a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

minimal representation of convex regions defined by analytic functions in this paper we are concerned with characterizing minimal representation of feasible regions defined by both linear and convex analytic constraints we say that a representation is minimal if every other representation has either more analytic nonlinear constraints or has the same number of analytic constraints and at least as many linear constraints we prove necessary and sufficient conditions for the representation to be minimal these are expressed in terms of the redundant constraints pseudo analytic constraints and implicit equality constraints in order to prove the minimal representation theorem we present results on the facets of the convex regions defined by analytic constraints finally we outline the steps of the procedure that could be used to determine a minimal representation

convex constrained programmes with unattained infima abstract we consider a problem of minimization of convex function f x over the convex region r where the objective function and the feasible region have a common direction of recession in cases when one of these directions is not in the constancy space of the objective function then the minimal solution is not achieved even if the function f x is bounded below over the region r many algorithms if applied to this class of programmes do not guarantee convergence to the global infimum our approach to this problem leads to derivation of the equation of the feasible parametrized curve c t such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region r we show that if all functions defining the program are analytic then c t is also an analytic function the equation of the curve can be successfully used to determine the global infimum in particular unboundedness of the convex constrained programmes in cases when the application of classical methods such as the steepest descent method fails to converge to the global infimum

on the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature the paper deals with an effective implementation of some algorithms for the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature here we discuss robust quantitative refinement of the karush kuhn tucker conditions extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components and plug them into the existing algorithms for the solution of the qpqc problems with r linear rate of convergence in the bounds on the spectrum the results are then extended to the problems with separable inequality and linear equality constraints the performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic tresca and coulomb friction discretized by up to one and half million nodal variables

feasibility in reverse convex mixed integer programming abstract in this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints where some or all of the variables are integer in particular we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility cf that is constraints that may affect a feasibility status of the system after some perturbation of the right hand sides furthermore we will investigate properties of the irreducible infeasible sets and infeasibility sets showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system are subsets of the set cf of constraints critical to feasibility

improved bounds on nonblocking 3 stage clos networks we consider a generalization of edge coloring bipartite graphs in which every edge has a weight in 0 1 and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1 for unit weights konig s theorem says that the number of colors needed is exactly the maximum degree for this generalization we show that 2 557 n o n colors are sufficient where n is the maximum total weight adjacent to any vertex improving the previously best bound of 2 833n o 1 due to du et al our analysis is interesting on its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one dimensional bin packing instance which we can prove allows perfect packing this question is motivated by the question of the rearrangeability of 3 stage clos networks in that context the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unit sized bins needed to pack the weights of the incident edges in that setting we are able to improve the bound to 2 5480 n o n also improving a bound of 2 5625n o 1 of du et al we also consider the online version of this problem in which edges have to be colored as soon as they are revealed in this context we can show that 5n colors are enough this contrasts with the best known lower bound of 3n 2 by tsai wang and hwang but improves upon the previous best upper bound of 5 75n obtained by gao and hwang additionally we show several improved bounds for more restricted versions of the problem these online bounds are achieved by simple and easy to implement algorithms inspired by the first fit heuristic for bin packing

a reduced gradient method for quadratic programs with quadratic constraints and lp constrained lp approximation problems abstract a dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l p constrained l p approximation problems the general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function these modifications permit blocks of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero

line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions many successful quasi newton methods for optimization are based on positive definite local quadratic approximations to the objective function that interpolate the values of the gradient at the current and new iterates line search termination criteria used in such quasi newton methods usually possess two important properties first they guarantee the existence of such a local quadratic approximation second under suitable conditions they allow one to prove that the limit of the component of the gradient in the normalized search direction is zero this is usually an intermediate result in proving convergence collinear scaling algorithms proposed initially by davidon in 1980 are natural extensions of quasi newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations and that they interpolate values of both the gradient and the function at the current and new iterates line search termination criteria that guarantee the existence of such a normal conic local approximation which also allow one to prove that the component of the gradient in the normalized search direction tends to zero are not known in this paper we propose such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions

approximation algorithms for single source unsplittable flow in the single source unsplittable flow problem we are given a network g a source vertex s and k commodities with sinks ti and real valued demands rho i 1 leq i leq k we seek to route the demand rho i of each commodity i along a single s ti flow path so that the total flow routed across any edge e is bounded by the edge capacity ue the conceptual difficulty of this np hard problem arises from combining packing constraints due to the existence of capacities with path selection in a graph of arbitrary topology in this paper we give a generic framework which yields approximation algorithms that are simpler than those previously known and achieve significant improvements upon the approximation ratios our framework with appropriate subroutines applies to all optimization versions previously considered and unlike previous work treats in a unified manner directed and undirected graphs we provide extensions of our algorithms which yield the best possible approximation guarantees for restricted sets of demand values and an associated scheduling problem

stochastic packing integer programs with few queries we consider a stochastic variant of the packing type integer linear programming problem which contains random variables in the objective vector we are allowed to reveal each entry of the objective vector by conducting a query and the task is to find a good solution by conducting a small number of queries we propose a general framework of adaptive and non adaptive algorithms for this problem and provide a unified methodology for analyzing the performance of those algorithms we also demonstrate our framework by applying it to a variety of stochastic combinatorial optimization problems such as matching matroid and stable set problems

controlled perturbations for quadratically constrained quadratic programs consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function the duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear however the objective function is not differentiable over the constraint region in this paper we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem based on this relationship we establish a perturbation theory to display that a well controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an optimal dual solution for an arbitrarily small number a simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding optimal primal solution moreover a numerical example is included to illustrate the potential of this controlled perturbation scheme

an optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications we propose a modification of our mpgp algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections possibly with the superrelaxation steplength the error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the hessian matrix in a given positive interval then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded we also show how to use the algorithm for the solution of separable and equality constraints the power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with tresca friction discretized by almost twelve millions nodal variables

the global packing number of a fat tree network data centers play an important role in today s internet development research to find scalable architecture and efficient routing algorithms for data center networks has gained popularity the fat tree architecture which is essentially a folded version of a clos network has proved to be readily implementable and is scalable in this paper we investigate routing on a fat tree network by deriving its global packing number and by presenting explicit algorithms for the construction of optimal load balanced routing solutions consider an optical network that employs wavelength division multiplexing in which every user node sets up a connection with every other user node the global packing number is basically the number of wavelengths required by the network to support such a traffic load under the restriction that each source to destination connection is assigned a wavelength that remains constant in the network in mathematical terms consider a bidirectional simple graph g and let n subseteq v g be a set of nodes a path system mathcal p of g with respect to n consists of n n 1 directed paths one path to connect each of the source destination node pairs in n the global packing number of a path system mathcal p denoted by phi g n mathcal p is the minimum integer k to guarantee the existence of a mapping phi mathcal p to 1 2 ldots k such that phi p neq phi widehat p if p and widehat p have common arc s the global packing number of g n denoted by phi g n is defined to be the minimum phi g n mathcal p among all possible path systems mathcal p in additional to wavelength division optical networks this number also carries significance for networks employing time division multiple access in this paper we compute by explicit route construction the global packing number of text t n n where t n denotes the topology of the n ary fat tree network and n is considered to be the set of all edge switches or the set of all supported hosts we show that the constructed routes are load balanced and require minimal link capacity at all network links

on the convergence of the method of analytic centers when applied to convex quadratic programs this work examines the method of analytic centers of sonnevend when applied to solve generalized convex quadratic programs where also the constraints are given by convex quadratic functions we establish the existence of a two sided ellipsoidal approximation for the set of feasible points around its center and show that a simple zero order algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor ofe with o sqrt m left ln varepsilon right iterations wherem is the number of inequality constraints here each iteration involves the computation of one newton step the bound of o sqrt m left ln varepsilon right newton iterations to guarantee an error reduction by a factor ofe in the objective function is as good as the one currently given forlinear programs however the algorithm considered here is of theoretical interest only full efficiency of the method can only be obtained when accelerating it by some higher order extrapolation scheme see e g the work of jarre sonnevend and stoer

a method for the parametric center problem with a strictly monotone polynomial time algorithm for linear programming given a system of linear inequalities ax b dt and equalities mx g ht where the right hand sides are parametrically deformed over the scalar t the parametric center problem is to trace the parametric family of approximate solutions x t to the center problems p t where p t is the problem maximize i 1m ln bi dit aix subject to ax b dt and mx g ht we present an algorithm for tracing the parametric family of solutions x t over some given range at the kth iterate of the algorithm the point x tk is an approximate solution to the center problem p tk in an appropriate measure of approximation the value of tk is increased to tk 1 and a newton step is used to generate x tk 1 which is an approximate solution to the center problem p tk 1 the sequence of values of t exhibit at least one of the following geometric rates of change tmax tk 1 tmax tk 1 1 128m tk 1 tmin tk tmin 1 1 128m where tmin tmax is the maximum minimum number t such t

extension of karmarkar s algorithm onto convex quadratically constrained quadratic problems convex quadratically constrained quadratic problems are considered it is shown that such problems can be transformed to aconic form the feasible set of the conic form is the intersection of a direct product of standard quadratic cones intersected with a hyperplane the analogue of a simplex and a linear subspace for a problem of such form the analogue of karmarkar s projective transformation and logarithmic barrier function are built this allows us to extend word by word the method of karmarkar for lp to qcqp and allows us to obtain a similar polynomial worst case bound for the number of iterations

exponential sums and newton polyhedra cohomology and estimates the basic objects of this study are exponential sums on a variety defined over a finite field fq q pa p char fq as we have remarked in some earlier articles 1 2 we find it more natural to begin with exponential sums on the torus gm n extend via the usual toric decomposition of an to exponential sums on affine n space and finally proceed via a standard character argument 4 to exponential sums on an affine variety defined over fq while this is the natural order of the work what we do in fact in the first part of this article is to combine the first two steps and deal with exponential sums on varieties v of the form v gm r x as r s n let f e fq xl xn xl xr be an arbitrary regular function on v then f is a sum of monomials and as such has a well defined newton polyhedron a ff at infinity this is the convex closure in rn of the lattice points which occur as exponents of the terms of f together with the origin we have indicated in our previous work 1 2 the description of some of the invariants of the associated l function in terms of properties of this polyhedron for example in 1 we showed how bounds for the degree and total degree of the l function associated with a general exponential sum on v can be expressed in terms of the volumes of a f and the intersections of a f and the various coordinate spaces in the present article assuming f is nondegenerate and commode with respect to af f we show these estimates are sharp our methods are p adic and are based on the work of dwork 11 12 our main accomplishment from which our other results follow is the extension of dwork s cohomology theory from smooth projective hypersurfaces in characteristic p to a general class of exponential sums given f regular on v we construct a complex of p adic banach spaces on which frobenius acts the alternating product of characteristic polynomials of frobenius describes the associated l function in fact when f is nondegenerate and commode with respect to a f the complex is acyclic in dimensions other than 0 and the characteristic

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

solving the discrete l p approximation problem by a method of centers abstract the discrete lp approximation problem is a basic problem in approximation theory and optimization this problem is normally solved by newton type methods which are complicated by the nondifferentiability of the gradient function for p 1 2 this paper discusses a scheme and its implementation for solving this problem by a method of analytic centers which provides a unified treatment for all p 1 and has a polynomial bound of complexity the original problem is reformulated as a constrained convex programming problem which admits a self concordant logarithmic barrier a special structure of the newton system derived from minimizing the potential function is utilized to reduce the amount of computation some other implementation techniques are also presented computational results show that the method of centers is robust for all p

nonblocking multirate networks an extension of the classical theory of connection networks is defined and studied this extension models systems in which multiple connections of differing data rates share the links within a network conditions under which the clos and cantor networks are strictly nonblocking for multirate traffic are determined the authors also determine conditions under which the benes network and variants of the cantor and clos networks are rearrangeable it is found that strictly nonblocking operation can be obtained for multirate traffic with essentially the same complexity as in the classical context

exponential sums on ⁿ ii we prove a vanishing theorem for the p adic cohomology of exponential sums on a n in particular we obtain new classes of exponential sums on an that have a single nonvanishing p adic cohomology group the dimension of this cohomology group equals a sum of milnor numbers

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

a logarithmic barrier function algorithm for quadratically constrained convex quadratic programming an interior point method for quadratically constrained convex quadratic programming is presented that is based on a logarithmic barrier function approach and terminates at a required accuracy of an approximate solution in polynomial time this approach generates a sequence of unconstrained optimization problems each of which is approximately solved by taking a single step in a newton direction

on computing the center of a convex quadratically constrained set we propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions the algorithm starts from an arbitrary initial point and approaches to the desired center by simultaneously reducing infeasibility or slackness of all constraints a partial newton step is taken at each iteration

a large step analytic center method for a class of smooth convex programming problems in this paper a large step analytic center method for smooth convex programming is proposed the method is a natural implementation of the classical method of centers it is assumed that the objective and constraint functions fulfil the so called relative lipschitz condition with lipschitz constant m 0 a great advantage of the method over the existing path following methods is that the steps can be made long by performing linesearches in this method linesearches are performed along the newton direction with respect to a strictly convex potential function when located far away from the central path when sufficiently close to this path a lower bound for the optimal value is updated it is proven that the number of iterations required by the algorithm to converge to an epsilon optimal solution is o 1 m 2 sqrt n ln epsilon or o 1 m 2 n ln epsilon depending on the updating scheme for the lower bound

identifying redundant constraints and implicit equalities in systems of linear constraints redundant constraints are constraints that can be omitted from a system of linear constraints without changing the feasible region implicit equalities are inequality constraints that can be replaced by equalities without changing the feasible region we prove some theorems concerning the identification of both kinds of constraints based on these theorems two methods are developed that allow identification of all redundant constraints and all implicit equalities computational experience on small problems indicates a behavior at least competitive with other methods in this field

an algorithm for convex quadratic programming that requires o n 3 5 l arithmetic operations a new interior point method for minimizing a convex quadratic function over a polytope is developed we show that our method requires o n3 5l arithmetic operations in our algorithm we construct a sequence pz0 pz1 pzk of nested convex sets that shrink towards the set of optimal solution s during iteration k we take a partial newton step to move from an approximate analytic center of pzk 1 to an approximate analytic center of pzk a system of linear equations is solved at each iteration to find the step direction the solution that is available after o ml iterations can be converted to an optimal solution our analysis indicates that inexact solutions to the linear system of equations could be used in implementing this algorithm

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

the demand matching problem we examine formulations for the well known b matching problem in the presence of integer demands on the edges a subset m of edges is feasible if for each node v the total demand of edges in m incident to v is at most bv we examine the system of star inequalities for this problem this system yields an exact linear description for b matchings in bipartite graphs for the demand version we show that the integrality gap for this system is at least 2 5 and at most 2 764 for general graphs the gap lies between 3 and 3 264 we also describe a 3 approximation algorithm 2 5 for bipartite graphs for the cardinality version of the problem a fully polynomial approximation scheme is also presented for the problem on a tree thus generalising a well known result for the knapsack problem recently the notion of demand matching has arisen in the design of clos network switches

on infeasibility of systems of convex analytic inequalities abstract in this paper we address the problem of the infeasibility of systems defined by convex analytic inequality constraints in particular we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints k that may affect the feasibility status of the system after some perturbation of the right hand sides we analyze properties of the irreducible sets as well as infeasibility sets in connection with the set k showing in particular that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one the results presented in this paper are generalizations of a theory developed for the systems of quadratic and linear inequality constraints

an analytic center based column generation algorithm for convex quadratic feasibility problems we consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities at each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut defined by a convex quadratic inequality is placed near the approximate center as the number of cuts increases the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem previously similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem our method differs from these early methods in that we use quadratic cuts in the computation instead of linear cuts moreover our method has a polynomial worst case complexity of o n ln frac 1 varepsilon on the total number of cuts to be used where n is the number of convex quadratic polynomial inequalities in the problem and varepsilon is the radius of the largest ball contained in the feasible set in contrast the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

on the best case performance of hit and run methods for detecting necessary constraints the hit and run methods are probabilistic algorithms that can be used to detect necessary nonredundant constraints in systems of linear constraints these methods construct random sequences of lines that pass through the feasible region these lines intersect the boundary of the region at twohit points each identifying a necessary constraint in order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration an indication of the best case performance of these methods can be determined by minimizing with respect to the hit probabilities the expected value of the number of iterations required to detect all necessary constraints we give a set of isolated strong local minimizers and prove that for two three and four necessary constraints the set of local minimizers is the complete set of global minimizers we conjecture that this is also the case for any number of necessary constraints the results in this paper also apply to sampling problems e g balls from an urn and to the coupon collector s problem

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

brief paper a predictive controller with artificial lyapunov function for linear systems with input state constraints this paper copes with the problem of satisfying input and or state hard constraints in set point tracking problems stability is guaranteed by synthesizing a lyapunov quadratic function for the system and by imposing that the terminal state lies within a level set of the function procedures to maximize the volume of such an ellipsoidal set are provided and interiorpoint methods to solve on line optimization are considered

on multirate rearrangeable clos networks in the multirate switching environment each connection request is associated with a bandwidth weight we consider a three stage clos network and assume that each link has a capacity of one after normalization the network is rearrangeable if for all possible sets of requests such that each input and output link generates a total weight not exceeding one there always exists a set of paths one for each request such that the sum of weights of all paths going through a link does not exceed the link capacity the question is to determine the minimum number of center switches which guarantees rearrangeability we obtain a lower bound of 11n 9 and an upper bound of 41n 16 we then extend the result for the three stage clos network to the multistage clos network finally we propose the weighted version of the edge coloring problem which somehow has escaped the literature associated with our switching network problem

minimal representation of quadratically constrained convex feasible regions in this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints we say that representation is minimal if every other representation has either more quadratic constraints or has the same number of quadratic constraints and at least as many linear constraints we will prove that a representation is minimal if and only if it contains no redundant constraints no pseudo quadratic constraints and no implicit equality constraints we define a pseudo quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints in order to prove the minimal representation theorem we also prove that if the surfaces of two quadratic constraints match on a ball then they match everywhere in this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints the redundant constraints can be identified using the hypersphere directions hd method

an interior point algorithm of o m ln e iterations for c 1 convex programming we present a theoretical result on a path following algorithm for convex programs the algorithm employs a nonsmooth newton subroutine it starts from a near center of a restricted constraint set performs a partial nonsmooth newton step in each iteration and converges to a point whose cost is within accuracy of the optimal cost in o sqrt m left ln varepsilon right iterations wherem is the number of constraints in the problem unlike other interior point methods the analyzed algorithm only requires a first order lipschitzian condition and a generalized hessian similarity condition on the objective and constraint functions therefore our result indicates the theoretical feasibility of applying interior point methods to certainc1 optimization problems instead ofc2 problems since the complexity bound is unchanged compared with similar algorithms forc2 convex programming the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

the minimum sum representation as an index of voting power we propose a new power index based on the minimum sum representation msr of a weighted voting game the msr oers a redesign of a voting game such that voting power as measured by the msr index becomes proportional to voting weight the msr index is a coherent measure of power that is ordinally equivalent to the banzhaf shapley shubik and johnston indices we provide a characterization for a bicameral meet as a weighted game or a complete game and show that the msr index is immune to the bicameral meet paradox we discuss the computation of the msr index using a linear integer program and the inverse msr problem of designing a weighted voting game with a given distribution of power

a combined homotopy interior point method for convex nonlinear programming abstract in this paper we present a new interior point method combined homotopy interior point method chip method for convex nonlinear programming without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set we prove that for any 0 an solution of the problem can be obtained by the chip method to our knowledge strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well known center path following method therefore the chip method essentially reduces the assumptions of the center path following method and can be applied to more general problems

an algorithm to determine boundedness of quadratically constrained convex quadratic programmes abstract in this paper we present an algorithm which can be used to determine whether or not a convex quadratic objective function is bounded from below over a feasible region defined by convex quadratic constraints if there are m constraints and n variables the algorithm terminates after at most min m 1 n 1 iterations each iteration requires the identification of implicit equality constraints in a system of homogeneous linear inequality and equality constraints and the identification can be completed by the solution of linear programmes in addition the algorithm has the advantage of providing a mechanism to reduce both the number of constraints and the dimension of the problem

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

infeasibility analysis for systems of quadratic convex inequalities in this paper we address the problem of the infeasibility of systems defined by quadratic convex inequality constraints in particular we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints k that may affect the feasibility status of the system after some perturbation of the right hand sides we show that all irreducible sets as well as infeasibility sets are subsets of the set k and that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one the results presented in this paper are also applicable to linear systems

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

an algorithm for set covering problem in this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent subgradient optimisation and linear programming computational results are presented for problems involving up to 400 rows and 4000 columns

quadratically constrained convex quadratic programmes faulty feasible regions abstract we are concerned with the problem of minimizing a convex quadratic function subject to convex quadratic constraints in particular we are concerned with the application of the method of analytic centres to problems having a faulty feasible region i e a region that is either unbounded or not full dimensional each iteration of a method of centres must determine an analytic centre of a truncated feasible region if the feasible region is faulty then the truncated region may have no interior it may have no analytic centre it may have an infinity of centres or it may have a unique centre thus the difficulty caused by faulty feasible regions is that even when the problem has a solution the method of centres may fail typically faulty regions have been dealt with by the direct method of adding more constraints and variables often involving a big m constant our method is unique in that it results in a reduction in the number of constraints and effectively in the number of variables

unboundedness of a convex quadratic function subject to concave and convex quadratic constraints abstract we present conditions for the existence of upper and lower bounds on convex quadratic objective functions subject to concave and convex quadratic constraints we also present techniques for determining whether or not the conditions are satisfied

an approximation algorithm for the generalized assignment problem the generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs each job is to be processed by exactly one machine processing jobj on machinei requires timep ij and incurs a cost ofc ij each machinei is available fort i time units and the objective is to minimize the total cost incurred our main result is as follows there is a polynomial time algorithm that given a valuec either proves that no feasible schedule of costc exists or else finds a schedule of cost at mostc where each machinei is used for at most 2t i time units we also extend this result to a variant of the problem where instead of a fixed processing timep ij there is a range of possible processing times for each machine job pair and the cost linearly increases as the processing time decreases we show that these results imply a polynomial time 2 approximation algorithm to minimize a weighted sum of the cost and the makespan i e the maximum job completion time we also consider the objective of minimizing the mean job completion time we show that there is a polynomial time algorithm that given valuesm andt either proves that no schedule of mean job completion timem and makespant exists or else finds a schedule of mean job completion time at mostm and makespan at most 2t

approximating the single source unsplittable min cost flow problem in the single source unsplittable min cost flow problem commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs the demand of each commodity must be routed along a single path so that the total flow through any edge is at most its capacity moreover the total cost must not exceed a given budget this problem has been introduced by kleinberg 7 and generalizes several np complete problems from various areas in combinatorial optimization such as packing partitioning scheduling load balancing and virtual circuit routing kolliopoulos and stein 9 and dinitz garg and goemans 4 developed algorithms improving the first approximation results of kleinberg for the problem of minimizing the violation of edge capacities and for other variants however known techniques do not seem to be capable of providing solutions without also violating the cost constraint we give the first approximation results with hard cost constraints moreover all our results dominate the best known bicriteria approximations finally we provide results on the hardness of approximation for several variants of the problem

multirate rearrangeable clos networks and a generalized edge coloring problem on bipartite graphs chung and ross siam j comput 20 1991 pp 726 736 conjectured that the minimum number m n r of middle stage switches for the symmetric 3 stage clos network c n m n r r to be rearrangeable in the multirate environment is at most 2n 1 this problem is equivalent to a generalized version of the bipartite graph edge coloring problem the best bounds known so far on this function m n r are 11n 9 leq m n r leq 41n 16 o 1 for n r geq 2 derived by du et al siam j comput 28 1999 pp 464 471 in this paper we make several contributions first we give evidence to show that even a stronger result might hold in particular we give a coloring algorithm to show that m n r leq lceil r 1 n 2 rceil which implies m n 2 leq lceil 3n 2 rceil stronger than the conjectured value of 2n 1 second we derive that m 2 r 3 by an elegant argument last we improve both the best upper and lower bounds given above lceil 5n 4 rceil leq m n r leq 2n 1 lceil r 1 2 rceil

purity of exponential sums on n ii we give a purity result for exponential sums of the type x2kn f x where k is a flnite fleld of characteristic p k c is a non trivial additive character and f 2 k x1 xn is a polynomial whose highest degree homogeneous form splits as a product of factors deflning a divisor with normal crossings in p ni1

approximating disjoint path problems using packing integer programs in a packing integer program we are given a matrix a and column vectors b c with nonnegative entries we seek a vector x of nonnegative integers which maximizes c t x subject to ax leq b the edge and vertex disjoint path problems together with their unsplittable flow generalization are np hard problems with a multitude of applications in areas such as routing scheduling and bin packing these two categories of problems are known to be conceptually related but this connection has largely been ignored in terms of approximation algorithms we explore the topic of approximating disjoint path problems using polynomial size packing integer programs motivated by the disjoint paths applications we introduce the study of a class of packing integer programs called column restricted we develop improved approximation algorithms for column restricted programs a result that we believe is of independent interest additional approximation algorithms for disjoint paths are presented that are simple to implement and achieve good performance when the input has a special structure

an optimal algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints an in a sense optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented its unique feature is an error bound in terms of bounds on the spectrum of the hessian of the cost function if applied to a class of problems with the spectrum of the hessians in a given positive interval the algorithm can find approximate solutions in a uniformly bounded number of simple iterations such as matrix vector multiplications moreover if the class of problems admits a sparse representation of the hessian it simply follows that the cost of the solution is proportional to the number of unknowns theoretical results are illustrated by numerical experiments

feasible partition problem in reverse convex and convex mixed integer programming abstract in this paper we consider the consistent partition problem in reverse convex and convex mixed integer programming in particular we will show that for the considered classes of convex functions both integer and relaxed systems can be partitioned into two disjoint subsystems each of which is consistent and defines an unbounded region the polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided