterization of the optimal limit of normalized maximal leakage is derived and asymptotically optimal encryption schemes are demonstrated furthermore the sample complexity of estimating maximal leakage from data is characterized up to subpolynomial factors finally the guessing framework used to define maximal leakage is used to give operational interpretations of commonly used leakage measures such as shannon capacity maximal correlation and local differential privacy

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

equivariant perturbation in gomory and johnson s infinite group problem i the one dimensional case we give an algorithm for testing the extremality of minimal valid functions for gomory and johnson s infinite group problem that are piecewise linear possibly discontinuous with rational breakpoints this is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions we also present an extreme function that is a piecewise linear function with some irrational breakpoints whose extremality follows from a new principle

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

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

a computational method for system of volterra fredholm integral equations adomian decomposition method has been applied for solving volterra fredholm equation by maleknejad k maleknejad m hadizadeh a new computational method for volterra fredholm integral equations comput math appl 37 1999 1 8 in 1999 in this paper we solve the system of nonlinear volterra fredholm integral equations by adomian decomposition method some examples are solved to illustrate the accuracy of this method

two row mixed integer cuts via lifting recently andersen et al 1 borozan and cornuejols 6 and cornuejols and margot 9 have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non negative continuous variables these inequalities are either split cuts or intersection cuts derived using maximal lattice free convex sets in order to use these inequalities to obtain cuts from two rows of a general simplex tableau one approach is to extend the system to include all possible non negative integer variables giving the two row mixed integer infinite group problem and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities in this paper we study the characteristics of these lifting functions we show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice free triangle with multiple integer points in the relative interior of one of its sides or a maximal lattice free triangle with integral vertices and one integer point in the relative interior of each side in the other cases maximal lattice free triangles with one integer point in the relative interior of each side and non integral vertices and maximal lattice free quadrilaterals non unique lifting functions may yield distinct extreme inequalities for the latter family of triangles we present sufficient conditions to yield an extreme inequality for the two row mixed integer infinite group problem

the third order term in the normal approximation for the awgn channel this paper shows that under the average error probability formalism the third order term in the normal approximation for the additive white gaussian noise channel with a maximal or equal power constraint is at least 1 2 log n o 1 this improves on the lower bound by polyanskiys poor verdu 2010 and matches the upper bound proved by the same authors

a geometric approach to cut generating functions the cutting plane approach to integer programming was initiated more than 40 years ago gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and balas introduced intersection cuts for the corner polyhedron this line of research was left dormant for several decades until relatively recently when a paper of andersen louveaux weismantel and wolsey generated renewed interest in the corner polyhedron and intersection cuts recent developments rely on tools drawn from convex analysis geometry and number theory and constitute an elegant bridge between these areas and integer programming we survey these results and highlight recent breakthroughs in this area

a note on the split rank of intersection cuts in this note we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts as a first step of this argument a polyhedral subset of the lattice free convex set that is used to generate the intersection cut is constructed we call this subset the restricted lattice free set it is then shown that lceil log 2 l rceil is a lower bound on the split rank of the intersection cut where l is the number of integer points lying on the boundary of the restricted lattice free set satisfying the condition that no two points lie on the same facet of the restricted lattice free set the use of this result is illustrated by obtaining a lower bound of lceil log 2 n 1 rceil on the split rank of n row mixing inequalities

convergence and error estimation of homotopy perturbation method for fredholm and volterra integral equations abstract in this paper an application of the homotopy perturbation method for solving fredholm and volterra integral equations of the second kind is presented discussed method consists in constructing the functional series sum of which determines the function giving the solution of considered problem conditions under which the constructed series is convergent are formulated and proved in the paper in general the composed series is rather fast convergent thanks to which calculation of few first terms ensures a very good approximation of the sought solution estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

the master equality polyhedron with multiple rows the master equality polyhedron mep is a canonical set that generalizes the master cyclic group polyhedron mcgp of gomory we recently characterized a nontrivial polar for the mep i e a polyhedron t such that an inequality defines a nontrivial facet of the mep if and only if its coefficient vector forms a vertex of t in this paper we study the mep when it is defined by m 1 rows we define the notion of a polaroid a set containing all nontrivial facet defining inequalities we show how to use linear programming lp to efficiently solve the separation problem for the mep when the polaroid has a compact polyhedral description we obtain such descriptions via subadditivity conditions when m 2 or m 3 and using lp duality show how to efficiently optimize over the mep these results yield a pseudo polynomial time lp based algorithm to solve the problem min cx ax b x geq bf 0 x in mathbb z n when a has at most three constraints for the mcgp and the mep defined by a single constraint the notions of two term subadditivity and valid inequalities for mep are essentially equivalent we show this is not true in the case of the mep when m 3 in fact we prove that subadditivity conditions with a sub exponential number of terms do not imply validity in particular when m 3 we show that four term subadditivity conditions are necessary and sufficient for validity

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

sufficiency of cut generating functions the concept of cut generating function has its origin in the work of gomory and johnson from the 1970s it has received renewed attention in the past few years recently conforti cornuejols daniilidis lemarechal and malick proposed a general framework for studying cut generating functions however they gave an example showing that not all cuts can be produced by cut generating functions in this framework they conjectured a natural condition under which cut generating functions might be sufficient this note settles this open problem

mixed integer sets from two rows of two adjacent simplex bases in andersen et al lecture notes in computer science vol 4513 springer berlin pp 1 15 2007 we studied a mixed integer set arising from two rows of a simplex tableau we showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables in this paper we generalize our findings and show that when upper bounds on the non basic variables are also considered further classes of facets arise that cannot be obtained from triangles and quadrilaterals specifically when exactly one upper bound on a non basic variable is introduced stronger inequalities that can be derived from pentagons involving up to six variables also appear