the quadratic shortest path problem complexity approximability and solution methods we consider the problem of finding a shortest path in a directed graph with a quadratic objective function the qspp we show that the qspp cannot be approximated unless p np for the case of a convex objective function an n approximation algorithm is presented where n is the number of nodes in the graph and apx hardness is shown furthermore we prove that even if only adjacent arcs play a part in the quadratic objective function the problem still cannot be approximated unless p np in order to solve the problem we first propose a mixed integer programming formulation and then devise an efficient exact branch and bound algorithm for the general qspp where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems in our computational experiments we solve to optimality different classes of instances with up to 1000 nodes

two parameter degenerate sliding bifurcations in filippov systems this paper is concerned with the extension to the case of codimension 2 degenerate sliding bifurcations of the theory of sliding bifurcations in filippov systems presented in m di bernardo p kowalczyk a nordmark bifurcations of dynamical systems with sliding derivation of normal form mappings physica d 170 2002 175 205 these bifurcations were detected in experimental systems such as the dry friction oscillator and turn out to be organising centres for branches of codimension 1 sliding bifurcations the analysis is carried out for generic n dimensional piecewise smooth systems the possible degenerate scenarios are classified it is shown that several branches of codimension 1 sliding bifurcations originate from the degenerate codimension 2 points such branches are appropriately classified in the degenerate crossing sliding case a friction oscillator is used as a representative example to illustrate and confirm the theoretical derivations the importance is discussed of the unfolding of the degenerate sliding bifurcations for the development of continuation techniques

the steiner traveling salesman problem with online edge blockages abstract we consider the online steiner traveling salesman problem in this problem we are given an edge weighted graph g v e and a subset d v of destination vertices with the optimization goal to find a minimum weight closed tour that traverses every destination vertex of d at least once during the traversal the salesman could encounter at most k non recoverable blocked edges the edge blockages are real time meaning that the salesman knows about a blocked edge whenever it occurs we first show a lower bound on the competitive ratio and present an online optimal algorithm for the problem while this optimal algorithm has non polynomial running time we present another online polynomial time near optimal algorithm for the problem experimental results show that our online polynomial time algorithm produces solutions very close to the offline optimal solutions

traveling salesman path problems in the traveling salesman path problem we are given a set of cities traveling costs between city pairs and fixed source and destination cities the objective is to find a minimum cost path from the source to destination visiting all cities exactly once in this paper we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once we first characterize traveling salesman walk perfect graphs graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities we show these graphs have a description by way of forbidden minors and also characterize them constructively we also address the asymmetric traveling salesman path problem atspp and give a factor o sqrt n approximation algorithm for this problem

bifurcations of dynamical systems with sliding derivation of normal form mappings this paper is concerned with the analysis of so called sliding bifurcations in n dimensional piecewise smooth dynamical systems with discontinuous vector field these novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible the derivation of appropriate normal form maps is detailed it is shown that the leading order term in the map depends on the particular bifurcation scenario considered this is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper a third order relay system serves as a numerical example

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

periodic p laplacian with nonlocal terms abstract in this paper we study the existence of non trivial periodic solutions for a periodic p laplacian with nonlocal terms based on the theory of leray schauder degree the key step is dealing with the degeneracy of the p laplacian and the logistic type terms arising in the right hand side of the equation

the steiner travelling salesman problem with correlated costs the steiner travelling salesman problem stsp is a variant of the tsp that is suitable for instances defined on road networks we consider an extension of the stsp in which the road traversal costs are both stochastic and correlated this happens for example when vehicles are prone to delays due to rush hours road works or accidents following the work of markowitz on portfolio selection we model this problem as a bi objective mean variance problem then we show how to approximate the efficient frontier via integer programming we also show how to exploit certain special structures in the correlation matrices computational results indicate that our approach is viable for instances with up to 100 nodes it turns out that minimum variance tours can look rather different from minimum expected cost tours

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

a cutting plane algorithm for the general routing problem the general routing problem grp is the problem of finding a minimum cost route for a single vehicle subject to the condition that the vehicle visits certain vertices and edges of a network it contains the rural postman problem chinese postman problem and graphical travelling salesman problem as special cases we describe a cutting plane algorithm for the grp based on facet inducing inequalities and show that it is capable of providing very strong lower bounds and in most cases optimal solutions

a cutting plane procedure for the travelling salesman problem on road networks abstract a cutting plane algorithm for the exact solution of the symmetric travelling salesman problem tsp is proposed the real tours on a usually incomplete road network which are in general non hamiltonian are characterized directly by an integer linear programming model the algorithm generates special cutting planes for this model computational results for real road networks with up to 292 visiting places are reported as well as for classical problems of the literature with up to 120 cities some of the latter problems have been solved for the first time with a pure cutting plane approach

a branch and bound algorithm for mixed zero one multiple objective linear programming abstract in this paper a branch and bound algorithm for the generation of the efficient set in mixed zero one multiple objective linear programming problems is presented the algorithm is developed as to take account of the multiple objectives in the node fathoming procedure in order to extend the algorithm s applicability to large sized problems from real life an interactive procedure is introduced which systematically reduces the number of efficient points and thus saves considerable computational effort without losing essential information the algorithm is tested in randomly generated problems along with a case study conceming the power generation sector

periodic solutions of non newtonian polytropic filtration equations with nonlinear sources in this paper the authors first consider the dirichlet boundary value problem to the non newtonian polytropic filtration equation of the form u t div u m p 2 u m h x t u a in wxrwith strong nonlinear sources the existence of nontrivial periodic solutions is established based on topological degree theory the authors also studied the dirichlet boundary value problem to the equation in the form u t div u m 1u p 2 u m 1u b x t u f x t in wxrwith weak nonlinear sources the existence is treated with leray schauder fixed point theory

a dynamic traveling salesman problem with stochastic arc costs we propose a dynamic traveling salesman problem tsp with stochastic arc costs motivated by applications such as dynamic vehicle routing in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed we formulate this as a dynamic program dp and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach we then apply approximate linear programming alp to overcome the dp s curse of dimensionality obtain a semi infinite linear programming lower bound and discuss its tractability we also analyze a rollout version of the price directed policy implied by our alp and derive worst case guarantees for its performance our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound

toward solving the steiner travelling salesman problem on urban road maps using the branch decomposition of graphs the steiner travelling salesman problem stsp is an important issue in intelligent transportation systems and has various practical applications such as travelling and parcel delivery in this study we consider the stsp in real world road maps i e given a road map with a set of service points or intersections selection of the representative vertices of a graph g to determine a closed route so a vehicle can visit each service point at least once with the minimum cost it has been theoretically proved that the stsp and its ancestor travelling salesman problem are np hard however it has been empirically demonstrated that they can be solved efficiently on planar graphs with small branchwidths therefore we propose a branch decomposition based extraction algorithm for the stsp in planar graphs our algorithm can solve the stsp in a planar graph g with a time complexity of 2o bw g no 1 where bw g is the branchwidth of g in the case of planar graphs with a small branchwidth our algorithm performs efficiently we evaluated our algorithm by applying it to real world urban road maps which demonstrated the suitability of our method for real world applications

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

shortest paths in stochastic networks with correlated link costs the objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs each link is assumed to be in one of two possible conditions conditional probability density functions for link travel times are assumed known for each condition conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip this problem is treated as a multistage adaptive feedback control process each stage is described by the physical state the location of the current decision point and the information state the service level of the previously traversed links proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided

existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms abstract in this paper the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of leray schauder s degree

the general routing polyhedron a unifying framework it is shown how to transform the general routing problem grp into a variant of the graphical travelling salesman problem gtsp this transformation yields a projective characterisation of the grp polyhedron using this characterisation it is shown how to convert facets of the gtsp polyhedron into valid inequalities for the grp polyhedron the resulting classes of inequalities subsume several previously published classes

equations for biological evolution in this paper we consider mathematical models inspired by the mechanisms of biological evolution we take populations which are subject to interaction and mutation in the cases we consider the interaction is through competition or through a prey predator relationship the models consider the specific characteristics as taking values in real intervals and the equations are of the integro differential type in the case of competition thanks to the fact that some of the equations have solutions which are quite explicit we succeed in proving the existence of attracting stationary solutions in the case of prey and predator using techniques of dynamical systems in infinite dimensional spaces we succeed in showing the existence of a global attractor which in some instances reduces to a point our analysis takes into account having distributions corresponding to all individuals having the same characteristics as possible populations

bifurcations in nonsmooth dynamical systems a review is presented of the one parameter nonsmooth bifurcations that occur in a variety of continuous time piecewise smooth dynamical systems motivated by applications a pragmatic approach is taken to defining a discontinuity induced bifurcation dib as a nontrivial interaction of a limit set with respect to a codimension one discontinuity boundary in phase space only dibs that are local are considered that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows three classes of systems are considered involving either state jumps jumps in the vector field or jumps in some derivative of the vector field a rich array of dynamics are revealed involving the sudden creation or disappearance of attractors jumps to chaos bifurcation diagrams with sharp corners and cascades of period adding for each kind of bifurcation identified where possible a kind of normal form or discontinuity mapping dm is given together with a canonical example and an application the goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators impact oscillators dc dc converters and problems in control theory

generalized boundary equilibria in n dimensional filippov systems the transition between persistence and nonsmooth fold scenarios abstract we analyze a codimension two bifurcation in generic n dimensional filippov autonomous piecewise smooth discontinuous systems when a hyperbolic equilibrium collides with the discontinuity boundary at generic codim 1 boundary equilibria be two different scenarios can occur the persistence of the stationary solution that changes from a standard equilibrium on one side of the discontinuity boundary into a pseudo equilibrium and the nonsmooth fold between the standard equilibrium and a coexisting pseudo equilibrium generalized codim 2 boundary equilibria gbe separate persistence and nonsmooth fold scenarios and are here shown to occur together with a fold bifurcation between pseudo equilibria plp in a two parameter plane the plp bifurcation curve emanates from the gbe point tangentially to the be curve the asymptotic of the plp curve locally to the gbe point is also derived and tested on a 4 dimensional ecological example

periodic doubly degenerate parabolic equation with nonlocal terms this paper is a remark on gao and wang 2007 4 under some weaker conditions than 4 we establish the existence of nontrivial nonnegative periodic solutions for a doubly degenerate parabolic equation with nonlocal terms the key is the use of de giorgi iteration technique and the theory of leray schauder degree

coexistence and optimal control problems for a degenerate predator prey model abstract in this paper we present a predator prey mathematical model for two biological populations which dislike crowding the model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts we provide conditions on the system ensuring the periodic coexistence namely the existence of two non trivial non negative periodic solutions representing the densities of the two populations we assume that the predator species is harvested if its density exceeds a given threshold a minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered

grazing and border collision in piecewise smooth systems a unified analytical framework a comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes this links grazings with border collisions in nonsmooth maps contrary to previous literature piecewise linear maps correspond only to nonsmooth discontinuity boundaries all other maps have either square root or 3 2 type singularities

bifurcations of piecewise smooth flows perspectives methodologies and open problems abstract in this paper the theory of bifurcations in piecewise smooth flows is critically surveyed the focus is on results that hold in arbitrarily but finitely many dimensions highlighting significant areas where a detailed understanding is presently lacking the clearest results to date concern equilibria undergoing bifurcations at switching boundaries and limit cycles undergoing grazing and sliding bifurcations after discussing fundamental concepts such as topological equivalence of two piecewise smooth systems discontinuity induced bifurcations are defined for equilibria and limit cycles conditions for equilibria to exist in n dimensions are given followed by the conditions under which they generically undergo codimension one bifurcations the extent of knowledge of their unfoldings is also summarized codimension one bifurcations of limit cycles and boundary intersection crossing are described together with techniques for their classification codimension two bifurcations are discussed with suggestions for further study

compact formulations of the steiner traveling salesman problem and related problems the steiner traveling salesman problem stsp is a variant of the tsp that is particularly suitable when routing on real life road networks the standard integer programming formulations of both the tsp and stsp have an exponential number of constraints on the other hand several compact formulations of the tsp i e formulations of polynomial size are known in this paper we adapt some of them to the stsp and compare them both theoretically and computationally it turns out that just by putting the best of the formulations into the cplex branch and bound solver one can solve instances with over 200 nodes we also briefly discuss the adaptation of our formulations to some related problems

a tight linearization and an algorithm for zero one quadratic programming problems this paper is concerned with the solution of linearly constrained zero one quadratic programming problems problems of this kind arise in numerous economic location decision and strategic planning situations including capital budgeting facility location quadratic assignment media selection and dynamic set covering a new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods an implicit enumeration algorithm which uses lagrangian relaxation benders cutting planes and local explorations is designed to exploit the strength of this linearization computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm

exploring variants of 2 opt and 3 opt for the general routing problem the general routing problem grp is the problem of finding a minimum length tour visiting a number of specified vertices and edges in an undirected graph in this paper we describe how the well known 2 opt and 3 opt local search procedures for node routing problems can be adapted to solve arc and general routing problems successfully two forms of the 2 opt and 3 opt approaches are applied to the grp the first version is similar to the conventional approach for the traveling salesman problem the second version includes a dynamic programming procedure and explores a larger neighborhood at the expense of higher running times extensive computational tests including ones on larger instances than previously reported in the arc routing literature are performed with variants of both algorithms in combination with the guided local search metaheuristic and the mechanisms of marking and neighbor lists the procedures systematically detect optimal or high quality solutions within limited computation time

positive periodic solutions for a system of anisotropic parabolic equations abstract we consider a system of two degenerate parabolic equations with nonlocal terms and dirichlet boundary conditions more precisely the degeneracy in each equation of the system is of the type r x laplacian where r x is a function depending on x where is a bounded smooth domain of r n the system models the diffusion and the interaction between two different biological species sharing the same territory the paper provides conditions on the parameters of the problem that guarantee the coexistence of a t periodic non negative solution u v with both non trivial u v

periodic solutions of evolutionp laplacian equations with nonlinear sources abstract in this paper we are concerned with the time periodic solutions to the evolution p laplacian equations of the form formula and formula with the dirichlet boundary value condition where p 2 0 is a bounded convex domain in r n with smooth boundary m x t is continuous on r periodic in t with period and positive in r the existence of nontrivial nonnegative periodic solutions is established provided that n 1 and p 1 p 1 p n