weak focus limit cycles and bifurcations for bounded quadratic systems abstract we study the bounded quadratic systems with either two weak foci or a weak focus of order 2 from the first case we obtain 1 1 configuration of limit cycles and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2 also we unfold the bounded quadratic system with a center and the maximum number of limit cycles in this case is two

a numerical scheme for a singular control problem investment consumption under proportional transaction costs abstract this paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs we consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption the problem is mainly governed by a time dependent hamilton jacobi bellman equation with gradient constraints we propose a numerical method which is composed of monte carlo simulation to take advantage of the high dimensional properties and finite difference method to approximate the gradients of the value function numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in dai et al 2009 and chen and dai 2013

global bifurcation of limit cycles in a family of polynomial systems in this paper we study the number of limit cycles in a family of polynomial systems using bifurcation methods and melnikov functions we obtain the maximal number of limit cycles in global bifurcation

rebalancing with linear and quadratic costs we consider a financial market consisting of one safe and one risky asset which offer constant investment opportunities taking into account both proportional transaction costs and linear price impact we derive optimal rebalancing policies for representative investors with constant relative risk aversion and a long horizon

bounded polynomial vector fields in r2 and rn abstract we characterize the bounded polynomial vector fields in r 2 additionally we provide a necessary condition but not sufficient which must be satisfied by bounded polynomial vector fields in r n

european option pricing with transaction costs the authors consider the problem of pricing european options in a market model similar to the black scholes one except that proportional transaction charges are levied on all sales and purchases o

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

a cross decomposition algorithm for capacitated facility location we present an implementation of the cross decomposition method to solve the capacitated facility location problem the method unifies benders decomposition and lagrangean relaxation into a single framework that involves successive solutions to a transportation problem and a simple plant location problem computational tests on problems with as many as 100 candidate facility locations and 200 customers provide evidence that the approach is superior to several other methods the specialized cross decomposition program is on a common set of test problems about ten times faster than several recently developed programs we also present a simple algorithm to strengthen benders cuts

homoclinic loop and multiple limit cycle bifurcation surfaces this paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces z and multiplicity two limit cycle bifurcation surfaces f for planar systems depending on two or more parameters it determines the side of or f on which limit cycles occur and it shows that if x and intersect then typically they do so at a flat contact

a comparison of heuristics and relaxations for the capacitated plant location problem abstract approaches proposed in the literature for the capacitated plant location problem are compared the comparison is based on new theoretical and computational results the main emphasis is on relaxations in particular dominance relations among the various relaxations found in the literature are identified in the computational study the relaxations are compared as a function of various characteristics of the test problems several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics both in computing time and in the quality of the solutions found

computation of lyapunov values for two planar polynomial differential systems the complex formulas of computing the lyapunov values are investigated for two planar polynomial systems with a pair of pure imaginary eigenvalues the relations between the two systems and their standard forms are discussed and the transformation formulas are given the corresponding complex formulas of computing the lyapunov values are obtained the results obtained here can be applied to judging the type of singular points and the number of limit cycles in planar polynomial systems

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

on an investment consumption model with transaction costs this paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log normal diffusions we suppose that transactions between the assets incur a cost proportional to the size of the transaction the problem is to maximize the total utility of consumption dynamic programming leads to a variational inequality for the value function existence and uniqueness of a viscosity solution are proved the variational inequality is solved by using a numerical algorithm based on policies iterations and multigrid methods numerical results are displayed for n 1 and n 2

an algorithm for solving large capacitated warehouse location problems abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon lagrangean relaxation of a mixed integer formulation of the problem feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the lagrangean relaxation by incorporating the lower bound and the reduction tests into a tree search procedure we are able to solve problems involving up to 500 potential warehouse locations and 1000 customers

using central prices in the decomposition of linear programs abstract this paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical dantzig wolfe algorithm this approach exploits a the relationship between the master program and the minimization of a convex piecewise linear function and b a recently proposed cutting plane algorithm for convex nondifferentiable optimization the new algorithm seeks to achieve deep cuts by generating them at or near to the analytic centre of a set of localization containing the solution therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence this algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems its implementation as a decomposition algorithm for large scale structured linear program seems to be a promising alternative to the standard dantzig wolfe approach which suffers from very slow convergence in many practical problems the new approach differs from the dantzig wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem but rather central prices i e analytic centres which balance them all this property seems to explain the good behaviour of the new algorithm

benders decomposition without separability a computational study for capacitated facility location problems abstract benders is one of the most famous decomposition tools for mathematical programming and it is the method of choice e g in mixed integer stochastic programming its hallmark is the capability of decomposing certain types of models into smaller subproblems each of which can be solved individually to produce local information notably cutting planes to be exploited by a centralized master problem as its name suggests the power of the technique comes essentially from the decomposition effect i e the separability of the problem into a master problem and several smaller subproblems in this paper we address the question of whether the benders approach can be useful even without separability of the subproblem i e when its application yields a single subproblem of the same size as the original problem in particular we focus on the capacitated facility location problem in two variants the classical linear case and a congested case where the objective function contains convex but non separable quadratic terms we show how to embed the benders approach within a modern branch and cut mixed integer programming solver addressing explicitly all the ingredients that are instrumental for its success in particular we discuss some computational aspects that are related to the negative effects derived from the lack of separability extensive computational results on various classes of instances from the literature are reported with a comparison with the state of the art exact and heuristic algorithms the outcome is that a clever but simple implementation of the benders approach can be very effective even without separability as its performance is comparable and sometimes even better than that of the most effective and sophisticated algorithms proposed in the previous literature

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

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

extensions to a lagrangean relaxation approach for the capacitated warehouse location problem abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon the lagrangean relaxation of a mixed integer formulation of the problem where we use subgradient optimisation in an attempt to maximise this lower bound problem reduction tests based upon this lower bound and the original problem are given incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers

a spectral method for an optimal investment problem with transaction costs under potential utility abstract this paper concerns the numerical solution of the finite horizon optimal investment problem with transaction costs under potential utility the problem is initially posed in terms of an evolutive hjb equation with gradient constraints in day and yi 2009 the problem is reformulated as a non linear parabolic double obstacle problem posed in one spatial variable and defined in an unbounded domain where several explicit properties and formulae are obtained the restatement of the problem in polar coordinates allows to pose the problem in one spatial variable in a finite domain avoiding some of the technical difficulties of the numerical solution of the previous statement of the problem if high precision is required the spectral numerical method proposed becomes more efficient than simpler methods as finite differences for example

lot size models with back logging strong reformulations and cutting planes we examine mixed integer programming reformulations of the uncapacitated lot sizing problem with backlogging first we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables leading to a known plant location reformulation and a shortest path reformulation each of these reformulations is strong in the sense that its linear programming relaxation solves the lot sizing problem secondly we attempt to treat the problem in the space of the original variables we give an implicit description of the convex hull of solutions and show how the problem of finding a violated cutting plane can be solved as a linear program we also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm the efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi item capacitated lot sizing problems with backlogging near optimal solutions have been found to problems with 8 periods and up to 100 times

upper and lower bounds for the single source capacitated location problem abstract the single source capacitated location problem is considered given a set of potential locations and the plant capacities it must be decided where and how many plants must be open and which clients must be assigned to each open plant a lagrangean relaxation is used to obtain lower bounds for this problem upper bounds are given by lagrangean heuristics followed by search methods and by one tabu search metaheuristic computational experiments on different sets of problems are presented

parabolic variational inequality with parameter and gradient constraints abstract this paper concerns a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints the method is to transform a two dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies moreover we analyze the behaviors of the free boundary surfaces the main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

a fully nonlinear free boundary problem for minimizing the ruin probability abstract in this paper we consider the problem of how to run an insurance company to minimize the ruin probability in a finite time horizon the value function v with the corresponding optimal reinsure strategy a is the solution of the following fully nonlinear equation v t 1 2 0 2 v x x v x inf 0 a 1 1 2 2 a 2 v x x a v x 0 under initial boundary condition v 0 t 1 v x 0 0 the main effort is to analyze the properties of the solution and the free boundary to show the optimal decision for the company

on lp viscosity solutions of bilateral obstacle problems with unbounded ingredients the global equi continuity estimate on l p viscosity solutions of bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous the existence of l p viscosity solutions is established via an approximation of given data the local holder continuity estimate on the first derivative of l p viscosity solutions is shown when the obstacles belong to c 1 beta and p n

quadratic models for generic local 3 parameter bifurcations on the plane the first chapter deals with singularities occurring in quadratic planar vector fields we make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated or hamiltonian or integrable or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers in the second chapter we provide quadratic models for all the known versal k parameter unfoldings with k 1 2 3 except for the nilpotent focus which cannot occur as a quadratic system we finally show that a certain type of elliptic points of codimension 4 does not have a quadratic versal unfolding introduction how complicated can local bifurcations be in quadratic systems or in other terms which phenomena can be described by germs of families of quadratic vector fields on the plane we do not give a complete answer to this question but we essentially treat two exemplary aspects the first one concerns the singularities themselves emphasizing their codimension as a quadratic system and as a general system the second one concerns the relation between the quadratic unfolding of the singularity and its versal unfolding as a general system the complete treatment of the first problem is described in chapter 1 we make a distinction between singularities having a finite codimension or having infinite codimension as a general system the singularities of the first kind are listed completely while those of the second kind are only checked to be nonisolated or isolated but hamiltonian or integrable or having an axis of symmetry after a linear coordinate change or approachable by centers these cases are of course not mutually distinct for the singularities of the first kind the quadratic codimension reveals to be the same as the general one it can reach at most 5 and this only happens if the vector field is homogeneous different kinds of singularities of codimension 4 however occur received by the editors december 1 1989 1980 mathematics subject classification 1985 revision primary 58fxx 34cxx

multiple limit cycle bifurcation surfaces and global families of multiple limit cycles abstract sufficient conditions are given for the local exitence of multiplicity m limit cycle bifurcation surfaces cm of planar analytic systems depending on n parameters with n m 2 in the generic case the surfaces c2 c3 and c4 are the familiar saddle node cusp and swallow tail bifurcation surfaces respectively the author s termination principle for global one parameter families of simple limit cycles of relatively prime planar analytic systems is generalized to a termination principle for global one parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of hopf bifurcation surfaces and or homoclinic or heteroclinic loop bifurcation surfaces

an irreversible investment problem with maintenance expenditure on a finite horizon free boundary analysis abstract this paper concerns a continuous time finite horizon optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty we assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity the objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint the problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies respectively we investigate behaviors of free boundaries study regularities of the value function and give optimal investment and maintenance policies as we know this is a first integral result for an investment maintenance problem with a finite time horizon due to use of partial differential equation pde technique

optimal trend following trading rules we develop an optimal trend following trading rule in a bull bear switching market where the drift of the stock price switches between two parameters corresponding to an uptrend bull market and a downtrend bear market according to an unobservable markov chain we consider a flnite horizon investment problem and aim to maximize the expected return of the terminal wealth we start by restricting to allowing at and long positions only and describe the trading decisions using a sequence of stopping times indicating the time of entering and exiting long positions assuming trading all available funds we show that the optimal trading strategy is a trend following system characterized by the conditional probability in the uptrend crossing two threshold curves the thresholds can be obtained by solving the associated hjb equations in addition we examine trading strategies with short selling in terms of an approximation simulations and empirical experiments are conducted and reported

on the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs we study the uniqueness of viscosity solutions of a hamilton jacobi bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

a branch and price algorithm for the capacitated facility location problem the capacitated facility location problem cflp is a well known combinatorial optimization problem with applications in distribution and production planning it consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs a number of solution approaches based on lagrangean relaxation and subgradient optimization has been proposed for this problem subgradient optimization does not provide a primal fractional optimal solution to the corresponding master problem however in order to compute optimal solutions to large or difficult problem instances by means of a branch and bound procedure information about such a primal fractional solution can be advantageous in this paper a stabilized column generation method is therefore employed in order to solve a corresponding master problem exactly the column generation procedure is then employed within a branch and price algorithm for computing optimal solutions to the cflp computational results are reported for a set of larger and difficult problem instances

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

an efficient tabu search procedure for the p median problem abstract in this paper we present a new solution heuristic for the p median problem the algorithm is based on tabu search principles and uses short term and long term memory as well as strategic oscillation and random tabu list sizes our proposed procedure is compared with two other move heuristics a well known interchange heuristic and a recent hybrid heuristic in computational tests on networks ranging in size up to 500 nodes the new heuristic is found to be superior with respect to the quality of solutions produced

separatrix and limit cycles of quadratic systems and dulac s theorem separatrix cycles for a planar quadratic vector field are studied the results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles

upper bounds for the number of zeroes for some abelian integrals consider the vector field x yg x y y xg x y where the set of critical points g x y 0 is formed by k straight lines not passing through the origin and parallel to one or two orthogonal directions we perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of k and n our approach is based on the explicit computation of the abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions when we apply our results for k 4 we recover or improve some results obtained in several previous works

the number of limit cycles for a family of polynomial systems in this paper the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods with the help of a computer algebra system e g maple 7 0 we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems 2 1 and 2 2 is 5n 5 1 1 n 2 for c 0 and n for c 0

numerical solution of an optimal investment problem with proportional transaction costs this paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite horizon optimal investment problem with proportional transaction costs the problem is initially posed in terms of an evolutive hjb equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable the proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain a characteristics method for time discretization to deal with the large gradients of the solution a newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle free boundary feature moreover piecewise linear lagrange finite elements for spatial discretization are considered numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in dai and yi 2009 6

a problem of singular stochastic control with optimal stopping in finite horizon in this paper we apply pde methods to solve a problem about minimizing the cost of a firm for operating a project which is a simple model of combined singular stochastic control and optimal stopping in finite horizon the value function v x t to the problem is governed by a time dependent hjb variational inequality with both gradient constraint and function constraint which leads to a time dependent stopping boundary and a moving boundary we will prove that two free boundaries are strictly monotonic and infinitely differentiable v partial xv partial tv are continuous and partial xx v is locally bounded based on these results we construct optimal control and optimal stopping

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

finite horizon optimal investment with transaction costs a parabolic double obstacle problem abstract this paper concerns optimal investment problem of a crra investor who faces proportional transaction costs and finite time horizon from the angle of stochastic control it is a singular control problem whose value function is governed by a time dependent hjb equation with gradient constraints we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies this enables us to make use of the well developed theory of obstacle problem to attack the problem the c 2 1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized

widely separated frequencies in coupled oscillators with energy preserving quadratic nonlinearity in this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics we make two assumptions for our system the first assumption is that the frequencies of the characteristic oscillations are widely separated and the second is that the nonlinear part of the vector field preserves the distance to the origin using the first assumption we prove that the reduced normal form of our system has an invariant manifold which exists for all values of the parameters this invariant manifold cannot be perturbed away by including higher order terms in the normal form using the second assumption we view the normal form as an energy preserving three dimensional system which is linearly perturbed restricting ourselves to a small perturbation the flow of the energy preserving system is used to study the flow in general we present a complete study of the flow of the energy preserving system and its bifurcations using these results we return to the dissipative system and provide the condition for having a hopf bifurcation of one of the two equilibria of the perturbed system we also numerically follow the periodic solution created via the hopf bifurcation and find a sequence of period doubling and fold bifurcations also a torus or neimark sacker bifurcation

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

the cyclicity of triangles and segments in quadratic systems abstract the paper deals with quadratic systems having separatrix contour in the form of a triangle or a conic segment the limit cycles appearing near this configuration after quadratic perturbation is studied the maximal number of limit cycles is 3 for the bounded triangle and 2 for the unbounded triangle and for the bounded segment

approximate solutions to large scale capacitated facility location problems this article presents a heuristic algorithm for solving large scale capacitated facility location problems the algorithm consists of solving a single source facility location problem and then solving a minimum cost network problem the single source problem is solved by applying an iterative procedure in which lower bounds on the optimal objective value are generated through lagrangian relaxation techniques upper bounds on the optimal objective value are obtained by modifying the lagrangian solution to restore feasibility the iterative procedure is terminated when the difference between the two bounds is within a prespecified tolerance further improvement of the objective value is attained by subjecting the single source solution to a network optimization code the heuristic algorithm has been tested on over 300 problems derived in one form or another from the kuehn hamburger test problems the testing has also been undertaken on the problems of van roy which were generated using netgen computational results indicate that our approach can be effective in producing near optimal solutions for problems with as many as 1000 customers and 100 potential facility locations most problems converged within 3 percent of the optimal value requiring only a modest amount of computation on an ibm 3090

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

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

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given

capacitated facility location valid inequalities and facets we examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem in particular we derive necessary and sufficient conditions for a family of effective capacity inequalities to be facet defining and further results on a more general family called submodular inequalities