approaching 3 2 for the s t path tsp we show that there is a polynomial time algorithm with approximation guarantee 3 2 e for the s t path tsp for any fixed e 0 it is well known that wolsey s analysis of christofide algorithm also works for the s t path tsp with its natural lp relaxation except for the narrow cuts in which the lp solution has a value less than two a fixed optimum tour has either a single edge in a narrow cut then call the edge and the cut lonely or at least three then call the cut busy our algorithm guesses by dynamic programming lonely cuts and edges then we partition the instance into smaller instances and strengthen the lp requiring a value of at least three for busy cuts by setting up a k stage recursive dynamic program we can compute a spanning tree v s and an lp solution y such that o 2 k y is in the t join polyhedron where t is the set of vertices whose degree in s has the wrong parity

2 matchings the traveling salesman problem and the subtour lp a proof of the boyd carr conjecture determining the precise integrality gap for the subtour linear programming lp relaxation of the traveling salesman problem is a significant open question with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality boyd and carr boyd s carr r 2011 finding low cost tsp and 2 matching solutions using certain half integer subtour vertices discrete optim 8 525 539 prior version accessed june 27 2011 http www site uottawa ca sylvia recentpapers halftri pdf observe that we do not even know the worst case upper bound on the ratio of the optimal 2 matching to the subtour lp they conjecture the ratio is at most 10 9 in this paper we prove the boyd carr conjecture in the case that the support of a fractional 2 matching has no cut edge we can further prove that an optimal 2 matching has cost at most 10 9 times the cost of the fractional 2 matching

algorithms for a multi level network optimization problem in this paper we work on a multi level network optimization problem that integrates into the same model important aspects of i discrete facility location ii topological network design and iii network dimensioning potential applications for the model are discussed stressing its growing importance the multi level network optimization problem treated is defined and a mathematical programming formulation is presented we make use of a branch and bound algorithm based on lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem we conduct a set of computational experiments that indicate the quality of the proposed approach

technical note on traveling salesman games with asymmetric costs we consider cooperative traveling salesman games with nonnegative asymmetric costs satisfying the triangle inequality we construct a stable cost allocation with budget balance guarantee equal to the held karp integrality gap for the asymmetric traveling salesman problem using the parsimonious property and a previously unknown connection to linear production games we also show that our techniques extend to larger classes of network design games we then provide a simple example showing that our cost allocation does not necessarily achieve the best possible budget balance guarantee even among cost allocations stable for the game defined by the held karp relaxation and discuss its implications on future work on traveling salesman games

on the stabilization of the homogeneous bilinear systems abstract the problem of stabilizing the single input homogeneous bilinear systems is considered a special class of these systems is considered the matrix of the linear i e control independent term of the right hand side is semitable while the matrix of the bilinear term is skew symmetric two types of stabilization are investigated constant feedback global asymptotic stabilization and practical stabilization by a family of the linear feedbacks it will be shown for the planar case that the above system is always practically stabilizable by a family of the linear feedbacks for n 3 the solution of both stabilization problems depends on the mutual position of the expanding direction of the linear part and the direction of the infinitesimal rotation defined by the bilinear term

an efficient approximation algorithm for the survivable network design problem the survivable network design problem sndp is to construct a minimum cost subgraph satisfying certain given edge connectivity requirements the first polynomial time approximation algorithm was given by williamson et al combinatorica 15 1995 435 454 this paper gives an improved version that is more efficient consider a graph ofn vertices and connectivity requirements that are at mostk both algorithms find a solution that is within a factor 2k 1 of optimal fork 2 and a factor 2 of optimal fork 1 our algorithm improves the time from o k3n4 to o k 2 n 2 kn 2 sqrt log log n our algorithm shares features with those of williamson et al combinatorica 15 1995 435 454 but also differs from it at a high level necessitating a different analysis of correctness and accuracy our analysis is based on a combinatorial characterization of the redundant edges several other ideas are introduced to gain efficiency these include a generalization of padberg and rao s characterization of minimum odd cuts use of a representation of all minimum s t cuts in a network and a new priority queue system the latter also improves the efficiency of the approximation algorithm of goemans and williamson siam journal on computing 24 1995 296 317 for constrained forest problems such as minimum weight matching generalized steiner trees and others 1998 the mathematical programming society inc published by elsevier science b v

enhanced formulations and branch and cut for the two level network design problem with transition facilities abstract we consider a new combinatorial optimization problem that combines network design and facility location aspects given a graph with two types of customers and two technologies that can be installed on the edges the objective is to find a minimum cost subtree connecting all customers while the primary customers are served by a primary subtree that is embedded into the secondary subtree in addition besides fixed link installation costs facility opening costs associated to each node where primary and secondary subtree connect have to be paid the problem is called the two level network design problem with transition facilities tlndf we first model the problem on an extended graph where an additional set of arcs corresponds to the installation of node facilities and propose a cut set based model for the tlndf that is defined on this extended graph we present several theoretical results relating families of cut set inequalities on the extended graph with subfamilies of cut set inequalities on the original graph we then show how a standard multi commodity flow model defined on the original graph can be strengthened using disaggregation by technology we prove that the disaggregated compact formulation on the original graph provides the same lower bound as the cut set formulation on the extended graph we develop a branch and cut algorithm for solving the tlndf the performance of this algorithm is improved by separating subfamilies of cut set inequalities on the original graph our computational study confirms the efficiency and applicability of the new approach

a note on the prize collecting traveling salesman problem we study the version of the prize collecting traveling salesman problem where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible we present an approximation algorithm with constant bound the algorithm is based on christofides algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers feasible for the original problem

multicommodity flow models for spanning trees with hop constraints abstract in this paper we compare the linear programming relaxations of undirected and directed multicommodity flow formulations for the terminal layout problem with hop constraints hop constraints limit the number of hops links between the computer center and any terminal in the network these constraints model delay constraints since a smaller number of hops decreases the maximum delay transmission time in the network they also model reliability constraints because with a smaller number of hops there is a lower route loss probability hop constraints are easily modelled with the variables involved in multicommodity flow formulations we give some empirical evidence showing that the linear programming relaxation of such formulations give sharp lower bounds for this hop constrained network design problem on the other hand these formulations lead to very large linear programming models therefore for bounding purposes we also derive several lagrangean based procedures from a directed multicommodity flow formulation and present some computational results taken from a set of instances with up to 40 nodes

input to state stability of exponentially stabilized semilinear control systems with inhomogeneous perturbations in this paper we investigate the robustness of state feedback stabilized semilinear systems subject to inhomogeneous perturbations in terms of input to state stability we consider a general class of exponentially stabilizing feedback controls which covers sampled discrete feedbacks and discontinuous mappings as well as classical feedbacks and derive a necessary and sufficient condition for the corresponding closed loop systems to be input to state stable with exponential decay and linear dependence on the perturbation this condition is easy to check and admits a precise estimate for the constants involved in the input to state stability formulation applying this result to an optimal control based discrete feedback yields an equivalence between open loop asymptotic null controllability and robust input to state state feedback stabilizability

a dynamic programming algorithm for the local access telecommunication network expansion problem abstract in this paper we consider the local access telecommunication network expansion problem in which growing demand can be satisfied by expanding cable capacities and or installing concentrators in the network the problem is known to be np hard we prove that the problem is weakly np hard and present a pseudo polynomial dynamic programming algorithm for the problem with time complexity o nb 2 and storage requirements o nb where n refers to the size of the network and b to an upper bound on concentrator capacity the cost structure in the network is assumed to be decomposable but may be non convex non concave and node and edge dependent otherwise this allows for incorporation of many aspects occurring in practical planning problems computational results indicate that the algorithm is very efficient and can solve medium to large scale problems to optimality within fractions of seconds to minutes

designing reliable tree networks with two cable technologies in this paper we introduce a minimal spanning tree problem with generalized hop constraints and primary connectivity constraints the problem is concerned with reliability requirements in a centralized network design problem where two different cable technologies are available the problem is shown to be np hard and as a consequence we derive lower bounding and upper bounding schemes for the problem we formulate the problem as a directed multicommodity flow model due to the size of the corresponding lp model we use lagrangean relaxation together with subgradient optimization to derive lower bounds a lagrangean heuristic is developed to construct feasible solutions the theoretically best lower bound associated to the lagrangean scheme quite often improves on the value of the corresponding lp bound since the relaxed problem does not satisfy the integrality property in fact using the heuristic these bounds can be proved to be even optimal for many of the cases tested these results are taken from a set of complete graphs with up to 40 nodes we also examine a few variations of the main model in particular we shall discuss several different ways of modeling the primary connectivity constraints one outcome of our discussion is that we shall derive an extended and compact representation of the convex hull of directed rooted subtrees when the underlying graph is series parallel

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

reducing the hierarchical network design problem abstract the hierarchical network design problem hndp can be seen as a minimum spanning tree problem involving two weight functions it is shown that the hndp is a special case of the directed steiner tree problem theorems are given to obtain optimal arcs that are used in a lagrangean relaxation of the linear programming model reduction tests reducing the size of the problem graph and or eliminating variables from this program are specialized and developed for the hndp the presented techniques are illustrated on example problems from the literature and computational results for the reduction tests are given

a dual ascent procedure for large scale uncapacitated network design the fixed charge network design problem arises in a variety of problem contexts including transportation communication and production scheduling we develop a family of dual ascent algorithms for this problem this approach generalizes known ascent procedures for solving shortest path plant location steiner network and directed spanning tree problems our computational results for several classes of test problems with up to 500 integer and 1 98 million continuous variables and constraints show that the dual ascent procedure and an associated drop add heuristic generate solutions that in almost all cases are guaranteed to be within 1 to 4 of optimality moreover the procedure requires no more than 150 seconds on an ibm 3083 computer the test problems correspond to dense and sparse networks including some models that arise in freight transport

bounds on self avoiding walks on directed square lattices the manhattan lattice is the covering lattice of the l lattice and the author has studied self avoiding walks on this pair of lattices upper and lower bounds on the connective constant mu have been obtained as have series analysis estimates

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

a partition based relaxation for steiner trees the steiner tree problem is a classical np hard optimization problem with a wide range of practical applications in an instance of this problem we are given an undirected graph g v e a set of terminals r subseteq v and non negative costs c e for all edges e in e any tree that contains all terminals is called a steiner tree the goal is to find a minimum cost steiner tree the vertices v backslash r are called steiner vertices the best approximation algorithm known for the steiner tree problem is a greedy algorithm due to robins and zelikovsky siam j discrete math 19 1 122 134 2005 it achieves a performance guarantee of 1 frac ln 3 2 approx 1 55 the best known linear programming lp based algorithm on the other hand is due to goemans and bertsimas math program 60 145 166 1993 and achieves an approximation ratio of 2 2 r in this paper we establish a link between greedy and lp based approaches by showing that robins and zelikovsky s algorithm can be viewed as an iterated primal dual algorithm with respect to a novel lp relaxation the lp used in the first iteration is stronger than the well known bidirected cut relaxation an instance is b quasi bipartite if each connected component of g backslash r has at most b vertices we show that robins and zelikovsky s algorithm has an approximation ratio better than 1 frac ln 3 2 for such instances and we prove that the integrality gap of our lp is between frac 8 7 and frac 2b 1 b 1

heuristics lps and trees on trees network design analyses we study a class of models known as overlay optimization problems composed of base and overlay subproblems linked by the requirement that the overlay solution be contained in the base solution in some telecommunication settings a feasible base solution is a spanning tree and the overlay solution is an embedded steiner tree or path for the general overlay optimization problem we describe a composite heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and the overlay subproblems and establish worst case performance guarantees as a function of the worst case bounds for the subproblems for the composite heuristic as well as an lp relaxation of the model for certain special cases both the heuristic and the lp relaxation have a worst case bound of 4 3 we extend this analysis to multiple overlays on the same base solution producing the first known worst case bounds approximately proportional to the square root of the number of commodities for the uncapacitated multicommodity network design problem in a companion paper we develop heuristic performance guarantees for various new multi tier survivable network design models that incorporate both multiple facility types and differential node connectivity levels

a dual based algorithm for multi level network design given an undirected network with l possible facility types for each edge and a partition of the nodes into l levels or grades the multi level network design mlnd problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade this problem generalizes the well known steiner network problem and the hierarchical network design problem and has applications in telecommunication transportation and electric power distribution network design in a companion paper we studied alternative model formulations for a two level version of this problem and analyzed the worst case performance of several heuristics based on steiner network and spanning tree solutions this paper develops a dual based algorithm for the mlnd problem the method first performs problem preprocessing to fix certain design variables and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value we report extensive computational results on large random two level test problems containing up to 500 nodes and 5 000 edges with varying cost structures the integer programming formulation of the largest of these problems has 20 000 integer variables and over 5 million constraints our tests indicate that the dual based algorithm is very effective producing solutions that are within 0 9 of optimality

universality of surface exponents of self avoiding walks on a manhattan lattice the authors use phenomenological renormalization group techniques to study the bulk and surface properties of self avoiding walks on a manhattan lattice they find that as is the case for the bulk exponents the underlying bond directionality does not change the universality class of the surface exponents relative to undirected lattices they also obtain an estimate for the position of the binding transition

critical behavior at the theta point of self avoiding walks on a manhattan lattice the authors study the collapse of self attracting self avoiding walks on a manhattan lattice by means of phenomenological renormalization group techniques their results support a recent prediction of nonuniversality for the entropic exponent gamma the surface exponents are in the same universality class as the theta transition on undirected lattices thus differing from those of the theta point they use a two parameter renormalization group to obtain estimates of the crossover exponent

minimum weight two connected spanning networks we consider the problem of constructing a minimum weight two connected network spanning all the points in a setv we assume a symmetric nonnegative distance functiond defined onv v which satisfies the triangle inequality we obtain a structural characterization of optimal solutions specifically there exists an optimal two connected solution whose vertices all have degree 2 or 3 and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components these are the strongest possible conditions on the structure of an optimal solution since we also show thatany two connected graph satisfying these conditions is theunique optimal solution for a particular choice of canonical distances satisfying the triangle inequality we use these properties to show that the weight of an optimal traveling salesman cycle is at most 4 3 times the weight of an optimal two connected solution examples are provided which approach this bound arbitrarily closely in addition we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points

the lyapunov spectrum of families of time varying matrices for l families of time varying matrices centered at an unperturbed matrix the lyapunov spectrum contains the floquet spectrum obtained by considering periodically varying piecewise constant matrices on the other hand it is contained in the morse spectrum of an associated flow on a vector bundle a closer analysis of the floquet spectrum based on geometric control theory in projective space and in particular on control sets is performed introducing a real parameter p 0 which indicates the size of the li perturbation we study when the floquet spectrum the morse spectrum and hence the lyapunov spectrum all coincide this holds if an inner pair condition is satisfied for all up to at most countably many p values

a branch and cut algorithm for the hierarchical network design problem the hierarchical network design problem consists of locating a minimum cost bi level network on a graph the higher level sub network is a path visiting two or more nodes the lower level sub network is a forest connecting the remaining nodes to the path we optimally solve the problem using an ad hoc branch and cut procedure relaxed versions of a base model are solved using an optimization package and if binary variables have fractional values or if some of the relaxed constraints are violated in the solution cutting planes are added once no more cuts can be added branch and bound is used the method for finding valid cutting planes is presented finally we use different available test instances to compare the procedure with the best known published optimal procedure with good results in none of the instances we needed to apply branch and bound but only the cutting planes

probabilistic analysis of the held and karp lower bound for the euclidean traveling salesman problem we analyze probabilistically the classical held karp lower bound derived from the 1 tree relaxation for the euclidean traveling salesman problem etsp we prove that if n points are identically and independently distributed according to a distribution with bounded support and absolutely continuous part fxdx over the d cube the held karp lower bound on these n points is almost surely asymptotic to hkdn d 1 d fx d 1 d dx where hkd is a constant independent of n the result suggests a probabilistic explanation of the observation that the lower bound is very close to the length of the optimal tour in practice since the etsp is almost surely asymptotic to tspdn d 1 d fx d 1 d dx the techniques we use exploit the polyhedral description of the held karp lower bound and the theory of subadditive euclidean functionals

modeling and heuristic worst case performance analysis of the two level network design problem this paper studies a multi facility network synthesis problem called the two level network design tlnd problem that arises in the topological design of hierarchical communication transportation and electric power distribution networks we are given an undirected network containing two types of nodes primary and secondary and fixed costs for installing either a primary or a secondary facility on each edge primary nodes require higher grade interconnections than secondary nodes using the more expensive primary facilities the tlnd problem seeks a minimum cost connected design that spans all the nodes and connects primary nodes via edges containing primary facilities the design can use either primary or secondary edges to connect the secondary nodes the tlnd problem generalizes the well known steiner network problem and the hierarchical network design problem in this paper we study the relationship between alternative model formulations for this problem e g directed and undirected models an

design of survivable networks with vulnerability constraints abstract we consider the network design problem with vulnerability constraints ndpvc which simultaneously addresses resilience against failures network survivability and bounds on the lengths of each communication path hop constraints solutions to the ndpvc are subgraphs containing a path of length at most h st for each commodity s t and a path of length at most h s t between s and t after at most k 1 edge failures we first show that a related and well known problem from the literature the hop constrained survivable network design problem k hsndp that addresses the same two measures produces solutions that are too conservative in the sense that they might be too expensive in practice or may even fail to provide feasible solutions we also explain that the reason for this difference is that mengerian like theorems not hold in general when considering hop constraints three graph theoretical characterizations of feasible solutions to the ndpvc are derived and used to propose integer linear programming formulations in a computational study we compare these alternatives with respect to the lower bounds obtained from the corresponding linear programming relaxations and their capability of solving instances to proven optimality in addition we show that in many cases the solutions produced by solving the ndpvc are cheaper than those obtained by the related khsndp

numerical stabilization of bilinear control systems extremal lyapunov exponents for bilinear control systems with constrained control values are computed numerically by solving discounted optimal control problems based on this computation a numerical algorithm to calculate stabilizing control functions is developed

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

survivable networks linear programming relaxations and the parsimonious property we consider the survivable network design problem the problem of designing at minimum cost a network with edge connectivity requirements as special cases this problem encompasses the steiner tree problem the traveling salesman problem and thek edge connected network design problem we establish a property referred to as the parsimonious property of the linear programming lp relaxation of a classical formulation for the problem the parsimonious property has numerous consequences for example we derive various structural properties of these lp relaxations we present some algorithmic improvements and we perform tight worst case analyses of two heuristics for the survivable network design problem

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

on the held karp relaxation for the asymmetric and symmetric traveling salesman problems a long standing conjecture in combinatorial optimization says that the integrality gap of the famous held karp relaxation of the metric stsp symmetric traveling salesman problem is precisely 4 3 in this paper we show that a slight strengthening of this conjecture implies a tight 4 3 integrality gap for a linear programming relaxation of the metric atsp asymmetric traveling salesman problem our main tools are a new characterization of the integrality gap for linear objective functions over polyhedra and the isolation of hard to round solutions of the relaxations

optimizing splitter and fiber location in a multilevel optical ftth network due to the emergence of bandwidth requiring services telecommunication operators are being compelled to renew their fix access network most of them favoring the fiber to the home ftth technology this paper focuses on the optimization of ftth deployment which is of prime importance due to the economic stakes the key design issue here is locating splitters and routing fi bers in an existing network infrastructure to which is associated a graph with given capacities on the edges no assumption is made on the structure of the graph first we propose a mixed integer formulation for this decision problem then valid inequalities and problem size reduction schemes are presented finally efficiency of solving approaches is assessed through extensive numerical tests performed on orange real life data

minimal and maximal lyapunov exponents of bilinear control systems abstract for a bilinear control system with bounded and unbounded controls x a 0 u 0 x graphics u i a i u 0 x in r d where u0 t r l compact ui t r for i 1 m the extremal exponential growth rates of the solutions x x0 u are analyzed if x0 u lim supt 1 t log x t x0 u then k supu u supx0 0 x0 u and k infu u infx0 0 x0 u are the maximal and minimal respectively lyapunov exponents of the system this paper gives several characterizations of these rates together with the corresponding uniform concepts with respect to the initial value or the control we describe the situations in which k and k and characterize the sets of initial values from which k and k can actually be realized the techniques are applied to high gain stabilizalion and the example of the linear oscillator with parameter controlled restoring force is treated in detail finally we indicate how the results can be used for feedback stabilization of linear systems when the feedback is allowed to be time varying but restricted to certain types of bounded gain matrices

asymptotic controllability and exponential stabilization of nonlinear control systems at singular points we discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points using a discounted optimal control approach we construct discrete feedback laws minimizing the lyapunov exponent of the linearization thus we obtain an equivalence result between uniform exponential controllability and uniform exponential stabilizability by means of a discrete feedback law

input to state dynamical stability and its lyapunov function characterization we present a new variant of the input to state stability iss property which is based on using a one dimensional dynamical system for building the class spl kscr spl lscr function for the decay estimate and for describing the influence of the perturbation we show the relation to the original iss formulation and describe characterizations by means of suitable lyapunov functions as applications we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs

on asymptotically stabilizing feedback control of bilinear systems asymptotically stabilizing feedback controls for a class of bilinear systems are discussed a simple modification which strengthens a result of slemrod 1 is noted and illustrated by example