a hybrid breakout local search and reinforcement learning approach to the vertex separator problem abstract the vertex separator problem vsp is an np hard problem which emerges from a variety of important domains and applications in this paper we present an improved breakout local search for vsp named bls rle the distinguishing feature of this approach is a new parameter control mechanism that draws upon ideas from reinforcement learning theory to reach an interdependent decision on the number and on the type of perturbation moves the mechanism complies with the principle of first carrying out intensification and then employing minimal diversification only if needed it uses a dedicated sampling strategy for a rapid convergence towards a limited set of parameter values that appear to be the most convenient for the given state of search extensive experimental evaluations and statistical comparisons on a wide range of benchmark instances show significant improvement in performance of the proposed algorithm over the existing bls algorithm for vsp indeed out of the 422 tested instances bls rle was able to attain the best known solution in 93 8 of the cases which is around 20 higher compared to the existing bls in addition we provide detailed analyses to evaluate the importance of the key elements of the proposed method and to justify the degree of diversification introduced during perturbation

explicit streamline diffusion finite element methods for the compressible euler equations in conservation variables the paper concerns the streamline diffusion finite element method applied to one and two dimensional gas flow described by the inviscid euler equations in conservation variables we point out that the streamline diffusion method is a natural finite element analogue to upstream type finite difference volume schemes and in fact constitutes a general framework for a large class of them we study explicit implementations of the method and derive different choices of stabilizing streamline diffusion matrices in particular we propose a consistent fully multidimensional version a brief review of the theoretical background to the method is presented and some numerical results in two dimensions are given

an adaptive linear approximation algorithm for copositive programs we study linear optimization problems over the cone of copositive matrices these problems appear in nonconvex quadratic and binary optimization for instance the maximum clique problem and other combinatorial problems can be reformulated as such problems we present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit in contrast to previous approximation schemes our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not using these approximations we derive an adaptive linear approximation algorithm for copositive programs numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

excluded grid minors and efficient polynomial time approximation schemes two of the most widely used approaches to obtain polynomial time approximation schemes ptass on planar graphs are the lipton tarjan separator based approach and baker s approach in 2005 demaine and hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial time approximation schemes eptass for several problems including c onnected d ominating s et and f eedback v ertex s et in this work we unify the two strengthened approaches to combine the best of both worlds we develop a framework allowing the design of eptas on classes of graphs with the subquadratic grid minor sqgm property roughly speaking a class of graphs has the sqgm property if for every graph g from the class the fact that g contains no t t grid as a minor guarantees that the treewidth of g is subquadratic in t for example the class of planar graphs and more generally classes of graphs excluding some fixed graph as a minor have the sqgm property at the heart of our framework is a decomposition lemma stating that for most bidimensional problems on a graph class g with the sqgm property there is a polynomial time algorithm that given a graph g e g as input and an e 0 outputs a vertex set x of size e opt such that the treewidth of g x is f e here opt is the objective function value of the problem in question and f is a function depending only on e this allows us to obtain eptass on apex minor free graphs for all problems covered by the previous framework as well as for a wide range of packing problems partial covering problems and problems that are neither closed under taking minors nor contractions to the best of our knowledge for many of these problems including c ycle p acking f p acking f d eletion m ax l eaf s panning t ree or p artial r d ominating s et no eptass even on planar graphs were previously known we also prove novel excluded grid theorems in unit disk and map graphs without large cliques using these theorems we show that these classes of graphs have the sqgm property based on the developed framework we design eptass and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs

simple finite element based computation of distance functions in unstructured grids a distance field is a representation of the closest distance from a point to a given surface distance fields are widely used in applications ranging from computer vision physics and computer graphics and have been the subject of research of many authors in the last decade most of the methods for computing distance fields are devoted to cartesian grids while little attention has been paid to unstructured grids finite element methods are well known for their ability to deal with partial differential equations in unstructured grids therefore we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the eikonal property 1 a simple algorithm to develop the computations is also presented and its efficiency demonstrated through various unstructured grid examples we observed that the presented algorithm has processing times proportional to the number of mesh nodes

a quadratically constrained quadratic optimization model for completely positive cone programming we propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values the objective function can be any quadratic form the constraints of each problem are described in terms of quadratic forms with no linear terms and all constraints are homogeneous equalities except one inhomogeneous equality where a quadratic form is set to be a positive constant for the equality constraints a hierarchy of copositvity condition is assumed this model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations in particular it generalizes the recent results on quadratic optimization problems by burer and the set semidefinite representation by eichfelder and povh optim lett 7 2013 pp 1373 1386

the vertex separator problem a polyhedral investigation the vertex separator vs problem in a graph g v e asks for a partition of v into nonempty subsets a b c such that there is no edge between a and b and c is minimized subject to a bound on max a b we give a mixed integer programming formulation of the problem and investigate the vertex separator polytope vsp the convex hull of incidence vectors of vertex separators necessary and sufficient conditions are given for the vsp to be full dimensional central to our investigation is the relationship between separators and dominators several classes of valid inequalities are investigated along with the conditions under which they are facet defining for the vsp some of our proofs combine in new ways projection with lifting in a companion paper we develop a branch and cut algorithm for the vs problem based on the inequalities discussed here and report on computational experience with a wide variety of vs problems drawn from the literature and inspired by various applications

the unique games conjecture integrality gap for cut problems and embeddability of negative type metrics into l 1 in this article we disprove a conjecture of goemans and linial namely that every negative type metric embeds into e1 with constant distortion we show that for an arbitrarily small constant 0 for all large enough n there is an n point negative type metric which requires distortion at least log log n 1 6 to embed into e1 surprisingly our construction is inspired by the unique games conjecture ugc establishing a previously unsuspected connection between probabilistically checkable proof systems pcps and the theory of metric embeddings we first prove that the ugc implies a super constant hardness result for the nonuniform s parsest c ut problem though this hardness result relies on the ugc we demonstrate nevertheless that the corresponding pcp reduction can be used to construct an integrality gap instance for s parsest c ut towards this we first construct an integrality gap instance for a natural sdp relaxation of u nique g ames then we simulate the pcp reduction and translate the integrality gap instance of u nique g ames to an integrality gap instance of s parsest c ut this enables us to prove a log log n 1 6 integrality gap for s parsest c ut which is known to be equivalent to the metric embedding lower bound

partitioning a graph into small pieces with applications to path transversal given a graph g v e and an integer k in mathbb n we study k vertex separator resp k edge separator where the goal is to remove the minimum number of vertices resp edges such that each connected component in the resulting graph has less than k vertices we also study k path transversal where the goal is to remove the minimum number of vertices such that there is no simple path of length k our main results are the following improved approximation algorithms

convex relaxations for nonconvex quadratically constrained quadratic programming matrix cone decomposition and polyhedral approximation we present a decomposition approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming qcqp we first develop a general conic program relaxation for qcqp based on a matrix decomposition scheme and polyhedral piecewise linear underestimation by employing suitable matrix cones we then show that the convex conic relaxation can be reduced to a semidefinite programming sdp problem in particular we investigate polyhedral underestimations for several classes of matrix cones including the cones of rank 1 and rank 2 matrices the cone generated by the coefficient matrices the cone of positive semidefinite matrices and the cones induced by rank 2 semidefinite inequalities we demonstrate that in general the new sdp relaxations can generate lower bounds at least as tight as the best known sdp relaxations for qcqp moreover we give examples for which tighter lower bounds can be generated by the new sdp relaxations we also report comparison results of different convex relaxation schemes for nonconvex qcqp with convex quadratic linear constraints nonconvex quadratic constraints and 0 1 constraints

eigenvalue bounds spectral partitioning and metrical deformations via flows we present a new method for upper bounding the second eigenvalue of the laplacian of graphs our approach uses multi commodity flows to deform the geometry of the graph we embed the resulting metric into euclidean space to recover a bound on the rayleigh quotient using this we show that every n vertex graph of genus g and maximum degree d satisfies 2 g eo g 1 3d n this recovers the o d n bound of spielman and teng for planar graphs and compares to kelner s bound of o g 1 poly d n but our proof does not make use of conformal mappings or circle packings we are thus able to extend this to resolve positively a conjecture of spielman and teng by proving that 2 g e o dh6log h n whenever g is kh minor free this shows in particular that spectral partitioning can be used to recover o sn sized separators in bounded degree graphs that exclude a fixed minor we extend this further by obtaining nearly optimal bounds on 2 for graphs that exclude small depth minors in the sense of plotkin rao and smith consequently we show that spectral algorithms find separators of sublinear size in a general class of geometric graphs moreover while the standard sweep algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree our approach produces a vector that works for arbitrary graphs this yields an alternate proof of the well known nonplanar separator theorem of alon seymour and thomas that states that every excluded minor family of graphs has o sn node balanced separators

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

improved approximation algorithms for minimum weight vertex separators we develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces unlike in the edge case we show that embeddings into l 1 and even euclidean embeddings are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes we obtain an o sqrt log n approximation for minimum ratio vertex cuts in general graphs based on a new semidefinite relaxation of the problem and a tight analysis of the integrality gap which is shown to be theta sqrt log n we also prove an optimal o log k approximate max flow min vertex cut theorem for arbitrary vertex capacitated multicommodity flow instances on k terminals for uniform instances on any excluded minor family of graphs we improve this to o 1 and this yields a constant factor approximation for minimum ratio vertex cuts in such graphs previously this was known only for planar graphs and for general excluded minor families the best known ratio was o log n these results have a number of applications we exhibit an o sqrt log n pseudoapproximation for finding balanced vertex separators in general graphs in fact we achieve an approximation ratio of o sqrt log opt where opt is the size of an optimal separator improving over the previous best bound of o log opt likewise we obtain improved approximation ratios for treewidth in any graph of treewidth k we show how to find a tree decomposition of width at most o k sqrt log k whereas previous algorithms yielded o k log k for graphs excluding a fixed graph as a minor which includes e g bounded genus graphs we give a constant factor approximation for the treewidth this in turn can be used to obtain polynomial time approximation schemes for several problems in such graphs

the all or nothing flow problem in directed graphs with symmetric demand pairs we study the approximability of the all or nothing multicommodity flow problem in directed graphs with symmetric demand pairs symanf the input consists of a directed graph g v e g v e and a collection of unordered pairs of nodes mathcal m left s 1t 1 s 2t 2 ldots s kt k right m s1t1 s2t2 sktk a subset mathcal m m of the pairs is routable if there is a feasible multicommodity flow in g g such that for each pair s it i in mathcal m siti m the amount of flow from s i si to t i ti is at least one and the amount of flow from t i ti to s i si is at least one the goal is to find a maximum cardinality subset of the given pairs that can be routed our main result is a poly logarithmic approximation with constant congestion for symanf we obtain this result by extending the well linked decomposition framework of chekuri et al 2005 to the directed graph setting with symmetric demand pairs we point out the importance of studying routing problems in this setting and the relevance of our result to future work

approximate tree decompositions of planar graphs in linear time many algorithms have been developed for np hard problems on graphs with small treewidth k for example all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth it turns out that the bottleneck of many algorithms for np hard problems is the computation of a tree decomposition of width o k in particular by the bidimensional theory there are many linear extended monadic second order problems that can be solved on n vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width o k can be found in such a time we present the first algorithm that on n vertex planar graphs with treewidth k finds a tree decomposition of width o k in such a time in more detail our algorithm has a running time of o n k 2 log k we show the result as a special case of a result concerning so called weighted treewidth of weighted graphs

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

large deformation frictional contact mechanics continuum formulation and augmented lagrangian treatment absract a complete methodology for the formulation and solution of unilateral contact problem with non associated threshold friction between solids undergoing large deformations is presented the approach includes a continuum contact mechanics model combined with an augmented lagrangian method for treating the contact and friction inequality constraints the main developments are a precise statement of the incremental contact minimization problem the exact computations of the first and second variations of the normal gap and the tangential slip velocity and an extension of the augmented lagrangian formulation for frictional contact problems from the discrete to the continuum framework the corresponding saddle point functional first variation mixed virtual work and second variation exact linearization are given the development are illustrated by numerical simulations of frictional contact problems between rigid elastic and elastoplastic bodies a new severe benchmark for large slip frictional contact elements is also proposed

an augmented lagrangian method for discrete large slip contact problems reference lmaf article 1993 001view record in web of science record created on 2005 09 14 modified on 2017 05 10

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

spectral properties of hypergraph laplacian and approximation algorithms the celebrated cheeger s inequality alon and milman 1985 alon 1986 establishes a bound on the edge expansion of a graph via its spectrum this inequality is central to a rich spectral theory of graphs based on studying the eigenvalues and eigenvectors of the adjacency matrix and other related matrices of graphs it has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph in this article we introduce a new hypergraph laplacian operator generalizing the laplacian matrix of graphs in particular the operator is induced by a diffusion process on the hypergraph such that within each hyperedge measure flows from vertices having maximum weighted measure to those having minimum since the operator is nonlinear we have to exploit other properties of the diffusion process to recover the cheeger s inequality that relates hyperedge expansion with the second eigenvalue of the resulting laplacian however we show that higher order spectral properties cannot hold in general using the current framework since higher order spectral properties do not hold for the laplacian operator we instead use the concept of procedural minimizers to consider higher order cheeger like inequalities for any k n we give a polynomial time algorithm to compute an o log r approximation to the kth procedural minimizer where r is the maximum cardinality of a hyperedge we show that this approximation factor is optimal under the sse hypothesis introduced by raghavendra and steurer 2010 for constant values of k moreover using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion we show that all our results for hypergraphs extend to vertex expansion in graphs

residual free bubbles for embedded dirichlet problems abstract we examine a stabilized method employing residual free bubbles for enforcing dirichlet constraints on embedded finite element interfaces in particular we focus attention on problems where the underlying mesh is not explicitly fitted to the geometry of the interface the residual free bubble problem is derived for a simple case and extensions are discussed we show that under certain conditions stabilization only requires knowledge of the residual free bubble on the interface we then examine methods to approximate the residual free bubble a series of benchmark tests are used to demonstrate the accuracy and robustness of the method comparisons are made to nitsche s method employing an eigenvalue estimate for the global stability parameter particular emphasis is placed on the accuracy of flux evaluations on the interface finally we employ the method to simulate an evolving interface problem motivated by resin transfer molding

a finite element based level set method for structural optimization a finite element based level set method is implemented for structural optimization the streamline diffusion finite element method is used for solving both the level set equation and the reinitialization equation the lumped scheme is addressed and the accuracy is compared with the conventional finite difference based level set method a dirichlet boundary condition is enforced during the reinitialization to prevent the boundary from drifting numerical examples of minimum mean compliance design illustrate the reliability of the proposed optimization method copyright 2009 john wiley sons ltd

constant factor approximations of branch decomposition and largest grid minor of planar graphs in o n 1 time we give constant factor approximation algorithms for computing the optimal branch decompositions and largest grid minors of planar graphs for a planar graph g with n vertices let bw g be the branchwidth of g and gm g the largest integer g such that g has a gxg grid as a minor let c 1 be a fixed integer and a b arbitrary constants satisfying a c 1 and b 2c 1 we give an algorithm which constructs in o n 1 1 clogn time a branch decomposition of g with width at most abw g we also give an algorithm which constructs a gxg grid minor of g with g gm g b in o n 1 1 clogn time the constants hidden in the big o notations are proportional to c a c 1 and c b 2c 1 respectively

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

decomposing combinatorial auctions and set packing problems combinatorial auctions allow bidders to bid on bundles of items rather than just on single items the winner determination problem in combinatorial auctions is the problem of determining the allocation of items to bidders such that the sum of the accepted bid prices is maximized this problem is equivalent to the well known maximum weight set packing problem even though these problems are np hard in general they can be solved in polynomial time on instances whose associated item graphs have bounded treewidth called structured item graphs however the tractability of determining whether for a given problem instance a structured item graph of fixed treewidth exists and if so computing one efficiently was an open problem in this article we solve this problem by proving that deciding the existence of structured item graphs is computationally intractable even for treewidth 3 motivated by this unfavorable complexity result we investigate other structural restrictions and we show that the notion of hypertree decomposition a well studied measure of hypergraph cyclicity turns out to be most useful here indeed we show that the winner determination problem is solvable in polynomial time on instances whose dual auction hypergraphs have bounded hypertree width our solution method is based on encoding winner determination via a constraint satisfaction optimization problem and on exhibiting an algorithm to solve this latter problem efficiently for such structurally restricted instances the class of tractable instances identified by our approach while being efficiently recognizable properly contains the class of instances having a structured item graph moreover on the larger class our method solves winner determination with the same asymptotic complexity as the best algorithm proposed in the literature for the subclass of structured item graphs hypertree decompositions can equally profitably be applied to the maximum weight independent set problem which is the dual problem of maximum weight set packing

the vertex separator problem algorithms and computations this is a companion paper to our polyhedral study 1 of the vertex separator vs problem given an undirected graph g the vs problem consists in identifying a minimum weight vertex set whose removal disconnects g subject to bounds on the size of the resulting components in this paper we describe versions of a branch and cut algorithm based on the results of that polyhedral study it uses two families of cuts symmetric and asymmetric for which we develop polynomial time greedy separation routines a heuristic to generate feasible separators is also used a computational experiment on several data sets from the literature compares the performance of three versions of our algorithm to that of the commercial mip solver xpress this experiment throws a sharp light on the role of cut density known to software developers but never before documented in the literature it convincingly shows that the practical usefulness of cuts in integer programming depends not only on their strength but also on their sparsity everything else being equal the smaller the cut support the better the power of the inequalities proposed here is well illustrated by the computational tests on dense graphs this is in accordance with the previous observation since the support of these cuts tends to decrease with graph density

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

approximation algorithms for digraph width parameters several problems that are np hard on general graphs are efficiently solvable on graphs with bounded treewidth efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs directed treewidth dag width and kelly width are some such notions which generalize treewidth whereas directed pathwidth generalizes pathwidth each of these digraph width measures have an associated decomposition structure in this paper we present approximation algorithms for all these digraph width parameters in particular we give an o log n approximation algorithm for directed treewidth and an o log 3 2 n approximation algorithm for directed pathwidth dag width and kelly width our algorithms construct the corresponding decompositions whose widths agree with the above mentioned approximation factors

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

a numerical strategy for investigating the kinetic response of stimulus responsive hydrogels we present a strategy for obtaining numerical solutions to a system of non linear coupled evolution equations describing volume transitions in stimulus responsive hydrogels srhs the theory underlying our sharp interface model of phase transitions in srhs is provided along with the assumptions leading to the specialized formulation that is the starting point for the numerical method the discrete formulation is then developed with a hybrid extended finite element level set method xfe lsm domain integral methodologies are used consistently to extract interfacial quantities such as the mechanical driving traction the jump in the normal component of the solute flux and requisite geometric information several benchmark studies are provided to demonstrate the accuracy and robustness of the numerical strategy we then investigate various features of srh kinetics including the regimes of unstable and stable phase transitions surface pattern formation and bulk phase separation

thermomechanical modeling of regressing heterogeneous solid propellants a numerical framework based on the generalized finite element method gfem is developed to capture the coupled effects of thermomechanical deformations and thermal gradients on the regression rate of a heterogeneous solid propellant the thermomechanical formulation is based on a multiplicative split of the deformation gradient and regression of the heterogeneous solid propellant is simulated using the level set method a spatial mesh convergence study is performed on a non regressing solid heterogeneous propellant system to examine the consistency of the coupled thermomechanical gfem solver the overall accuracy spatial and temporal of the coupled thermomechanical solver for regressing solid propellants is obtained from a periodic sandwich propellant configuration where the effects of thermomechanical deformations on its regression rate is investigated finally the effects of thermomechanical deformations in a regressing two dimensional heterogeneous propellant pack are studied and time average regression rates are reported

continuous quadratic programming formulations of optimization problems on graphs four np hard optimization problems on graphs are studied the vertex separator problem the edge separator problem the maximum clique problem and the maximum independent set problem we show that the vertex separator problem is equivalent to a continuous bilinear quadratic program this continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem the maximum clique problem and the maximum independent set problem all of these formulations when expressed as maximization problems are shown to follow from the convexity properties of the objective function along the edges of the feasible set an algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators computational results are given

exploiting symmetry in copositive programs via semidefinite hierarchies copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization an important line of research is to use semidefinite programming to approximate conic programming over the copositive cone two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs and the lack of information about the quality of the semidefinite programming approximations these drawbacks are an inevitable consequence of the intractability of the generic problems that such approaches attempt to solve to address such drawbacks we develop customized solution approaches for highly symmetric copositive programs which arise naturally in several contexts for instance symmetry properties of combinatorial problems are typically inherited when they are addressed via copositive programming as a result we are able to compute new bounds for crossing number instances in complete bipartite graphs

model based control of dynamic frictional contact problems using the example of hot rolling abstract we consider hot forming processes in which a metal solid body is deformed by several rolls in order to obtain a desired final shape to minimize cutting scrap and to ensure that this shape satisfies the required tolerances as precisely as possible we formulate an optimal control problem where we use the trajectories of the rolls as control functions the deformation of the solid body is described through the basic equations of nonlinear continuum mechanics which are here coupled with an elasto viscoplastic material model based on a multiplicative split of the deformation gradient we assume that the deformations of the rolls can be neglected thus we add unilateral frictional contact boundary conditions resulting in an evolutionary quasi variational inclusion the associated control to observation map is non differentiable due to changes of state between elastic and plastic material behavior contact and separation and stick and slip motion yet we still want to apply gradient based methods to solve the optimal control problem and therefore have to make sure that derivatives of cost functional and constraints exist to resolve this issue we first regularize all non differentiabilities and subsequently apply the direct differentiation method to obtain sensitivity information finally we formulate a suitable algorithm and discuss numerical results for a real world example to illustrate its capability

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

mip reformulations of the probabilistic set covering problem in this paper we address the following probabilistic version psc of the set covering problem min cx mathbb p ax ge xi ge p x in 0 1 n where a is a 0 1 matrix xi is a random 0 1 vector and p in 0 1 is the threshold probability level we introduce the concepts of p inefficiency and polarity cuts while the former is aimed at deriving an equivalent mip reformulation of psc the latter is used as a strengthening device to obtain a stronger formulation simplifications of the mip model which result when one of the following conditions hold are briefly discussed a is a balanced matrix a has the circular ones property the components of xi are pairwise independent the distribution function of xi is a stationary distribution or has the disjunctive shattering property we corroborate our theoretical findings by an extensive computational experiment on a test bed consisting of almost 10 000 probabilistic instances this test bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location warehouse location and k median models our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve psc and in many cases can reduce hours of computing time to a fraction of a second

exceptional family and solvability of copositive complementarity problems abstract in this paper by extending the concept of exceptional family to complementarity problems over the cone of symmetric copositive real matrices we propose an existence theorem of a solution to the copositive complementarity problem extensions of isac carbone s condition karamardian s condition weakly properness and coercivity are also introduced several applications of these results are presented and we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone copositive complementarity problems

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

conservative discretization of contact impact problems for nearly rigid bodies abstract a discretization scheme for contact impact problems related to modeling of gears is proposed the problem is first discretized in time and then a variational formulation for the resulted one step problem is developed a finite element discretization completes the discretization process the scheme is a reinterpretation of the general simo laursen chavla algorithm in the context of the rigid body motion superimposed with small elastic deformation it conserves precisely the linear momentum and total energy and approximately the angular momentum the discretization method is illustrated with two numerical examples the standard 1d impact problem for an elastic rod and a 2d model problem of an elastic wheel bouncing within a constraining box

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

faster approximation schemes and parameterized algorithms on odd h minor free graphs we improve the running time of the general algorithmic technique known as baker s approach 1994 1 on h minor free graphs from o n f h to o f h n o 1 the numerous applications include e g a 2 approximation for coloring and ptases for various problems such as dominating set and max cut where we obtain similar improvements on classes of odd minor free graphs which have gained significant attention in recent time we obtain a similar acceleration for a variant of the structural decomposition theorem proved by demaine et al 2010 20 we use these algorithms to derive faster 2 approximations furthermore we present the first ptases and subexponential fpt algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique we also introduce a technique to derive nearly subexponential parameterized algorithms on h minor free graphs our technique applies in particular to problems such as steiner tree directed subgraph with a property directed longest path and connected independent dominating set on some or all proper minor closed graph classes we obtain as a corollary that all problems with a minor monotone subexponential kernel and amenable to our technique can be solved in subexponential fpt time onh minor free graphs this results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

an assumed gradient finite element method for the level set equation the level set equation is a non linear advection equation and standard finite element and finite difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution we recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front interface being captured as with the original level set equation the use of an extensional velocity helps maintain this signed distance function for some interface evolution problems this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve further we find that sufficient accuracy is available through a standard galerkin formulation without any stabilization or discontinuity capturing terms several numerical experiments are conducted to assess the ability of the proposed assumed gradient level set method to capture the correct solution particularly in the presence of discontinuities in the extensional velocity or level set gradient we examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a hamilton jacobi equation with convex non convex hamiltonian importantly discretizations based on structured and unstructured finite element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well copyright 2005 john wiley sons ltd

a partition of unity based finite element method for level sets level set methods have recently gained much popularity to capture discontinuities including their possible propagation typically the partial differential equations that arise in level set methods in particular the hamilton jacobi equation are solved by finite difference methods however finite difference methods are less suited for irregular domains moreover it seems slightly awkward to use finite differences for the capturing of a discontinuity while in a subsequent stress analysis finite elements are normally used for this reason we here present a finite element approach to solving the governing equations of level set methods after a review of the governing equations the initialization of the level sets the discretization on a finite domain and the stabilization of the resulting finite element method will be discussed special attention will be given to the proper treatment of the internal boundary condition which is achieved by exploiting the partition of unity property of finite element shape functions finally a quantitative analysis including accuracy analysis is given for a one dimensional example and a qualitative example is given for a two dimensional case with a curved discontinuity

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

a copositive programming approach to graph partitioning we consider 3 partitioning the vertices of a graph into sets s 1 s 2 and s 3 of specified cardinalities such that the total weight of all edges joining s 1 and s 2 is minimized this problem is closely related to several np hard problems like determining the bandwidth or finding a vertex separator in a graph we show that this problem can be formulated as a linear program over the cone of completely positive matrices leading in a natural way to semidefinite relaxations of the problem we show in particular that the spectral relaxation introduced by helmberg e 1995 can equivalently be formulated as a semidefinite program finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound

unilateral problems in nonlinear three dimensional elasticity in 1977 john ball obtained a series of very important existence results in hyperelasticity the objective of the present paper is to extend his results in two directions first by considering unilateral boundary conditions corresponding to contact without friction secondly by allowing locking constraints in the set of admissible deformations

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