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

coarse stability and bifurcation analysis using time steppers a reaction diffusion example evolutionary pattern forming partial differential equations pdes are often derived as limiting descriptions of microscopic kinetic theory based models of molecular processes e g reaction and diffusion the pde dynamic behavior can be probed through direct simulation time integration or more systematically through stability bifurcation calculations time stepper based approaches like the recursive projection method shroff g m keller h b 1993 siam j numer anal 30 1099 1120 provide an attractive framework for the latter we demonstrate an adaptation of this approach that allows for a direct effective coarse bifurcation analysis of microscopic kinetic based models this is illustrated through a comparative study of the fitzhugh nagumo pde and of a corresponding lattice boltzmann model

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

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

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

fuzzy guaranteed cost sampled data control of nonlinear systems coupled with a scalar reaction diffusion process a fuzzy guaranteed cost sampled data control problem is addressed in this paper for a class of nonlinear coupled systems of an n dimensional lumped parameter system modeled by ordinary differential equation ode with a scalar reaction diffusion process represented by parabolic partial differential equation pde a takagi sugeno t s fuzzy coupled ode pde model is initially proposed to accurately represent the nonlinear coupled ode pde system based on the t s fuzzy coupled ode pde model a fuzzy sampled data controller is subsequently developed via the lyapunov s direct method to not only locally exponentially stabilize the nonlinear coupled system but also provide an upper bound of the given cost function moreover a suboptimal fuzzy sampled data control design is also addressed in the sense of minimizing the upper bound of the cost function the main contribution of this study lies in that a new parameterized linear matrix inequality lmi technique is proposed to reduce the conservativeness of the method and an lmi based fuzzy suboptimal sampled data control design is developed for the nonlinear coupled ode pde system based on this parameterized lmi technique simulation results on the sampled data control of hypersonic rocket car are provided to illustrate the effectiveness and merit of the design method

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

h _ infty fuzzy observer based control for a class of nonlinear distributed parameter systems with control constraints an hinfin fuzzy observer based control design is proposed for a class of nonlinear parabolic partial differential equation pde systems with control constraints for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement in the proposed control scheme galerkin s method is initially applied to the pde system to derive a nonlinear ordinary differential equation ode system that accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently represented by the takagi sugeno t s fuzzy model then based on the t s fuzzy model a fuzzy observer based controller is developed to stabilize the nonlinear pde system and achieve an optimized hinfin disturbance attenuation performance for the finite dimensional slow system while control constraints are respected the outcome of the hinfin fuzzy observer based control problem is formulated as a bilinear matrix inequality bmi optimization problem a local optimization algorithm that treats the bmi as a double linear matrix inequality is presented to solve this bmi optimization problem finally the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness

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

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

split closure and intersection cuts in the seventies balas introduced intersection cuts for a mixed integer linear program milp and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation in the early nineties cook kannan and schrijver introduced the split closure of a milp and showed that the split closure is a polyhedron in this paper we show that the split closure can be obtained using only intersection cuts we give two different proofs of this result one geometric and one algebraic the result is then used to provide a new proof of the fact that the split closure of a milp is a polyhedron finally we extend the result to more general disjunctions

on mixing inequalities rank closure and cutting plane proofs we study the mixing inequalities that were introduced by gunluk and pochet math program 90 2001 pp 429 457 we show that a mixing inequality which mixes n mir inequalities has mir rank at most n if it is a type i mixing inequality and at most n 1 if it is a type ii mixing inequality we also show that these bounds are tight for n 2 given a mixed integer set p i p cap z i where p is a polyhedron and z i x in mathbb r n x i in mathbb z forall i in i we define mixing inequalities for p i we show that the elementary mixing closure of p with respect to i can be described using a bounded number of mixing inequalities each of which has a bounded number of terms this implies that the elementary mixing closure of p is a polyhedron finally we show that any mixing inequality can be derived via a polynomial length mir cutting plane proof combined with results of dash on the complexity of cutting plane proofs using split cuts ibm research report rc 24082 oct 2006 and pudlak j symbolic logic 62 1997 pp 981 998 this implies that there are valid inequalities for a certain mixed integer set that cannot be obtained via a polynomial size mixing cutting plane proof

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

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

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

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

cutting planes for multistage stochastic integer programs this paper addresses the problem of finding cutting planes for multistage stochastic integer programs we give a general method for generating cutting planes for multistage stochastic integer programs based on combining inequalities that are valid for the individual scenarios we apply the method to generate cuts for a stochastic version of a dynamic knapsack problem and for stochastic lot sizing problems we give computational results which show that these new inequalities are very effective in a branch and cut algorithm

fuzzy boundary control design for a class of nonlinear parabolic distributed parameter systems this paper deals with the problem of fuzzy boundary control design for a class of nonlinear distributed parameter systems which are described by semilinear parabolic partial differential equations pdes both distributed measurement form and collocated boundary measurement form are considered a takagi sugeno t s fuzzy pde model is first applied to accurately represent the semilinear parabolic pde system based on the t s fuzzy pde model two types of fuzzy boundary controllers which are easily implemented since only boundary actuators are used are proposed to ensure the exponential stability of the resulting closed loop system sufficient conditions of exponential stabilization are established by employing the lyapunov direct method and the vector valued wirtinger s inequality and presented in terms of standard linear matrix inequalities finally the advantages and effectiveness of the proposed control methodology are demonstrated by the simulation results of two examples

disjunctive programming and relaxations of polyhedra given a polyhedron l with h facets whose interior contains no integral points and a polyhedron p recent work in integer programming has focused on characterizing the convex hull of p minus the interior of l we show that to obtain such a characterization it suffices to consider all relaxations of p defined by at most n h 1 among the inequalities defining p this extends a result by andersen cornuejols and li

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

convergence of a finite element approximation to a state constrained elliptic control problem we consider an elliptic optimal control problem with pointwise state constraints the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation the corresponding minima are shown to converge in l 2 to the exact control as the discretization parameter tends to zero furthermore error bounds for the control and the state are obtained in both two and three space dimensions finally we present numerical examples which confirm our analytical findings

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

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

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

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

a convex analysis perspective on disjunctive cuts we treat with tools from convex analysis the general problem of cutting planes separating a point from a closed convex set p crucial for this is the computation of extreme points in the so called reverse polar set introduced by e balas in 1979 in the polyhedral case this enables the computation of cuts that define facets of p we exhibit three equivalent optimization problems to compute such extreme points one of them corresponds to selecting a specific normalization to generate cuts we apply the above development to the case where p is the closed convex hull of a union and more particularly a union of polyhedra case of disjunctive cuts we conclude with some considerations on the design of efficient cut generators the paper also contains an appendix reviewing some fundamental concepts of convex analysis

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

reduce and split cuts improving the performance of mixed integer gomory cuts mixed integer gomory cuts have become an integral part of state of the art software for solving mixed integer linear programming problems therefore improvements in the performance of these cutting planes can be of great practical value in this paper we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed integer gomory cuts this is motivated by the fact that in a mixed integer gomory cut the coefficient of an integer variable lies between 0 and 1 whereas for a continuous variable there is no upper bound the heuristic tries to reduce the coefficients of the continuous variables we call the resulting cuts reduce and split cuts we found that on several test problems reduce and split cuts can substantially enhance the performance of a branch and bound algorithm

barrier methods for optimal control problems with convex nonlinear gradient state constraints in this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems governed by partial differential equations we will derive existence of solutions together with first order optimality conditions afterwards we show continuity of the central path together with convergence rates depending on the interior point parameter

notions of maximality for integral lattice free polyhedra the case of dimension three lattice free sets and their applications for cutting plane methods in mixed integer optimization have been studied in recent literature the family of all integral lattice free polyhedra that are not properly contained in another integral lattice free polyhedron has been of particular interest we call these polyhedra d maximal for fixed d the family of d maximal integral lattice free polyhedra is finite up to unimodular equivalence in view of possible applications in cutting plane theory one would like to have a classification of this family this is a challenging task already for small dimensions in contrast the subfamily of all integral lattice free polyhedra that are not properly contained in any other lattice free set which we call d maximal lattice free polyhedra allow a rather simple geometric characterization hence the question was raised for which dimensions the notions of d maximality and d maximality are equivalent this was known to be the case for dimensions one and two on the

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

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

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

spatially piecewise fuzzy control design for sampled data exponential stabilization of semilinear parabolic pde systems this paper employs a takagi sugeno t s fuzzy partial differential equation pde model to solve the problem of sampled data exponential stabilization in the sense of spatial l infty norm vert cdot vert infty for a class of nonlinear parabolic distributed parameter systems dpss where only a few actuators and sensors are discretely distributed in space initially a t s fuzzy pde model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear dpss subsequently a static sampled data fuzzy local state feedback controller is constructed based on the t s fuzzy pde model by constructing an appropriate lyapunov krasovskii functional candidate and employing vector valued wirtinger s inequalities a variation of vector valued poincare wirtinger inequality in one dimensional spatial domain as well as a vector valued agmon s inequality it is shown that the suggested sampled data fuzzy controller exponentially stabilizes the nonlinear dpss in the sense of vert cdot vert infty if sufficient conditions presented in term of standard linear matrix inequalities lmis are fulfilled moreover an lmi relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled data fuzzy controller finally the satisfactory and better performance of the suggested sampled data fuzzy controller are demonstrated by numerical simulation results of two examples

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

exponential stabilization for 1 d linear ito type state dependent stochastic parabolic pde systems via static output feedback abstract this paper deals with the problem of exponential stabilization for 1 d linear stochastic parabolic partial differential equation pde systems with state multiplicative noise in the form of ito type a static output feedback sof control scheme is proposed to stabilize the stochastic pde system in a stochastic framework via locally collocated piecewise uniform actuators and sensors the closed loop well posedness analysis is provided by virtue of the semi group theory based on the infinite dimensional infinitesimal operator and lyapunov technique the sof controller design is developed to guarantee the exponential stability of the resulting closed loop system in the mean square sense finally simulation results on a stochastic heat equation are presented to verify the effectiveness of the proposed design method

sequence independent lifting for mixed integer programming we show that superadditive lifting functions lead to sequence independent lifting of inequalities for general mixed integer programming as an application we note that mixed integer rounding mir may be viewed as sequence independent lifting consequently we obtain facet conditions for mir inequalities for mixed integer knapsacks

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

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

fuzzy control for nonlinear time delay distributed parameter systems under spatially point measurements this paper introduces a fuzzy control fc under spatially point measurements for nonlinear time delay distributed parameter systems dpss described by parabolic partial differential difference equations pddes first a takagi sugeno t s fuzzy pdde model is employed to represent the nonlinear time delay dpss second with the aid of the t s fuzzy pdde model an fc design under spatially point measurements is developed in the formulation of linear matrix inequalities by constructing an appropriate lyapunov functional which can stabilize exponentially the time delay dpss this stabilization condition can be applied to either slowing varying time delay or fast varying one finally simulation results of a numerical example are provided to illustrate the effectiveness of the proposed method

stabilization of bernoulli euler beams by means of a pointwise feedback force we study the energy decay of a bernoulli euler beam which is subject to a pointwise feedback force we show that both uniform and nonuniform energy decay may occur the uniform or nonuniform decay depends on the boundary conditions in the case of nonuniform decay in the energy space we give explicit polynomial decay estimates valid for regular initial data our method consists of deducing the decay estimates from observability inequalities for the associated undamped problem via sharp trace regularity results

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

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

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

exponential pointwise stabilization of semilinear parabolic distributed parameter systems via the takagi sugeno fuzzy pde model this paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the takagi sugeno t s fuzzy partial differential equation pde model where a finite number of actuators are active only at some specified points of the spatial domain these actuators are referred to as pointwise actuators three cases of state feedback are respectively considered in this study as follows full state feedback piecewise state feedback and collocated pointwise state feedback it is initially assumed that a t s fuzzy pde model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic pde system based on the obtained t s fuzzy pde model lyapunov based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector valued wirtinger s inequality to guarantee locally exponential pointwise stabilization of the semilinear pde system and presented in terms of standard linear matrix inequalities lmis moreover the favorable property offered by sharing all the same premises in the t s fuzzy pde models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback a parameterized lmi is introduced for this case to enhance the stabilization ability of the fuzzy controller finally the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples

a priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time in this paper we consider an optimal control problem which is governed by a linear parabolic equation and is subject to state constraints pointwise in time optimal order error estimates are developed for a space time finite element discretization of this problem numerical examples confirm the theoretical results as a by product of our analysis we derive a new regularity result for the optimal control

static collocated piecewise fuzzy control design of quasi linear parabolic pde systems subject to periodic boundary conditions this paper presents a lyapunov and partial differential equation pde based methodology to solve static collocated piecewise fuzzy control design of quasi linear parabolic pde systems subject to periodic boundary conditions two types of piecewise control i e globally piecewise control and locally piecewise control are considered respectively a takagi sugeno t s fuzzy pde model that is constructed via local sector nonlinearity method is first employed to accurately describe spatiotemporal dynamics of quasi linear pdes based on the t s fuzzy pde model a static collocated piecewise fuzzy feedback controller is constructed to guarantee the locally exponential stability of the resulting closed loop system sufficient conditions for the existence of such fuzzy controller are developed by applying vector valued poincare wirtinger inequality and its variants and a linear matrix inequality lmi relaxation technique these sufficient conditions are presented in terms of standard lmis finally the performance of the suggested fuzzy controller is illustrated by numerical simulation results of a nonlinear pde system described by quasi linear fitzhugh nagumo equation with periodic boundary conditions

brief paper wirtinger s inequality and lyapunov based sampled data stabilization discontinuous lyapunov functionals appeared to be very efficient for sampled data systems fridman 2010 naghshtabrizi hespanha teel 2008 in the present paper new discontinuous lyapunov functionals are introduced for sampled data control in the presence of a constant input delay the construction of these functionals is based on the vector extension of wirtinger s inequality these functionals lead to simplified and efficient stability conditions in terms of linear matrix inequalities lmis the new stability analysis is applied to sampled data state feedback stabilization and to a novel sampled data static output feedback problem where the delayed measurements are used for stabilization

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

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

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

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

exponential stabilization for a class of nonlinear parabolic pde systems via fuzzy control approach this paper deals with the exponential stabilization problem for a class of nonlinear spatially distributed processes that are modeled by semilinear parabolic partial differential equations pdes for which a finite number of actuators are used a fuzzy control design methodology is developed for these systems by combining the pde theory and the takagi sugeno t s fuzzy model based control technique initially a t s fuzzy parabolic pde model is proposed to accurately represent a semilinear parabolic pde system then based on the t s fuzzy model a lyapunov technique is used to design a continuous fuzzy state feedback controller such that the closed loop pde system is exponentially stable with a given decay rate the stabilization condition is presented in terms of a set of spatial differential linear matrix inequalities sdlmis furthermore a recursive algorithm is presented to solve the sdlmis via the existing linear matrix inequality optimization techniques finally numerical simulations on the temperature profile control of a catalytic rod are given to verify the effectiveness of the proposed design method

guaranteed cost design for controlling semilinear parabolic pde systems with mobile collocated actuators and sensors this paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation pde systems using mobile collocated actuators and sensors

h_ infty fuzzy control of nonlinear systems under unreliable communication links this paper investigates the problem of h infin fuzzy control of nonlinear systems under unreliable communication links the nonlinear plant is represented by a takagi sugeno t s fuzzy model and the control strategy takes the form of parallel distributed compensation the communication links existing between the plant and controller are assumed to be imperfect that is data packet dropouts occur intermittently which appear typically in a network environment and stochastic variables satisfying the bernoulli random binary distribution are utilized to model the unreliable communication links attention is focused on the design of h infin controllers such that the closed loop system is stochastically stable and preserves a guaranteed h infin performance two approaches are developed to solve this problem based on the quadratic lyapunov function and the basis dependent lyapunov function respectively several examples are provided to illustrate the usefulness and applicability of the developed theoretical results

parameterized linear matrix inequality techniques in fuzzy control system design this paper proposes different parameterized linear matrix inequality plmi characterizations for fuzzy control systems these plmi characterizations are in turn relaxed into pure lmi programs which provides tractable and effective techniques for the design of suboptimal fuzzy control systems the advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations

branching on general disjunctions this paper considers a modification of the branch and cut algorithm for mixed integer linear programming where branching is performed on general disjunctions rather than on variables we select promising branching disjunctions based on a heuristic measure of disjunction quality this measure exploits the relation between branching disjunctions and intersection cuts in this work we focus on disjunctions defining the mixed integer gomory cuts at an optimal basis of the linear programming relaxation the procedure is tested on instances from the literature experiments show that for a majority of the instances the enumeration tree obtained by branching on these general disjunctions has a smaller size than the tree obtained by branching on variables even when variable branching is performed using full strong branching

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

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

a priori error estimates for optimal control problems with pointwise constraints on the gradient of the state we analyze a finite element approximation of an elliptic optimal control problem with pointwise bounds on the gradient of the state variable we derive convergence rates if the control space is discretized implicitly by the state equation in contrast to prior work we obtain these results directly from classical results for the w 1 error of the finite element projection without using adjoint information if the control space is discretized directly we first prove a regularity result for the optimal control to control the approximation error based on which we then obtain analogous convergence rates

an optimization approach for well targeted transcranial direct current stimulation transcranial direct current stimulation tdcs is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes the aim of this study is to introduce and analyze a novel optimization method for safe and well targeted multiarray tdcs for optimization we consider an optimal control problem for a laplace equation with neumann boundary conditions with control and pointwise gradient state constraints we prove well posedness results for the proposed methods and provide computer simulation results in a highly realistic six compartment geometry adapted hexahedral head model for discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown for numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality target intensity and orientation the numerical results reveal that the optimized current flow fields show significantly higher focality and in most cases higher directional agreement to the target vector in comparison to standard bipolar electrode montages

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

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

unbounded convex sets for non convex mixed integer quadratic programming this paper introduces a fundamental family of unbounded convex sets that arises in the context of non convex mixed integer quadratic programming it is shown that any mixed integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family some fundamental properties of the convex sets are derived along with connections to some other well studied convex sets several classes of valid and facet inducing inequalities are also derived

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

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

fuzzy state space modeling and robust observer based control design for nonlinear partial differential systems in this paper a robust fuzzy control design is proposed for the stabilization of nonlinear partial differential systems npdss based on galerkin s method a takagi sugeno t s fuzzy pds is first proposed to model an npds then the t s fuzzy pds can be represented by a finite dimensional t s fuzzy subsystem in controlled mode and a coupled infinite dimensional t s fuzzy subsystem in residual mode therefore the npds can be partitioned into a finite dimensional t s fuzzy slow state space subsystem to be controlled and a coupled infinite dimensional fast residual subsystem to be tolerated based on the small gain theorem a robust fuzzy observer based controller is developed to tolerate the coupled residual subsystem to asymptotically stabilize the npds furthermore based on the dissipative theory an hinfin control design is proposed to attenuate the effects of external disturbances and measurement noises on the robust stabilization of npdss the matlab linear matrix inequality toolbox can be employed to efficiently solve the optimal hinfin fuzzy observer based control design problem of npdss finally a simulation example is given to illustrate the design procedure and confirm the performance of the proposed robust fuzzy observer based control method for the perturbative npdss

finite element approximation of elliptic control problems with constraints on the gradient we consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state we present a tailored finite element approximation to this optimal control problem where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order raviart thomas mixed finite element pointwise bounds on the gradient variable are enforced in the elements of the triangulation controls are not discretized error bounds for control and state are obtained in two and three space dimensions a numerical example confirms our analytical findings

mingling mixed integer rounding with bounds mixed integer rounding mir is a simple yet powerful procedure for generating valid inequalities for mixed integer programs when used as cutting planes mir inequalities are very effective for mixed integer programming problems with unbounded integer variables for problems with bounded integer variables however cutting planes based on lifting techniques appear to be more effective this is not surprising as lifting techniques make explicit use of the bounds on variables whereas the mir procedure does not in this paper we describe a simple procedure which we call mingling for incorporating variable bound information into mir by explicitly using the variable bounds the mingling procedure leads to strong inequalities for mixed integer sets with bounded variables we show that facets of mixed integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure in particular the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of marchand and wolsey math program 85 15 33 1999 as well as the continuous integer knapsack cover and pack inequalities of atamturk math program 98 145 175 2003 ann oper res 139 21 38 2005 in addition mingling inequalities give a generalization of the two step mir inequalities of dash and gunluk math program 105 29 53 2006 under some conditions

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

a c0 interior penalty method for the dirichlet control problem governed by biharmonic operator an energy space based dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a c0 interior penalty method is proposed and analyzed an abstract a priori error estimate is derived under the minimal regularity conditions the abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular further an optimal order l2 norm error estimate is derived numerical experiments illustrate the theoretical findings

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

characterization of the split closure via geometric lifting abstract we analyze split cuts from the perspective of cut generating functions via geometric lifting we show that cuts a natural higher dimensional generalization of the k cuts of cornuejols et al give all the split cuts for the mixed integer corner relaxation as an immediate consequence we obtain that the k cuts are equivalent to split cuts for the 1 row mixed integer relaxation further we show that split cuts for finite dimensional corner relaxations are restrictions of split cuts for the infinite dimensional relaxation in a final application of this equivalence we exhibit a family of pure integer programs whose split closure has arbitrarily bad integrality gap this complements the mixed integer example provided by basu et al 2011

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