smoothness of transition maps in singular perturbation problems with one fast variable abstract this paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field one fast multiple slow directions orbits connecting both sections are canard orbits i e they first move rapidly towards the attracting part of a critical surface then travel a distance near this critical surface even beyond the point where the orbit enters the repelling part of the critical surface and finally repel away from the surface we prove that the transition map is smooth in a transcritical situation however where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold the smoothness of the transition map may be limited due to resonance phenomena that are revealed by blowing up the turning point we present a polynomial example in r 3

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis

model boundary approximation method as a unifying framework for balanced truncation and singular perturbation approximation we show that two widely accepted model reduction techniques balanced truncation bt and balanced singular perturbation approximation bspa can be derived as limiting approximations of a carefully constructed parameterization of linear time invariant systems by employing the model boundary approximation method mbam 1 we also show that mbam provides a novel way to interpolate between bt and bspa by exploring the set of approximations on the boundary of the model manifold which is associated with the specific choice of model parameterization and initial condition and is embedded in a sample space of measured outputs between the elements that correspond to the two model reduction techniques this paper suggests similar types of approximations may be obtainable in topologically similar places i e on certain boundaries on the associated model manifold of nonlinear systems if analogous parameterizations can be achieved therefore extending these widely accepted model reduction techniques to nonlinear systems 1

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

dynamic transcritical bifurcations in a class of slow fast predator prey models abstract we study the stability loss delay phenomenon in a dynamic transcritical bifurcation in a class of three dimensional prey and predator systems the dynamics of the predator is assumed to be slow comparatively to the dynamics of the prey as an application a well known model considered by clark will be discussed

a note on the selection of benders cuts a new cut selection criterion for benders cuts is proposed and computationally analyzed the results show that the new criterion is more robust and often considerably faster than the standard ones

a cross decomposition scheme with integrated primal dual multi cuts for two stage stochastic programming investment planning problems we describe a cross decomposition algorithm that combines benders and scenario based lagrangean decomposition for two stage stochastic programming investment planning problems with complete recourse where the first stage variables are mixed integer and the second stage variables are continuous the algorithm is a novel cross decomposition scheme and fully integrates primal and dual information in terms of primal dual multi cuts added to the benders and the lagrangean master problems for each scenario the potential benefits of the cross decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions in the original formulation where the underlying lp relaxation is weak the cross decomposition method outperforms multi cut benders decomposition if the formulation is improved with the addition of tightening constraints the performance of both decomposition methods improves but cross decomposition clearly remains the best method for large scale problems

almost planar homoclinic loops in r3 in this paper we study homoclinic loops of vector fields in 3 dimensional space when the two principal eigenvalues are real of opposite sign which we call almost planar we are interested to have a theory for higher codimension bifurcations almost planar homoclinic loop bifurcations generically occur in two versions non twisted and twisted loops we consider high codimension homoclinic loop bifurcations under generic conditions the generic condition forces the existence of a 2 dimensional topological invariant ring non necessarily unique which is a topological cylinder in the non twisted case and a topological mobius band in the twisted case if the third eigenvalue is negative resp positive the limit set resp limit set of trajectories starting in a neighborhood of the loop is contained in the invariant ring the dynamics is then given by fixed points or periodic points of period 2 of a 1 dimensional map which admits a nice asymptotic expansion allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non twisted case

stabilization of a class of slow fast control systems at non hyperbolic points in this document we deal with the local asymptotic stabilization problem of a class of slow fastsystems or singularly perturbed ordinary differential equations the systems studied here have the following properties 1 they have one fast and an arbitrary number of slow variables and 2 they have a non hyperbolic singularity at the origin of arbitrary degeneracy our goal is to stabilize such a point the presence of the aforementioned singularity complicates the analysis and the controller design in particular the classical theory of singular perturbations cannot be used we propose a novel design based on geometric desingularization which allows the stabilization of a non hyperbolic point of singularly perturbed control systems our results are exemplified on a didactic example and on an electric circuit

on maximum bifurcation delay in real planar singularly perturbed vector fields abstract in singularly perturbed vector fields where the unperturbed vector field has a curve of singularities a critical curve orbits tend to be attracted towards or repelled away from this curve depending on the sign of the divergence of the vector field at the curve when at some point this sign bifurcates from negative to positive orbits will typically be repelled away immediately after passing the bifurcation point turning point atypical behaviour is nevertheless observed as well when orbits follow the critical curve for some distance after the turning point before they repel away from it a delay in the bifurcation is present interesting are systems that have a maximum bifurcation delay i e there is a point on the critical curve beyond which orbits cannot stay close to the critical curve this behaviour is known to appear in some systems in dimension 3 see e benoit ed dynamic bifurcations in lecture notes in mathematics vol 1493 springer verlag berlin 1991 and it is commonly believed that it is not an issue in real planar systems beside making the observation that it does appear in non analytic planar systems it is shown that whenever bifurcation delay appears it has no non trivial maximum for analytic planar vector fields the proof is based on the notion of family blow up at the turning point on formal power series in terms of blow up variables the study of their gevrey properties and analytic continuation of their borel transform these results complement existing results concerning the equivalence of local and global canard solutions in a fruchard r schafke overstability and resonance ann inst fourier grenoble 53 1 2003 227 264

the entry exit function and geometric singular perturbation theory this work was partly supported by fwo project g093910 de maesschalck and nsf grant dms 1211707 schecter we thank the referee for a careful reading of the paper which led to a number of clarifications and corrections

cross decomposition for mixed integer programming many methods for solving mixed integer programming problems are based either on primal or on dual decomposition which yield respectively a benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via lagrangean relaxation these methods exploit either the primal or the dual structure of the problem we propose a new approach cross decomposition which allows exploiting simultaneously both structures the development of the cross decomposition method captures profound relationships between primal and dual decomposition it is shown that the more constraints can be included in the langrangean relaxation provided the duality gap remains zero the fewer the benders cuts one may expect to need if the linear programming relaxation has no duality gap only one benders cut is needed to verify optimality

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

accelerating the regularized decomposition method for two stage stochastic linear problems practical improvements of the regularized decomposition algorithm for two stage stochastic problems are presented they are associated with the primal simplex method for solving subproblems a penalty formulation of the subproblems is used which facilitates crash and warm starts and allows more freedom when creating the model the computational results are highly encouraging

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

transition time analysis in singularly perturbed boundary value problems the paper deals with the boundary value problem x xj x2 0 with x 0 a x t b for a b t 0 and e 0 close to zero it is shown that for t sufficiently big the problem has exactly three solutions two of which reach negative values solutions reaching negative values occur for t t e 0 and we show that asymptotically for e 0 t e ln e i e lim6 o t e 1 the main tools are transit time analysis in the lienard plane and normal form techniques as such the methods are rather qualitative and useful in other similar problems introduction in this paper the authors will apply desingularization techniques to a singularly perturbed boundary value problem by doing so we will be able to explain apparent transition phenomena in a geometric way the equation we study is given by

on using approximations of the benders master problem abstract in this paper we study some approximate ways of solving the benders master problem in benders decomposition the applications of lagrange duality and surrogate duality to the benders master problem are found to be equivalent we also find that the lagrangean dual of the benders master problem cannot yield better bounds than the partial lagrangean dual of the original problem compared to solving the benders master problem exactly the approximations often yields worse bounds and a lack of controllability which might prohibit the generation of necessary benders cuts for a milp problem this might yield worse bounds than its lp relaxation a convergence test originating from cross decomposition can be used to check the progress of a method using such approximations

a note on vibrations of generally restrained beams exact frequency and normal mode shape expressions are derived in this note for generally restrained bernoulli euler beams with unsymmetrical translations and rotations at either end the eigenfrequencies and mode shape parameters are presented for a wide range of restraint parameters several degenerate cases have been studied and some of these have been compared with those available in the published literature it is believed that the results presented in this paper will be of use in design of beams shafts and piping under dynamic considerations

a regularized decomposition method for minimizing a sum of polyhedral functions a problem of minimizing a sum of many convex piecewise linear functions is considered in view of applications to two stage linear programming where objectives are marginal values of lower level problems it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise linear induced feasibility constraints we propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component the master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts the method goes through nonbasic points in general and is finitely convergent without any nondegeneracy assumptions next we present a special technique for solving the regularized master problem that uses an active set strategy and qr factorization and exploits the structure of the master finally some numerical evidence is given

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

partial linearization for planar nonautonomous differential equations abstract we prove a partial linearization theorem for planar nonautonomous differential equations with one center like direction and one hyperbolic direction gap conditions are formulated in terms of the dichotomy spectral intervals the result is applied to the duffing van der pol oscillator under a nonautonomous parametric perturbation

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

regular and slow fast codimension 4 saddle node bifurcations abstract using geometric singular perturbation theory including the family blow up as one of the main techniques we prove that the cyclicity i e maximum number of limit cycles in both regular and slow fast unfoldings of nilpotent saddle node singularity of codimension 4 is 2 the blow up technique enables us to use the well known results for slow fast codimension 1 and 2 hopf bifurcations slow fast bogdanov takens bifurcations and slow fast codimension 3 saddle and elliptic bifurcations

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

limit cycles near hyperbolas in quadratic systems abstract in this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane we study a strip of hyperbolas occurring in some quadratic systems we deal with the cyclicity of the degenerate graphics d i 2 a from the programme set up in f dumortier r roussarie c rousseau hilbert s 16th problem for quadratic vector fields j differential equations 110 1994 86 133 to solve the finiteness part of hilbert s 16th problem for quadratic systems techniques from geometric singular perturbation theory are combined with the use of the bautin ideal we also rely on the theory of darboux integrability

multi layer canard cycles and translated power functions abstract the paper deals with two dimensional slow fast systems and more specifically with multi layer canard cycles these are canard cycles passing through n layers of fast orbits with n 2 the canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n we prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

multiple canard cycles in generalized lienard equations the paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields the results deal with any number of parameters proofs are based on the techniques introduced in canard cycles and center manifolds f dumortier and r roussarie 1996 mem amer math soc 121 the presentation is limited to generalized lienard equations ex x c x x c 0

adaptive multicut aggregation for two stage stochastic linear programs with recourse outer linearization methods for two stage stochastic linear programs with recourse such as the l shaped algorithm generally apply a single optimality cut on the nonlinear objective at each major iteration while the multicut version of the algorithm allows for several cuts to be placed at once in general the l shaped algorithm tends to have more major iterations than the multicut algorithm however the trade offs in terms of computational time are problem dependent this paper investigates the computational trade offs of adjusting the level of optimality cut aggregation from single cut to pure multicut specifically an adaptive multicut algorithm that dynamically adjusts the aggregation level of the optimality cuts in the master program is presented and tested on standard large scale instances from the literature computational results reveal that a cut aggregation level that is between the single cut and the multicut can result in substantial computational savings over the single cut method

a demand shifting feasibility algorithm for benders decomposition abstract benders decomposition is a popular method for solving problems by resource directive decomposition often the resource allocations from the master problem lead to infeasible subproblems as resources are insufficient to meet demand this generally requires the use of feasibility cuts to reach a feasible solution which can be computationally expensive for problems in which subproblems have limited capacity we propose an efficient algorithm which shifts excess demand to other sources of capacity the advantages of the algorithm are that it is quick requires only one solution of each subproblem in each benders iteration and does not add any feasibility cuts into the master problem a computational study is performed on a fleet sizing problem to illustrate the algorithm s efficiency when compared to the traditional feasibility cut method

polynomial normal forms of constrained differential equations with three parameters abstract we study generic constrained differential equations cdes with three parameters thereby extending takens s classification of singularities of such equations in this approach the singularities analyzed are the swallowtail the hyperbolic and the elliptic umbilics we provide polynomial local normal forms of cdes under topological equivalence generic cdes are important in the study of slow fast sf systems many properties and the characteristic behavior of the solutions of sf systems can be inferred from the corresponding cde therefore the results of this paper show a first approximation of the flow of generic sf systems with three slow variables

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

dynamic analysis of a multi span uniform beam carrying a number of various concentrated elements abstract this paper employs the numerical assembly method nam to determine the exact frequency response amplitudes of a multiple span beam carrying a number of various concentrated elements and subjected to a harmonic force and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force first the coefficient matrices for the intermediate concentrated elements pinned support applied force left end support and right end support of a beam are derived next the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method fem finally the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix the graph of dynamic response amplitudes versus various exciting frequencies gives the frequency response curve for any point of a multiple span beam carrying a number of various concentrated elements for the case of zero harmonic force the above mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained

l shaped decomposition of two stage stochastic programs with integer recourse we consider two stage stochastic programming problems with integer recourse the l shaped method of stochastic linear programming is generalized to these problems by using generalized benders decomposition nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques finite convergence of the method is established when gomory s fractional cutting plane algorithm or a branch and bound algorithm is applied

a geometric analysis of the lagerstrom model problem abstract lagerstrom s model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low reynolds number by the method of matched asymptotic expansions in this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems in particular invariant manifold theory as an essential part of the dynamics takes place near a line of non hyperbolic equilibria a blow up transformation is introduced to resolve these singularities this approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem in particular the source of the logarithmic switchback phenomenon is identified

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

improving benders decomposition using a genetic algorithm we develop and investigate the performance of a hybrid solution framework for solving mixed integer linear programming problems benders decomposition and a genetic algorithm are combined to develop a framework to compute feasible solutions we decompose the problem into a master problem and a subproblem a genetic algorithm along with a heuristic are used to obtain feasible solutions to the master problem whereas the subproblem is solved to optimality using a linear programming solver over successive iterations the master problem is refined by adding cutting planes that are implied by the subproblem we compare the performance of the approach against a standard benders decomposition approach as well as against a stand alone solver cplex on miplib test problems

a benders decomposition based heuristic for the hierarchical production planning problem abstract in this paper a heuristic procedure for determining good feasible solutions to a multi item dynamic production planning problem is described and evaluated the problem which is modelled as a mixed integer linear program has a hierarchical structure where production items are aggregated into families and families into product types a decomposition of the original problem creates a trivial subproblem providing feasible solutions on the type level and a master problem which separates into one uncapacitated lot sizing problem for each family the latter problems can be solved by dynamic programming with very little computational effort besides the nice structure of both the sub and master problem the dual information from the subproblem can be given interesting economic interpretations as distinguished from hax and meal s hierarchical framework for production planning and also they hybrid approach suggested by graves in 1982 there exists a direct information flow from the family level problem to the type level problem in form of a production plan the heuristic is essentially an application of benders decomposition the only difference being that the benders cuts are relaxed using lagrangean multipliers a subgradient procedure is used to update the lagrangean multipliers one iteration of this heuristic requires less computational effort compared to grave s algorithm this is due to the fact that the benders subproblem can be solved by inspection while graves algorithm involves solving an lp problem on the type level for thirtysix medium size test problems the average deviation from optimum is 2 34 and the deviation ranges between 0 00 and 5 95

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

analysis of a slow fast system near a cusp singularity this paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe our analysis consists of two main parts first we recall a formal normal form suitable for systems as the one studied here afterwards taking advantage of this normal form we investigate the transition near the cusp singularity by means of the blow up technique our contribution relies heavily in the usage of normal form theory allowing us to refine previous results

inexact cuts in benders decomposition benders decomposition is a well known technique for solving large linear programs with a special structure in particular it is a popular technique for solving multistage stochastic linear programming problems early termination in the subproblems generated during benders decomposition assuming dual feasibility produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution we describe an inexact cut algorithm prove its convergence under easily verifiable assumptions and discuss a corresponding dantzig wolfe decomposition algorithm the paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

singular perturbations and vanishing passage through a turning point abstract the paper deals with planar slow fast cycles containing a unique generic turning point we address the question on how to study canard cycles when the slow dynamics can be singular at the turning point we more precisely accept a generic saddle node bifurcation to pass through the turning point it reveals that in this case the slow divergence integral is no longer the good tool to use but its derivative with respect to the layer variable still is we provide general results as well as a number of applications we show how to treat the open problems presented in artes et al 2009 1 and dumortier and rousseau 2009 13 dealing respectively with the graphics di 2 a and df 1 a from dumortier et al 1994 14

large scale mixed integer programming benders type heuristics abstract the limited success of exact methods for solving integer programming problems has prompted the development of heuristic procedures which have performed surprisingly well in their search for near optimal solutions the present paper constitutes an attempt to generalize these procedures to the mixed integer case the approach is based on the utilization of the benders partitioning method bpm which separates the integer from the continuous variables a number of alternatives are presented for integrating ip heuristics in the bpm thus alleviating its main limitation the necessity of solving a sequence of integer master problems the rationale and its usefulness are illustrated on large scale applications from the fields of power systems and network flows

accelerated multigrid for graph laplacian operators we consider multigrid type techniques for the numerical solution of large linear systems whose coefficient matrices show the structure of weighted graph laplacian operators we combine ad hoc coarser grid operators with iterative techniques used as smoothers empirical tests suggest that the most effective smoothers have to be of krylov type with subgraph preconditioners while the projectors which define the coarser grid operators have to be designed for maintaining as much as possible the graph structure of the projected matrix at the inner levels the main theoretical contribution of the paper is the characterization of necessary and sufficient conditions for preserving the graph structure in this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they may behave unsatisfactorily for unstructured graphs furthermore we report and discuss several numerical experiments showing that our approach is effective even in very difficult cases where the known approaches are rather slow as a conclusion the main advantage of the proposed approach is the robustness since our multigrid type technique behaves uniformly well in all cases without requiring either the setting or the knowledge of critical parameters as it happens when using the best known preconditioned krylov methods

an improved benders decomposition algorithm for the tree of hubs location problem abstract the tree of hubs location problem is a particularly hard variant of the so called hub location problems when solving this problem by a benders decomposition approach it is necessary to deal with both optimality and feasibility cuts while modern implementations of the benders decomposition method rely on pareto optimal optimality cuts or on rendering feasibility cuts based on minimal infeasible subsystems a new cut selection scheme is devised here under the guiding principle of extracting useful information even when facing infeasible subproblems the proposed algorithm outperforms two other modern variants of the method and it is capable of optimally solving instances five times larger than the ones previously reported on the literature

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

on the convergence of cross decomposition cross decomposition is a recent method for mixed integer programming problems exploiting simultaneously both the primal and the dual structure of the problem thus combining the advantages of dantzig wolfe decomposition and benders decomposition finite convergence of the algorithm equipped with some simple convergence tests has been proved stronger convergence tests have been proposed but not shown to yield finite convergence in this paper cross decomposition is generalized and applied to linear programming problems mixed integer programming problems and nonlinear programming problems with and without linear parts using the stronger convergence tests finite exact convergence is shown in the first cases unbounded cases are discussed and also included in the convergence tests the behaviour of the algorithm when parts of the constraint matrix are zero is also discussed the cross decomposition procedure is generalized by using generalized benders decomposition in order to enable the solution of nonlinear programming problems

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

free vibrations of beams with general boundary conditions a simple and unified approach is presented for the vibration analysis of a generally supported beam the flexural displacement of the beam is sought as the linear combination of a fourier series and an auxiliary polynomial function the polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives as a result not only is it always possible to expand the displacement in a fourier series for beams with any boundary conditions but also the solution converges at a much faster speed the reliability and robustness of the proposed technique are demonstrated through numerical examples

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

the period function of classical lienard equations in this paper we study the number of critical points that the period function of a center of a classical lienard equation can have centers of classical lienard equations are related to scalar differential equations x x f x x 0 with f an odd polynomial let us say of degree 2 1 we show that the existence of a finite upperbound on the number of critical periods only depending on the value of can be reduced to the study of slow fast lienard equations close to their limiting layer equations we show that near the central system of degree 2 1 the number of critical periods is at most 2 2 we show the occurrence of slow fast lienard systems exhibiting 2 2 critical periods elucidating a qualitative process behind the occurrence of critical periods it all provides evidence for conjecturing that 2 2 is a sharp upperbound on the number of critical periods we also show that the number of critical periods multiplicity taken into account is always even

benders decomposition and column and row generation for solving large scale linear programs with column dependent rows in a recent work muter birbil and bulbul 2013a identified and characterized a general class of linear programming lp problems known as problems with column dependent rows cdr problems these lps feature two sets of constraints with mutually exclusive groups of variables in addition to a set of structural linking constraints in which variables from both groups appear together in a typical cdr problem the number of linking constraints grows very quickly with the number of variables which motivates generating both columns and their associated linking constraints simultaneously on the fly in this paper we expose the decomposable structure of cdr problems via benders decomposition however this approach brings on its own theoretical challenges one group of variables is generated in the benders master problem while the generation of the linking constraints is relegated to the benders subproblem along with the second group of variables a fallout of this separation is that only a partial description of the dual of the benders subproblem is available over the course of the algorithm we demonstrate how the pricing subproblem for the column generation applied to the benders master problem does also update the dual polyhedron and the existing benders cuts in the master problem to ensure convergence ultimately a novel integration of benders cut generation and the simultaneous generation of columns and constraints yields a brand new algorithm for solving large scale cdr problems we illustrate the application of the proposed method on a time constrained routing problem our numerical experiments confirm the outstanding performance of the new decomposition method

solving an electricity generating capacity expansion planning problem by generalized benders decomposition this paper discusses application of generalized benders decomposition in a model for planning least cost investments in electricity generating capacity subject to probabilistic reliability constraints the planning problem is decomposed into a set of subproblems each representing the operation of a set of generating plants of fixed capacity in 1 year and a master problem representing optimal capacity investments over the entire planning horizon the subproblems are solved using a procedure called probabilistic simulation which calculates the expected cost of operating the generating system the reliability level and dual multipliers reflecting the value of small changes in the plant capacities the master problem is a linear program which uses these dual multipliers to approximate the nonlinear cost and reliability functions the solution to the capacity expansion problem is found by iteratively solving the master problem and the subproblems

slow divergence integral on a mobius band abstract the slow divergence integral has proved to be an important tool in the study of slow fast cycles defined on an orientable two dimensional manifold e g r 2 the goal of our paper is to study 1 canard cycle and 2 canard cycle bifurcations on a non orientable two dimensional manifold e g the mobius band by using similar techniques our focus is on smooth slow fast models with a hopf breaking mechanism the same results can be proved for a jump breaking mechanism and non generic turning points the slow fast bifurcation problems on the mobius band require the study of the 2 return map attached to such 1 and 2 canard cycles we give a simple sufficient condition expressed in terms of the slow divergence integral for the existence of a period doubling bifurcation near the 1 canard cycle we also prove the finite cyclicity property of singular 1 and 2 homoclinic loops regular 1 homoclinic loops of finite codimension have been studied by guimond

a priori optimization of the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is defined on a graph g v e where v is the vertex set and e is the edge set each vertex vi has a probability pi of being present with each edge vi vj is associated a distance or cost cij in a first stage an a priori hamiltonian tour on g is designed the list of present vertices is then revealed in a second stage the a priori tour is followed by skipping the absent vertices the ptsp consists of determining a first stage solution that minimizes the expected cost of the second stage tour the problem is formulated as an integer linear stochastic program and solved by means of a branch and cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function problems involving up to 50 vertices are solved to optimality

classical lienard equations of degree n 6 can have n 12 2 limit cycles abstract based on geometric singular perturbation theory we prove the existence of classical lienard equations of degree 6 having 4 limit cycles it implies the existence of classical lienard equations of degree n 6 having at least n 1 2 2 limit cycles this contradicts the conjecture from lins de melo and pugh formulated in 1976 where an upperbound of n 1 2 limit cycles was predicted this paper improves the counterexample from dumortier panazzolo and roussarie 2007 by supplying one additional limit cycle from degree 7 on and by finding a counterexample of degree 6 we also give a precise system of degree 6 for which we provide strong numerical evidence that it has at least 3 limit cycles

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

benders decomposition for large scale uncapacitated hub location this paper describes an exact algorithm capable of solving large scale instances of the well known uncapacitated hub location problem with multiple assignments the algorithm applies benders decomposition to a strong path based formulation of the problem the standard decomposition algorithm is enhanced through the inclusion of several features such as the use of a multicut reformulation the generation of strong optimality cuts the integration of reduction tests and the execution of a heuristic procedure extensive computational experiments were performed to evaluate the efficiency and robustness of the algorithm computational results obtained on classical benchmark instances with up to 200 nodes and on a new and more difficult set of instances with up to 500 nodes confirm the efficiency of the algorithm

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

planning and scheduling by logic based benders decomposition we combine mixed integer linear programming milp and constraint programming cp to solve an important class of planning and scheduling problems tasks are allocated to facilities using milp and scheduled using cp and the two are linked via logic based benders decomposition tasks assigned to a facility may run in parallel subject to resource constraints cumulative scheduling we solve problems in which the objective is to minimize cost makespan or total tardiness we obtain significant computational speedups of several orders of magnitude for the first two objectives relative to the state of the art in both milp and cp we also obtain better solutions and bounds for problems than cannot be solved to optimality

the implicit hitting set approach to solve combinatorial optimization problems with an application to multigenome alignment we develop a novel framework the implicit hitting set approach for solving a class of combinatorial optimization problems the explicit hitting set problem is as follows given a set u and a family s of subsets of u find a minimum cardinality set that intersects hits every set in s in the implicit hitting set problem ihsp the family of subsets s is not explicitly listed its size is generally exponential in terms of the size of u instead it is given via a polynomial time oracle that verifies if a given set h is a hitting set or returns a set in s that is not hit by h many np hard problems can be straightforwardly formulated as implicit hitting set problems we show that the implicit hitting set approach is valuable in developing exact and heuristic algorithms for solving this class of combinatorial optimization problems specifically we provide a generic algorithmic strategy which combines efficient heuristics and exact methods to solve any ihsp given an instance of an ihsp the propose

asymptotic analysis of subcritical hopf homoclinic bifurcation abstract this paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical hopf bifurcation with a homoclinic orbit of the hopf equilibrium our work is motivated by our previous analysis of a hodgkin huxley neuron model which possesses a subcritical hopf bifurcation j guckenheimer r harris warrick j peck a willms j comput neurosci 4 1997 257 277 in this model the hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium that is the hopf bifurcation appears to be close to a homoclinic bifurcation as well this model of the lateral pyloric lp cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike frequency adaptation in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent the presence of a subcritical hopf bifurcation in this model was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non oscillatory solution the analysis presented here explains the apparent proximity of homoclinic and hopf bifurcations we also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical hopf bifurcation that may be close to a codimension two subcritical hopf homoclinic bifurcation

gevrey properties of real planar singularly perturbed systems abstract by applying geometric techniques to real analytic singularly perturbed vector fields on the plane we develop a way to give a bound on the gevrey type of the taylor development of canard manifolds at degenerate planar turning points by blowing up the phase space at the turning point we find asymptotic estimates even when such expansions w r t traditional phase space variables do not exist the asymptotic estimates are then used to give a sufficient and necessary condition on the existence of local canard solutions

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

benders decomposition for production routing under demand uncertainty the production routing problem prp is a generalization of the inventory routing problem and concerns the production and distribution of a single product from a production plant to multiple customers using capacitated vehicles in a discrete and finite time horizon in this study we consider the stochastic prp with demand uncertainty in two stage and multistage decision processes the decisions in the first stage include production setups and customer visit schedules while the production and delivery quantities are determined in the subsequent stages we introduce formulations for the two problems which can be solved by a branch and cut algorithm to handle a large number of scenarios we propose a benders decomposition approach which is implemented in a single branch and bound tree and enhanced through lower bound lifting inequalities scenario group cuts and pareto optimal cuts for the multistage problem we also use a warm start procedure that relies on the solution of the simpler two stage prob

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h