small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

some properties of the injective tensor product of lp 0 1 and a banach space abstract let x be a real or complex banach space and 1 l p 0 1 x the injective tensor product of lp 0 1 and x has the radon nikodym property resp the analytic radon nikodym property the near radon nikodym property contains no copy of c0 is weakly sequentially complete if and only if x has the same property and each continuous linear operator from lp 0 1 to x is compact

locally convex spaces containing no copy of c0 abstract we show that locally cinvex spaces which contain no copy of c 0 have a series of very nice properties

on mathematical programming with indicator constraints in this paper we review the relevant literature on mathematical optimization with logical implications i e where constraints can be either active or disabled depending on logical conditions to hold in the case of convex functions the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its original dimension we concentrate on the attempt of avoiding the issue of dealing with large nlps in particular we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms then we significantly extend these special cases in two different directions one involving more general convex sets and the other with disjunctions involving three terms computational experiments comparing disjunctive programming formulations in the original space of variables with straightforward bigm ones show that the former are computationally viable and promising

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

boolean algebras and uniform convergence of series abstract several classical results on uniform convergence of unconditionally cauchy series are generalized to weakly unconditionally cauchy series this uniform convergence is characterized through subalgebras and subfamilies of p n a generalization of the orlicz pettis theorem is also proved by mean of subalgebras of p n

on the value of binary expansions for general mixed integer linear programs we study the use of binary variables in reformulating general mixed integer linear programs we show that binary reformulations result in problems for which almost all the binary variables replacing a general integer variable need to be explored during branching we also give computational results on the performance of such reformulations in solving the mixed integer programs which support our theoretical results

an approach to the conditional error rate principle with nuisance parameters summary in the presence of nuisance parameters the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters to use the conditional error rate principle with nuisance parameters we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters we show that the optimization problem that defines such a test can be solved efficiently by existing algorithms

convex relaxations of non convex mixed integer quadratically constrained programs extended formulations this paper addresses the problem of generating strong convex relaxations of mixed integer quadratically constrained programming miqcp problems miqcp problems are very difficult because they combine two kinds of non convexities integer variables and non convex quadratic constraints to produce strong relaxations of miqcp problems we use techniques from disjunctive programming and the lift and project methodology in particular we propose new methods for generating valid inequalities from the equation y x x t we use the non convex constraint y x x t preccurlyeq 0 to derive disjunctions of two types the first ones are directly derived from the eigenvectors of the matrix y x x t with positive eigenvalues the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction we also use the convex sdp constraint y x x t succcurlyeq 0 to derive convex quadratic cuts and we combine both approaches in a cutting plane algorithm we present computational results to illustrate our findings

strong formulations for quadratic optimization with m matrices and indicator variables we study quadratic optimization with indicator variables and an m matrix i e a psd matrix with non positive off diagonal entries which arises directly in image segmentation and portfolio optimization with transaction costs as well as a substructure of general quadratic optimization problems we prove under mild assumptions that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem to strengthen the formulation we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex hull descriptions of these sets we also describe strong conic quadratic valid inequalities preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation

finding multiple solutions to general integer linear programs integer linear programming ilp problems occur frequently in many applications in practice alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function this study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ilp problem numerical examples in real applications are presented to demonstrate the usefulness of the proposed method

cutting planes for optimization of convex functions over nonconvex sets we derive linear inequality characterizations for sets of the form mathop rm conv x q in mathbb r d times mathbb r q ge q x x in mathbb r d mathop rm int p where q is convex and differentiable and p subset mathbb r d we show that in several cases our characterization leads to polynomial time separation algorithms that operate in the original space of variables in particular when q is a positive definite quadratic and p is a polyhedron or an ellipsoid

two term disjunctions on the second order cone balas introduced disjunctive cuts in the 1970s for mixed integer linear programs several recent papers have attempted to extend this work to mixed integer conic programs in this paper we study the structure of the convex hull of a two term disjunction applied to the second order cone and develop a methodology to derive closed form expressions for convex inequalities describing the resulting convex hull our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex possibly nonlinear valid inequalities that correspond to these linear inequalities we identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull in particular our results on two term disjunctions on the second order cone generalize related results on split cuts by modaresi kilinc and vielma and by andersen and jensen

minimizing lipschitz continuous strongly convex functions over integer points in polytopes this paper is about the minimization of lipschitz continuous and strongly convex functions over integer points in polytopes our results are related to the rate of convergence of a black box algorithm that iteratively solves special quadratic integer problems with a constant approximation factor despite the generality of the underlying problem we prove that we can find efficiently with respect to our assumptions regarding the encoding of the problem a feasible solution whose objective function value is close to the optimal value we also show that this proximity result is the best possible up to a factor polynomial in the encoding length of the problem

shrinking and boundedly complete schauder frames in frechet spaces abstract we study schauder frames in frechet spaces and their duals as well as perturbation results we define shrinking and boundedly complete schauder frames on a locally convex space study the duality of these two concepts and their relation with the reflexivity of the space we characterize when an unconditional schauder frame is shrinking or boundedly complete in terms of properties of the space several examples of concrete schauder frames in function spaces are also presented

weak property y0 and regularity of inductive limits abstract an inductive limit e t ind en tn is regular if and only if it satisfies the weak property y0 i e each weakly unconditionally cauchy series in e t is contained and is a weakly unconditionally cauchy series in some en tn in particular an lf space e t ind en tn is regular if and only if every weakly unconditionally cauchy series kxk is a c series i e for any scalar sequence k c0 the series k kxk is convergent furthermore for inductive limits of frechet spaces containing no copy of c0 a number of characteristic conditions of regularity are given

a disjunctive cutting plane procedure for general mixed integer linear programs in this paper we develop a cutting plane algorithm for solving mixed integer linear programs with general integer variables a novel feature of the algorithm is that it generates inequalities at all optimal vertices of the lp relaxation at each iteration the cutting planes generated in the procedure are found by considering a natural generalization of the 0 1 disjunction used by balas ceria and cornuejols in the context of solving binary mixed integer linear programs 3 4

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

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

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

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

projected perspective reformulations with applications in design problems the perspective relaxation pr is a general approach for constructing tight approximations to mixed integer nonlinear programs minlp with semicontinuous variables the pr of a minlp can be formulated either as a mixed integer second order cone program mi socp provided that the original objective function is socp representable or as a semi infinite minlp in this paper we show that under some further assumptions rather restrictive but satisfied in several practical applications the pr of a mixed integer quadratic program miqp can also be reformulated as a piecewise quadratic program qp ultimately yielding a qp relaxation of roughly the same size of the standard continuous relaxation furthermore if the original problem has some exploitable structure then this structure is typically preserved in the reformulation thus allowing the construction of specialized approaches for solving the pr we report on implementing these ideas on two miqps with appropriate structure a sensor placement problem and a quadratic cost single commodity network design problem

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

intersection cuts for nonlinear integer programming convexification techniques for structured sets we study the generalization of split k branch split and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure we introduce two techniques to give precise characterizations of such convex hulls and use them to construct split k branch split and intersection cuts for several classes of non polyhedral sets in particular we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality we also give simple formulas for k branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets

further results on an abstract model for branching and its application to mixed integer programming a key ingredient in branch and bound b b solvers for mixed integer programming mip is the selection of branching variables since poor or arbitrary selection can affect the size of the resulting search trees by orders of magnitude a recent article by le bodic and nemhauser math program 166 1 2 369 405 2017 investigated variable selection rules by developing a theoretical model of b b trees from which they developed some new effective scoring functions for mip solvers in their work le bodic and nemhauser left several open theoretical problems solutions to which could guide the future design of variable selection rules in this article we first solve many of these open theoretical problems we then implement an improved version of the model based branching rules in scip 6 0 a state of the art academic mip solver in which we observe an 11 geometric average time and node reduction on instances of the miplib 2017 benchmark set that require large b b trees

a hierarchy of relaxations for nonlinear convex generalized disjunctive programming we propose a framework to generate alternative mixed integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations we extend the concept of basic steps defined for disjunctive linear programs to the nonlinear case a basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts we show that the strength of the relaxations increases as the number of conjuncts decreases leading to a hierarchy of relaxations we prove that the tightest of these relaxations allows in theory the solution of the disjunctive convex program as a nonlinear programming problem we present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed integer program finally we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs gdp this methodology is validated through a set of numerical examples

on the use of binary programming for sensor scheduling in this paper we propose two myopic sensor scheduling algorithms for target tracking scenarios in which there is a tradeoff between tracking performance and sensor usage costs specifically we consider the problem of activating the lowest cost combination of at most l sensors that maintains a desired squared error accuracy in the target s position estimate for sensors that provide position information only we develop a binary 0 1 mixed integer programming formulation for the scheduling problem and solve it using a linear programming relaxation based branch and bound technique for sensors that provide both position and velocity information we pose the scheduling problem as a binary convex programming problem and solve it using the outer approximation algorithm we apply our scheduling procedures in a network of sensors where the sensor usage costs correspond to network energy consumption our simulation results demonstrate that scheduling using binary programming allows us to obtain optimal solutions to scheduling involving up to 50 70 sensors typically in the order of seconds

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

unconditionally convergent series of operators on banach spaces abstract we show that any series n knof operators inl x y that is unconditionally convergent in the weak operator topology and satisfies the condition that n f knis a compact operator for every index setf n is unconditionally convergent in the uniform operator topology if and only ifx the dual space of the banach spacex contains no copy ofc0

cuts for mixed 0 1 conic programming in this we paper we study techniques for generating valid convex constraints for mixed 0 1 conic programs we show that many of the techniques developed for generating linear cuts for mixed 0 1 linear programs such as the gomory cuts the lift and project cuts and cuts from other hierarchies of tighter relaxations extend in a straightforward manner to mixed 0 1 conic programs we also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts gomory cuts for mixed 0 1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems e g we show that all the subtour elimination inequalities for the traveling salesman problem are rank 1 gomory cuts with respect to a single semidefinite constraint we also include results from our preliminary computational experiments with these cuts

the chvatal gomory closure of a strictly convex body in this paper we prove that the chvatal gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron thus we generalize a result of schrijver schrijver a 1980 on cutting planes ann discrete math 9 291 296 which shows that the chvatal gomory closure of a rational polyhedron is a polyhedron

how to convexify the intersection of a second order cone and a nonconvex quadratic a recent series of papers has examined the extension of disjunctive programming techniques to mixed integer second order cone programming for example it has been shown by several authors using different techniques that the convex hull of the intersection of an ellipsoid mathcal e e and a split disjunction l x j x j u le 0 l xj xj u 0 with l u l

convex programming for disjunctive convex optimization given a finite number of closed convex sets whose algebraic representation is known we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets we derive an algebraic characterization of the feasible region in a higher dimensional space and propose a solution procedure akin to the interior point approach for convex programming

perspective reformulations of mixed integer nonlinear programs with indicator variables we study mixed integer nonlinear programs minlp s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables an indicator variable when it is turned off forces some of the decision variables to assume fixed values and when it is turned on forces them to belong to a convex set many practical minlps contain integer variables of this type we first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints and a convex hull description of this set is derived we then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions further we show that for many classes of problems the convex hull can be expressed via conic quadratic constraints and thus relaxations can be solved via second order cone programming our work is closely related with the earlier work of ceria and soares math program 86 595 614 1999 as well as recent work by frangioni and gentile math program 106 225 236 2006 and akturk et al oper res lett 37 187 191 2009 finally we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role in particular we present computational results for three applications quadratic facility location network design with congestion and portfolio optimization with buy in thresholds that show the power of the reformulation technique

a branch and cut method for 0 1 mixed convex programming we generalize the disjunctive approach of balas ceria and cornuejols 2 and devevlop a branch and cut method for solving 0 1 convex programming problems we show that cuts can be generated by solving a single convex program we show how to construct regions similar to those of sherali and adams 20 and lovasz and schrijver 12 for the convex case finally we give some preliminary computational results for our method

resolvent estimates and decomposable extensions of generalized cesaro operators we determine the spectrum of generalized cesaro operators with essentially rational symbols acting on various spaces of analytic functions including hardy spaces weighted bergman and dirichlet spaces then we show that in all cases these operators are subdecomposable c 2009 elsevier inc all rights reserved

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

exact solutions to a class of stochastic generalized assignment problems abstract this paper deals with a stochastic generalized assignment problem with recourse only a random subset of the given set of jobs will require to be actually processed an assignment of each job to an agent is decided a priori and once the subset of jobs which have to be executed is known reassignments can be performed if there are overloaded agents we construct a convex approximation of the objective function that is sharp at all feasible solutions we then present three versions of an exact algorithm to solve this problem based on branch and bound techniques optimality cuts and a special purpose lower bound numerical results are reported

on the spectrum of the cesaro operator on spaces of analytic functions this paper concerns the cesaro operator acting on various spaces of analytic functions on the unit disc the remarkable fact that this operator is subnormal when acting on the hardy space h2 has lead to extensive studies of its spectral picture on other spaces of this type we present some of the methods that have been used to obtain information about the spectrum of the cesaro operator acting on hardy and bergman spaces and give a unified approach to these problems which also yields new results in this direction in particular we prove that the cesaro operator is subdecomposable on h1 and on the standard weighted bergman spaces 0

a class of integral operators on spaces of analytic functions abstract we determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operators t f z z 1 1 z 0 z f 1 1 d acting on either the analytic besov spaces or other banach spaces of analytic functions on the unit disc including the classical hardy and weighted bergman spaces as well as certain dirichlet spaces and generalized bloch spaces our results unify and extend recent work by aleman and persson 2 the current authors 4 and albrecht and miller 1

spectral properties of cesaro like operators on weighted bergman spaces abstract we determine spectral pictures and resolvent estimates for a class of integral operators on the weighted bergman spaces l a p d m for p 1 1 and show in particular these operators are subdecomposable moreover in the case that p 2 we show that some of these operators are subnormal some are normaloid and some are subscalar

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

abscissas of weak convergence of vector valued dirichlet series abstract the abscissas of convergence uniform convergence and absolute convergence of vector valued dirichlet series with respect to the original topology and with respect to the weak topology x x of a locally convex space x in particular of a banach space x are compared the relation of their coincidence with geometric or topological properties of the underlying space x is investigated cotype in the context of banach spaces and nuclearity and certain topological invariants for frechet spaces play a relevant role

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 hierarchy of relaxations for linear generalized disjunctive programming abstract generalized disjunctive programming gdp originally developed by raman and grossmann 1994 is an extension of the well known disjunctive programming paradigm developed by balas in the mid 70s in his seminal technical report balas 1974 this mathematical representation of discrete continuous optimization problems which represents an alternative to the mixed integer program mip led to the development of customized algorithms that successfully exploited the underlying logical structure of the problem the underlying theory of these methods however borrowed only in a limited way from the theories of disjunctive programming and the unique insights from balas work have not been fully exploited in this paper we establish new connections between the fields of disjunctive programming and generalized disjunctive programming for the linear case we then propose a novel hierarchy of relaxations to the original linear gdp model that subsumes known relaxations for this model and show that a subset of these relaxations are tighter than the latter we discuss the usefulness of these relaxations within the context of mip and illustrate these results on the classic strip packing problem

conic mixed integer rounding cuts a conic integer program is an integer programming problem with conic constraints many problems in finance engineering statistical learning and probabilistic optimization are modeled using conic constraints here we study mixed integer sets defined by second order conic constraints we introduce general purpose cuts for conic mixed integer programming based on polyhedral conic substructures of second order conic sets these cuts can be readily incorporated in branch and bound algorithms that solve either second order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch and bound tree central to our approach is a reformulation of the second order conic constraints with polyhedral second order conic constraints in a higher dimensional space in this representation the cuts we develop are linear even though they are nonlinear in the original space of variables this feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed integer programming the reformulation also allows the use of polyhedral methods for conic integer programming we report computational results on solving unstructured second order conic mixed integer problems as well as mean variance capital budgeting problems and least squares estimation problems with binary inputs our computational experiments show that conic mixed integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed integer programs and hence improving their solvability

perspective cuts for a class of convex 0 1 mixed integer programs we show that the convex envelope of the objective function of mixed integer programming problems with a specific structure is the perspective function of the continuous part of the objective function using a characterization of the subdifferential of the perspective function we derive perspective cuts a family of valid inequalities for the problem perspective cuts can be shown to belong to the general family of disjunctive cuts but they do not require the solution of a potentially costly nonlinear programming problem to be separated using perspective cuts substantially improves the performance of branch cut approaches for at least two models that either naturally or after a proper reformulation have the required structure the unit commitment problem in electrical power production and the mean variance problem in portfolio optimization