time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

an error based approach to complementary formulations of static field solutions the constitutive relationship between fields in electromagnetic media is cast in error form for well posed static problems the error splits systematically into complementary functionals error minimization provides a universal variational principle that generates complementary and dual solution formulations the error which is wholly attributed to the numerical constraints in approximate solutions is always defined and computable locally and globally

bounding the gap between the mccormick relaxation and the convex hull for bilinear functions we investigate how well the graph of a bilinear function b 0 1 n rightarrow mathbb r b 0 1 n r can be approximated by its mccormick relaxation in particular we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is at most c times the difference between the concave and convex envelopes answering a question of luedtke namazifar and linderoth we show that this factor c cannot be bounded by a constant independent of n more precisely we show that for a random bilinear function b we have asymptotically almost surely c geqslant sqrt n 4 c n 4 on the other hand we prove that c leqslant 600 sqrt n c 600n which improves the linear upper bound proved by luedtke namazifar and linderoth in addition we present an alternative proof for a result of misener smadbeck and floudas characterizing functions b for which the mccormick relaxation is equal to the convex hull

a fourth order finite difference solver for nerve conduction equation in rectangular domains a fourth order finite difference solver for the nerve conduction equation is presented that has mesh independent convergence rate and unconditional convergence for any time step size numerics are performed for a number of test problems which shows that the features of the nerve conduction equation are preserved the computational loads are roughly same as ones of normal second order scheme such as the crank nicolson scheme

fast approximation algorithms on maxcut k coloring and k color ordering for vlsi applications there are a number of vlsi problems that have a common structure we investigate such a structure that leads to a unified approach for three independent vlsi layout problems partitioning placement and via minimization along the line we first propose a linear time approximation algorithm on maxcut and two closely related problems k coloring and maximal k color ordering problem the k coloring is a generalization of the maxcut and the maximal k color ordering is a generalization of the k coloring for a graph g with e edges and n vertices our maxcut approximation algorithm runs in o e n sequential time yielding a nodebalanced maxcut with size at least w e w e n 2 improving the time complexity of o e log e known before building on the proposed maxcut technique and employing a height balanced binary decomposition we devise an o e n log k time algorithm for the k coloring problem which always finds a k partition of vertices such that the number of bad or defected edges does not exceed w e k n 1 n sup log k thus improving both the time complexity o enk and the bound e k known before the other related problem is the maximal k color ordering problem that has been an open problem we show the problem is np complete then present an approximation algorithm building on our k coloring structure a performance bound on maximal k color ordering cost 2kw e 3 is attained in o ek time the solution quality of this algorithm is also tested experimentally and found to be effective

on decomposability of multilinear sets we consider the multilinear set mathcal s defined as the set of binary points x y satisfying a collection of multilinear equations of the form y i prod i in i x i i in mathcal i where mathcal i denotes a family of subsets of 1 ldots n of cardinality at least two such sets appear in factorable reformulations of many types of nonconvex optimization problems including binary polynomial optimization a great simplification in studying the facial structure of the convex hull of the multilinear set is possible when mathcal s is decomposable into simpler multilinear sets mathcal s j j in j namely the convex hull of mathcal s can be obtained by convexifying each mathcal s j separately in this paper we study the decomposability properties of multilinear sets utilizing an equivalent hypergraph representation for multilinear sets we derive necessary and sufficient conditions under which mathcal s is decomposable into mathcal s j j in j based on the structure of pair wise intersection hypergraphs our characterizations unify and extend the existing decomposability results for the boolean quadric polytope finally we propose a polynomial time algorithm to optimally decompose a multilinear set into simpler subsets our proposed algorithm can be easily incorporated in branch and cut based global solvers as a preprocessing step for cut generation

formulation of the eddy current problem in multiply connected regions in terms of h in this paper various formulations for the eddy current problem are presented the formulations are based on solving directly for the magnetic field h and they differ from each other mainly by how the field on the boundary is treated the electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region this provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains besides the fivefold decomposition another essential tool in our approach is whitney elements as they provide the structure needed to retain consistency between the continuous and discrete problems the paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems 1998 john wiley sons ltd

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

perspective reformulations of the cta problem with l2 distances any institution that disseminates data in aggregated form has the duty to ensure that individual confidential information is not disclosed either by not releasing data or by perturbing the released data while maintaining data utility controlled tabular adjustment cta is a promising technique of the second type where a protected table that is close to the original one in some chosen distance is constructed the choice of the specific distance shows a trade off although the euclidean distance has been shown and is confirmed here to produce tables with greater utility it gives rise to mixed integer quadratic problems miqps with pairs of linked semi continuous variables that are more difficult to solve than the mixed integer linear problems corresponding to linear norms we provide a novel analysis of perspective reformulations prs for this special structure in particular we devise a projected pr p2r which is piecewise conic but simplifies to a nonseparable miqp when the instance is symmetric we then compare different formulations of the cta problem showing that the ones based on p2r most often obtain better computational results

concave extensions for nonlinear 0 1 maximization problems a well known linearization technique for nonlinear 0 1 maximization problems can be viewed as extending any polynomial in 0 1 variables to a concave function defined on 0 1 n some properties of this standard concave extension are investigated polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix the standard extension is proved to be identical to another type of concave extension defined as the lower envelope of a class of affine functions majorizing the given polynomial

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

experiments in quadratic 0 1 programming we present computational experience with a cutting plane algorithm for 0 1 quadratic programming without constraints our approach is based on a reduction of this problem to a max cut problem in a graph and on a partial linear description of the cut polytope

edge element formulations of eddy current problems abstract various formulations of eddy current problems in terms of scalar and vector potentials are reviewed in the paper the vector potentials are approximated by edge finite elements and the scalar potentials by nodal ones the formulations are ungauged leading in most cases to singular finite element equations systems special attention is paid to ensuring that the right hand sides of the equations systems are consistent the resulting numerical schemes prove to be as robust as the corresponding coulomb gauged approaches realized by nodal elements

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

the benefits of nodal and edge elements coupling for discretizing global constraints in dual magnetodynamic formulations the treatment of massive and stranded inductors is studied in the frame of dual magnetodynamic finite element h and a formulations on both sides nodal and edge finite elements are used and source fields are defined when needed as mathematical tools to be used directly in each formulation to lead to circuit relations for inductors simplified expressions of source fields are proposed

a finite element method with lagrange multipliers for low frequency harmonic maxwell equations the aim of this paper is to analyze a finite element method to solve the low frequency harmonic maxwell equations in a bounded domain containing conductors and dielectrics this system of partial differential equations is a model for the so called eddy currents problem after writing this problem in terms of the magnetic field it is discretized by nedelec edge finite elements on a tetrahedral mesh error estimates are easily obtained if the curl free condition is imposed on the elements in the dielectric domain then the curl free condition is imposed at a discrete level by introducing a piecewise linear multivalued potential the resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential moreover this approach leads to an important saving in computational effort problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account implementation issues are discussed including an amenable procedure to impose the boundary conditions by means of a lagrange multiplier finally the method is applied to solve a three dimensional model problem a cylindrical electrode surrounded by dielectric

numerical solution of eddy current problems in bounded domains using realistic boundary conditions abstract the aim of this paper is to analyze a finite element method to solve the low frequency harmonic maxwell equations in a bounded domain containing conductors and dielectrics and using realistic boundary conditions in that they can be easily measured these equations provide a model for the so called eddy currents the problem is formulated in terms of the magnetic field this formulation is discretized by using nedelec edge finite elements on a tetrahedral mesh error estimates are easily obtained when the curl free condition is imposed explicitly on the elements in the dielectric domain a multivalued magnetic scalar potential is introduced then to impose this curl free condition the discrete counterpart of this formulation leads to an important saving in computational effort problems related to the topology are also considered more precisely the possibility of having a non simply connected dielectric domain is taken into account finally the method is applied to solve two three dimensional model problems a test with a known analytical solution and the computation of the electromagnetic field in a metallurgical arc furnace

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

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

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

roof duality complementation and persistency in quadratic 0 1 optimization the paper is concerned with the primal problem of maximizing a given quadratic pseudo boolean function four equivalent problems are discussed the primal the complementation the discrete rhys lp and the weighted stability problem of a sam graph each of them has a relaxation the roof dual the quadratic complementation the continuous rhys lp and the fractional weighted stability problem of a sam graph the main result is that the four gaps associated with the four relaxations are equal furthermore a solution to any of these problems leads at once to solutions of the other three equivalent ones the four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem the optimal solutions of the roof dual define best linear majorantsp x off having the following persistency property if theith coefficient inp is positive negative thenx i 1 0 in every optimum of the primal problem several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal on the other hand a class of gap free functions properly including the supermodular ones is exhibited

approximation and intractability results for the maximum cut problem and its variants the maximum cut problem is known to be an important np complete problem with many applications the authors investigate this problem which they call the normal maximum cut problem and a variant of it which is referred to as the connected maximum cut problem they show that any n vertex e edge graph admits a cut with at least the fraction 1 2 1 2n of its edges thus improving the ratio 1 2 2 e known before it is shown that it is np complete to decide if a given graph has a normal maximum cut with at least a fraction 1 2 epsilon of its edges where the positive constant epsilon can be taken smaller than any value chosen the authors present an approximation algorithm for the normal maximum cut problem on any graph that runs in o e log e n log n p log p log n parallel time using p 1

magnetostatic field computations in terms of two component vector potentials in this paper the magnetostatic problem is stated in terms of two component electric and magnetic vector potentials an associated numerical method based on the adoption of edge elements is proposed this procedure overcomes the cancellation problems and the complexity of the interface conditions encountered by similar approaches in the presence of magnetic inhomogeneities and discontinuities of currents and magnetic fields

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

a formulation of the eddy current problem in the presence of electric ports the time harmonic eddy current problem with either voltage or current intensity excitation is considered we propose and analyze a new finite element approximation of the problem based on a weak formulation where the main unknowns are the electric field in the conductor a scalar magnetic potential in the insulator and for the voltage excitation problem the current intensity the finite element approximation uses edge elements for the electric field and nodal elements for the scalar magnetic potential and an optimal error estimate is proved some numerical results illustrating the performance of the method are also presented

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

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

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

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

numerical analysis of electric field formulations of the eddy current model this paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current subjected to boundary conditions involving only data easily available in applications two different cases are considered depending on the boundary data input current intensities or differences of potential weak formulations in terms of the electric field are given in both cases in the first one the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit in the second one the electric potentials are imposed by means of lagrange multipliers which are proved to represent the input current intensities optimal error estimates are proved in both cases and implementation issues are discussed finally numerical tests confirming the theoretical results are reported

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

deriving convex hulls through lifting and projection we consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting and projection technique to construct them the general procedure obtains the convex hulls as an intersection of semi infinite families of linear inequalities each derived using lifting techniques that are interpreted using convexification tools we demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization each family of inequalities yields a few linear nonlinear constraints fully characterized in the space of the original problem variables when the projection problems are amenable to a closed form solution in particular we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints x 1 b 1 x 2 b 2 ge x 3 and x 1 x 2 b 2 le x 3 where b 1 b 2 ge 1 as a consequence we obtain a closed form description of the convex hull of the bilinear equality x 1x 2 x 3 in the presence of variable bounds as an intersection of a few linear and nonlinear inequalities this explicit characterization hitherto unknown can improve relaxation techniques for factorable functions which utilize this equality to relax products of functions with known relaxations

matroid optimisation problems with nested non linear monomials in the objective function recently buchheim and klein discrete appl math 177 34 52 2014 suggested to study polynomial time solvable optimisation problems with linear objective functions combined with exactly one additional quadratic monomial they concentrated on special quadratic spanning tree or forest problems we extend their results to general matroid optimisation problems with a set of nested monomials in the objective function we study polytopes arising from the standard linearisation of the monomials our results provide insight on the polyhedral structure of matroid optimisation problems with arbitrary polynomial objective function with a focus on separation algorithms and strengthened cutting planes extending results by edmonds comb struct appl 69 87 1970 for the matroid polytope we present a complete description for the linearised polytope indeed apart from the standard linearisation one needs appropriately strengthened rank inequalities satisfying certain non separability conditions the separation problem of these extended rank inequalities reduces to a submodular function minimisation problem in the case of exactly one additional non linear monomial we completely characterise the facetial structure of the associated polytope

error bounds for monomial convexification in polynomial optimization convex hulls of monomials have been widely studied in the literature and monomial convexifications are implemented in global optimization software for relaxing polynomials however there has been no study of the error in the global optimum from such approaches we give bounds on the worst case error for convexifying a monomial over subsets of this implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately our main error bounds depend primarily on the degree of the monomial making them easy to compute since monomial convexification studies depend on the bounds on the associated variables in the second part we conduct an error analysis for a multilinear monomial over two different types of box constraints as part of this analysis we also derive the convex hull of a multilinear monomial over

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

the mortar edge element method in three dimensions application to magnetostatics the subject presented in this paper concerns the approximation of a magnetostatic problem where the primary unknown is the magnetic vector potential in a three dimensional system composed of two solid parts separated by an interface the approximation of the problem here is based on the mortar element method combined with edge elements on tetrahedral meshes that do not match at the interface in this paper we describe the proposed method and its implementation aspects together with some numerical results that illustrate how the method works

a polyhedral branch and cut approach to global optimization a variety of nonlinear including semidefinite relaxations have been developed in recent years for nonconvex optimization problems their potential can be realized only if they can be solved with sufficient speed and reliability unfortunately state of the art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software in this paper we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch and cut approach our algorithm exploits convexity either identified automatically or supplied through a suitable modeling language construct in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems we prove that if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function the convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited root node relaxations are computed for 87 problems from globallib and minlplib and detailed computational results are presented for globally solving 26 of these problems with baron 7 2 which implements the proposed techniques the use of cutting planes for these problems reduces root node relaxation gaps by up to 100 and expedites the solution process often by several orders of magnitude

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

a polyhedral study of binary polynomial programs we study the polyhedral convex hull of a mixed integer set x1d4ae defined by a collection of multilinear equations over the unit hypercube such sets appear frequently in the factorable reformulation of mixed integer nonlinear optimization problems in particular the set x1d4ae represents the feasible region of a linearized unconstrained binary polynomial optimization problem we define an equivalent hypergraph representation of the mixed integer set x1d4ae which enables us to derive several families of facet defining inequalities structural properties and lifting operations for its convex hull in the space of the original variables our theoretical developments extend several well known results from the boolean quadric polytope and the cut polytope literature paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub expressions

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

a new high order algorithm for a class of nonlinear evolution equation we derive a new finite difference scheme which is easily extended to fourth order accurate in both temporal and spatial dimensions it is shown through a discrete fourier analysis that the method is unconditionally stable for a 2d problem it requires only a regular seven point difference stencil similar to that used in the standard second order algorithms such as the crank nicolson algorithm numerical experiments are conducted to test its high accuracy and efficiency of the new algorithm

some results on the strength of relaxations of multilinear functions we study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions specifically we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach due to mccormick the standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently we show that for a multilinear function having a single product term this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero we then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms finally for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is always within a constant of the difference between the concave and convex envelopes these results along with numerical examples we provide give insight into how to construct strong relaxations of multilinear functions

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

efficient reduction of polynomial zero one optimization to the quadratic case we address the problem of optimizing a polynomial with real coefficients over binary variables we show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem the number of variables in the latter linearization is only slightly larger than in the former if polynomial constraints are present in the original problem then their linearized counterparts carry over to the linearized quadratic problem unchanged if the original problem formulation does not contain any constraints we obtain a reduction to unconstrained quadratic zero one optimization which is equivalent to the well studied max cut problem the separation problem for general unconstrained polynomial zero one optimization thus reduces to the separation problem for the cut polytope this allows us to transfer the entire knowledge gained for the latter polytope by intensive research and in particular the sophisticated separation techniques that have been developed we report preliminary experimental results obtained with a straightforward implementation of this approach

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

berge acyclic multilinear 0 1 optimization problems abstract the problem of optimizing a multilinear polynomial f in 0 1 variables arises in applications from many different areas we are interested in resolution methods based on reformulating the polynomial problem into an equivalent linear one an approach that attempts to draw benefit from the extensive literature in integer linear programming more precisely we characterize instances for which the classical standard linearization procedure guarantees integer optimal solutions we show that the standard linearization polytope ph is integer if and only if the hypergraph h defined by the higher degree monomials of f is berge acyclic or equivalently when the matrix defining ph is balanced this characterization follows from more general conditions that guarantee integral optimal vertices for a relaxed formulation depending on the sign pattern of the monomials of f

a modified backward time discretization for nonlinear parabolic equations using patch approximations a modified backward difference time discretization is presented for galerkin approximations for the nonlinear parabolic equation in mathbb r d q x u u t sum i j 1 d frac partial partial x i left a ij x u frac partial u partial x j right sum i 1 d b i x u frac partial u partial x i f x t u this procedure uses a local approximation of the coefficients based on patches of finite elements to update the coefficient matrix at each time step and allows for efficient use of such direct methods for solving linear systems of equations as nested dissection optimal order error estimates and almost optimal order work requirements are derived an alternating direction variant is also presented which has both optimal order error estimates and optimal order work requirements

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

a discrete weighted helmholtz decomposition and its application we propose a discrete weighted helmholtz decomposition for edge element functions the decomposition is orthogonal in a weighted l 2 inner product and stable uniformly with respect to the jumps in the discontinuous weight function as an application the new helmholtz decomposition is applied to demonstrate the quasi optimality of a preconditioned edge element system for solving a saddle point maxwell system in non homogeneous media by a non overlapping domain decomposition preconditioner i e the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain and more importantly it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media numerical experiments are presented to validate the effectiveness of the non overlapping domain decomposition preconditioner

the finite element numerical analysis for a class nonlinear evolution equations a complete stability and convergence analysis is given for two level scheme for the approximate solution of a class generalized nerve conduction equation defining a kind of truncation error and using a kind of technique optimal order estimates in l 2 is constructively obtained

cut polytopes boolean quadratic polytopes and nonnegative quadratic pseudo boolean functions in this paper we present a class of valid inequalities as well as a class of facets for the cut polytope of the complete graph it is shown that many of the known classes of valid inequalities e g triangle hypermetric and cycle inequalities belong to this class by specializing some of the parameters large classes of facets can be defined it is shown that each of these classes contains at least n 3n 3 facets where n 10 denotes the number of vertices and that the corresponding separation problems are polynomially solvable these results are based on a one to one correspondence established between the set of valid inequalities facets for the cut polytope of kn 1 and the set of nonnegative extremal quadratic pseudo boolean functions in n variables

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

an optimal domain decomposition preconditioner for low frequency time harmonic maxwell equations the time harmonic maxwell equations are considered in the low frequency case a finite element domain decomposition approach is proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure which is proven to converge the rate of convergence turns out to be independent of the mesh size showing that the preconditioner implicitly defined by the iterative procedure is optimal for obtaining this convergence result it has been necessary to prove a regularity theorem for dirichlet and neumann harmonic fields

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

upper and lower bounds for local electromagnetic quantities most engineering problems are solved by means of numerical methods that are able to provide only approximate solutions for which it would be extremely useful to have efficient error estimators upper and lower bounds for quantities of integral character like the stored magnetic energy or the ohmic power dissipated in the domain of interest had been clearly established along with the procedures to obtain them numerically however upper and lower bounds for local quantities would be of paramount interest in several fields of applications like non destructive testing or nuclear magnetic resonance we present here a procedure for the determination of upper and lower bounds of local field quantities namely the average value of a field component in an arbitrarily small region it is based on the introduction of an auxiliary field and is the natural extension of the method establishing the bounds of global quantities our technique can be used for any linear system in stationary conditions for which a virtual work principle can be applied its efficiency is demonstrated with the analysis of some stationary 2d and 3d electromagnetic problems 1998 john wiley sons ltd

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

quantifying double mccormick when using the standard mccormick inequalities twice to convexify trilinear monomials as is often the practice in modeling and software there is a choice of which variables to group first for the important case in which the domain is a nonnegative box we calculate the volume of the resulting relaxation as a function of the bounds defining the box in this manner we precisely quantify the strength of the different possible relaxations defined by all three groupings in addition to the trilinear hull itself as a by product we characterize the best double mccormick relaxation we wish to emphasize that in the context of spatial branch and bound for factorable formulations our results do not only apply to variables in the input formulation our results apply to monomials that involve auxiliary variables as well so our results apply to the product of any three possibly complicated expressions in a formulation