exterior point algorithms for nearest points and convex quadratic programs we consider the problem of finding the nearest point by euclidean distance in a simplicial cone to a given point and develop an exterior penalty algorithm for it each iteration in the algorithm consists of a single newton step following a reduction in the value of the penalty parameter proofs of convergence of the algorithm are given various other versions of exterior penalty algorithms for nearest point problems in nonsimplicial polyhedral cones and for convex quadratic programs all based on a single descent step following a reduction in the value of the penalty parameter per iteration are discussed the performance of these algorithms in large scale computational experiments is very encouraging it shows that the number of iterations grows very slowly if at all with the dimension of the problem

a general method to derive lyapunov functions for non linear systems abstract a method is proposed that derives suitable lyapunov functions for a general class of non linear system expressed in state variables as nfirst order non linear differential equations lyapunov functions for several third and fourth order nonlinear systems derived using the proposed method are identical to those obtained by ogurtsov 1958 1959 and barbasin 1960 by an analogy with linear systems by barbasin 1960 using the method of separation of variables and by siddiqee 1968 applying a qualitative approach

diagonal matrix scaling and linear programming a positive semidefinite symmetric matrix either has a nontrivial nonnegative zero or can be scaled by a positive diagonal matrix into a doubly quasi stochastic matrix this paper describes a simple path following newton algorithm of the complexity o sqrt n l iterations to either scale an n times n matrix or give a nontrivial nonnegative zero the latter problem is well known to be equivalent to linear programming

on dividing an amount according to individual claims or liabilities a basic principle of distributive justice states that if an allocation among a group of individuals is fair then it should be perceived as fair when restricted to each subgroup of individuals this consistency principle applies in particular to methods for allocating taxes among citizens according to their ability to pay every consistent continuous taxation method optimizes an additively separable objective function optimal allocations may be computed by a simple lagrange multiplier technique similar results hold for allocating assets among creditors according to their claims the techniques are illustrated by constructing an objective function for a bankruptcy method from the babylonian talmud

kinodynamic motion planning kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints in the general problem given a robot system we must find a minimal time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration we consider the simplified case of a point mass under newtonian mechanics together with velocity and acceleration bounds the point must be flown from a start to a goal amidst polyhedral obstacles in 2d or 3d although exact solutions to this problem are not known we provide the first provably good approximation algorithm and show that it runs in polynomial time

generalized scalings satisfying linear equations abstract we unify and generalize a broad class of problems referred in the literature as scaling problems by extending the applicability of a formulation suggested recently by bapat and raghavan specifically let a r n b r m and c r m n be given where a is strictly positive a c scaling of the vector a is defined to be a vector a r n with a i a i m k 1 u c kj k for some strictly positive vector u r m the problem of finding a c scaling of the vector a which satisfies the linear system cx b will be called a generalized scaling problem in this paper it is shown that previously studied matrix scaling problems e g finding scalings with prespecified row sums and column sums or finding scalings with row sums equaling the corresponding column sums or finding scalings of multidimensional matrices with prespecified margins are special instances of generalized scaling problems generalized scaling problems are reduced to convex optimization problems and the reduction is used to characterize solutions to develop necessary and sufficient conditions for their existence to establish uniqueness results and to characterize approximate solutions

relaxation methods for problems with strictly convex separable costs and linear constraints we consider the minimization problem with strictly convex possibly nondifferentiable separable cost and linear constraints the dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on gauss seidel relaxation we show that these methods yield the optimal primal solution and under additional assumptions an optimal dual solution to do this it is necessary to extend the classical gauss seidel convergence results because the dual cost may not be strictly convex and may have unbounded level sets

an axiomatic approach to proportionality between matrices given a matrix p 0 what does it mean to say that a matrix f of the same dimension whose row and column sums must fall between specific limits is proportional to p this paper gives an axiomatic solution to this question in two distinct contexts first for any real allocation matrix f second for any integer constrained apportionment matrix f in the case of f real the solution turns out to coincide with what has been variously called biproportional scaling and diagonal equivalence and has been much used in econometrics and statistics in the case of f integer the problem arises in the simultaneous apportionment of seats to regions and to parties and also in the rounding of tables of census data

numerical determination of domains of attraction for electrical power systems using the method of zubov a new approach to the determination of the domain of attraction of ordinary differential equations is applied to power systems taking into account constant damping and no saliency constant damping and saliency and variable damping and saliency the results are compared with the conventional methods and are found to give a better estimate of the stability boundary a model which includes the effects of constant damping and the velocity governor is also studied in this third order system cross sections of the stability surface for the principal planes are given again showing the superiority of the method

constructing roadmaps of semi algebraic sets i completeness abstract this paper describes preliminary work on an algorithm for planning collision free motions for a robot manipulator in the presence of obstacles the physical obstacles lead to forbidden regions in the robots configuration space and for collision free motion we need paths through configuration space which avoid these regions our method is to construct a certain one dimensional subset or roadmap of the space of allowable configurations if s denotes the set of allowable configurations the roadmap has the property that any connected component of s contains a single connected component of the roadmap it is also possible starting from an arbitrary point p s to rapidly construct a path from p to a point on the roadmap thus given any two points in s we can rapidly determine whether they lie in the same connected component of s and if they do we can return a candidate path between them we do not give a complete description of the algorithm here but we define the roadmap geometrically and verify that it has the necessary connectivity

domain of stability of synchronous generators by a cell mapping approach in this paper the theory of simple cell mapping is applied to the determination of domains of stability of synchronous generators the generators are represented by two dimensional differential dynamical systems including the effects of variable damping and saliency first the differential equation governing the behaviour of a generator is reduced to a cell to cell mapping a scheme of compactification is introduced in order to obtain a complete cell state space which consists of a finite number of cells the methodology of the simple cell mapping is then used to prepare a computer program using the program all the singularities of the cell mapping and the domain of stability can be determined readily the method is particularly attractive because it can easily take care of any non linearity different systems can be treated by the same program by simply changing the form of the differential equation in the program and or changing the system parameters three kinds of synchronous machine are studied i

a quadratically convergent polynomial algorithm for solving entropy optimization problems a potential reduction algorithm is developed for solving entropy optimization problems it is shown that the algorithm generates an epsilon optimal solution within at most o sqrt n log epsilon iterations where as usual n is the number of nonnegative variables and each iteration solves a system of linear equations under a computable criterion the algorithm is tuned to the pure newton method in a manner that leads to quadratic convergence while maintaining primal feasibility at each step a stopping criterion is derived which ensures that the objective function approaches its optimal value within any prescribed tolerance this applies for all entropy optimization problems having interior optimal solutions

polynomial methods for separable convex optimization in unimodular linear spaces with applications we consider the problem of minimizing a separable convex objective function over the linear space given by a system mx 0 with m a totally unimodular matrix in particular this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron we first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by goldberg and tarjan j assoc comput mach 36 1989 pp 873 886 and extended to some other problems t r ervolina and s t mccormick discrete appl math 46 1993 pp 133 165 a frank and a v karzanov technical report rr 895 m laboratoire artemis imag universite joseph fourier grenoble france 1992 t ibaraki a v karzanov and h nagamochi private communication 1993 m hadjiat technical report groupe intelligence artificielle faculte des sciences de luminy marseille france 1994 can be generalized to give a combinatorial method with geometric convergence for our problem we also generalize the computationally more efficient cancel and tighten method we then consider objective functions that are piecewise linear pure and piecewise quadratic or piecewise mixed linear and quadratic and we show how both methods can be implemented to find exact solutions in polynomial time strongly polynomial in the piecewise linear case these implementations are then further specialized for finding circulations and cocirculations in a network we finish by showing how to extend our methods to find optimal integer solutions to linear spaces of larger fractionality and to the case when the objective functions are given by approximate oracles

divisor based biproportional apportionment in electoral systems a real life benchmark study biproportional apportionment methods provide two way proportionality in electoral systems where the electoral region is subdivided into electoral districts the problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix and such that a set of row and column sum requirements are fulfilled in a divisor based method for biproportional apportionment the problem is solved by computing appropriate row and column divisors and by rounding the quotients we present a comprehensive experimental evaluation of divisor based biproportional apportionment in an electoral system context by performing experiments on real life benchmark instances election data with multimember districts we evaluate the general quality of divisor based apportionments with respect to e g deviation from quota reversal orderings and occurrences of ties for example we studied the frequency in which a party with a higher vote count in a district ended up with fewer seats in that district

on complexity of matrix scaling abstract line sun scaling problem for a nonnegative matrix a is to find positive definite diagonal matrices y z which result in prescribed row and column sums of the scaled matrix yaz the matrix balancing problem for a nonnegative square matrix a is to find a positive definite diagonal matrix x such that the row sums in the scaled matrix xax are equal to the corresponding column sums we demonstrate that versions of both these problems same as those of other scaling problems for non negative multiindex arrays can be reduced to a specific geometric programming problem for the latter problem we develop a polynomial time algorithm thus deriving polynomial time solvability of a number of generic scaling problems for nonnegative multiindex arrays our results extend those previously known for the problems of matrix balancing 3 and of double stochastic scaling of a square nonnegative matrix 2

on linear programming and matrix scaling over the algebraic numbers abstract adler and beling considered the linear programming problem over the real algebraic numbers they obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial time solvability using the ellipsoid method and under some models of computation based on a better notion of size than that used by adler and beling we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint matrix dimensions for these canonical problems as well as for the matrix scaling problem shown to be a more general problem than linear programming we obtain the necessary bounds demonstrate simple rounding schemes justify the applicability of two polynomial time interior point algorithms under some models of computation describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior point algorithms under an input model and give an alternative method to the traditional duality based approach for the conversion of a general linear programming problem into a feasibility problem

collision detection for moving polyhedra we consider the collision detection problem for a three dimensional solid object moving among polyhedral obstacles the configuration space for this problem is six dimensional and the traditional representation of the space uses three translational parameters and three angles typically euler angles the constraints between the object and obstacles then involve trigonometric functions we show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven dimensional space by simple manipulation the constraints may be projected down into a six dimensional space with no increase in complexity the algebraic form of the constraints greatly simplifies computation of collision points and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles

a vertex algorithm for collision detection abstract an algorithm for detecting the collision of moving objects is presented the algorithm applies to polyhedral objects that can be represented as convex hulls of finite number of vertices in two or three dimensional space this is then extended to a three or four dimensional space respectively to represent the objects and their motion nonlinear programming techniques are then employed to detect possible interference the algorithm detects in one step whether or not the objects will interfere during their motion which may involve pure translations or rotations or both a subalgorithm for computing the minimum distance between the objects is also presented this can be used to solve the interference detection problem or for more general applications

methods for scaling to doubly stochastic form abstract new methods for scaling square nonnegative matrices to doubly stochastic form are described a generalized version of the convergence theorem of sinkhorn and knopp 1967 is proved and applied to show convergence for these new methods tests indicate that one of the new methods has significantly better average and worst case behavior than the sinkhorn knopp method for one of the 3 3 examples of marshall and olkin 1968 sk requires 130 times as many operations as the new algorithm to achieve row and column sums 1 10 5

a comparative study of algorithms for matrix balancing the problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics urban planning statistics demography and stochastic modeling these problems are called matrix balancing problems we describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing matrix scaling and nonlinear optimization the algorithms we study are the ras algorithm a diagonal similarity scaling algorithm and a truncated newton algorithm for network optimization we present results from computational experiments with large scale problems based on producing consistent estimates of social accounting matrices for developing countries

an extended integral method to derive lyapunov functions for nonlinear systems in this paper chin s 1987 b integral method of deriving lyapunov functions for the investigation of nonlinear system stability is extended whereas in chin s method each weighting function includes only one state variable in the extended integral method the weighting functions can be any continuous functions of time some or all state variables and their time derivatives as a result of the less restrictive weighting functions the sensitivity of the method to the form of the state space model is reduced which gives the user more freedom in deriving the lyapunov functions the method also extends the integral method in the sense that it can be applied to both autonomous and non autonomous systems furthermore the intrinsic method of chin 1988 is shown to be a special case of the extended integral method many examples from the literature and from engineering applications are provided to show that the extended method is surprisingly easy to apply and results in lyapunov functions for more general

on the problem of multiple mobile robots cluttering a workspace we consider the problem of multiple mobile robots navigating in a common workspace initially based on the premise that the costs of operation of the robots can be ignored results regarding the probability of collision of two point robots in the workspace are derived obviously these results depend on the geometry of the workspace and hence they are derived for workspaces with various geometries the results are then extended for two robots of finite dimensions the question are k 1 robots better than k is then considered based on two modes of operation namely the batch scheduled mode and the list scheduled mode various theoretical results are derived if the costs of operation of the robots can be ignored based on various computational results obtained we conjecture that the answer to the question is in the affirmative in both modes of operation finally by using the model of computation which includes the costs of operation of the robots the problem is revisited we have shown that for various geometries and revenue to cost ratios there are optimal finite values for the number of robots that can operate within the workspace expressions for these optimal values have been derived

algorithms for proportional matrices in reals and integers letr be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixedh 0 closely related axiomatic approaches have been developed to ascribe meanings to the statements the real matrixf r and the integer matrixa r are proportional to a given matrixp 0

analysis of a classical matrix preconditioning algorithm we study a classical iterative algorithm for balancing matrices in the l norm via a scaling transformation this algorithm which goes back to osborne and parlett 8 reinsch in the 1960s is implemented as a standard preconditioner in many numerical linear algebra packages surprisingly despite its widespread use over several decades no bounds were known on its rate of convergence in this article we prove that for any irreducible n n real or complex input matrix a a natural variant of the algorithm converges in o n3log n e elementary balancing operations where measures the initial imbalance of a and e is the target imbalance of the output matrix the imbalance of a is vlog aiout aiin v where aiout aiin are the maximum entries in magnitude in the ith row and column respectively this bound is tight up to the log n factor a balancing operation scales the ith row and column so that their maximum entries are equal and requires o m n arithmetic operations on average where m is the number of nonzero elements in a thus the running time of the iterative algorithm is o n2m this is the first time bound of any kind on any variant of the osborne parlett reinsch algorithm we also prove a conjecture of chen that characterizes those matrices for which the limit of the balancing process is independent of the order in which balancing operations are performed

smb collision detection based on temporal coherence the paper presents a novel collision detection algorithm termed the sort moving boxes smb for large number of moving 2d 3d objects which are represented by their axis aligned bounding boxes aabbs the main feature of the algorithm is the full exploitation of the temporal coherence of the objects exhibited in a dynamic environment in the algorithm the aabbs are first projected to each cartesian axis the projected intervals on the axes are separately sorted by the diminishing increment sort dis and further divided into subsections by processing all the intervals within the subsections to check if they overlap a complete contact list can be built the smb is a fast and robust collision detection algorithm particularly for systems involving a large number of moving aabbs and also supports for the dynamic insertion and deletion of objects its performance in terms of both expected total detection time and memory requirements is proportional to the total number of aabbs n and is not influenced by size differences of aabbs the space size and packing density over a large range up to ten times difference the only assumption made is that the sorted list at one time step will remain an almost sorted list at the next time step which is valid for most applications whose movement and deformation of each aabb and the dynamic change of the total number n are approximately continuous

lyapunov function for power system with transfer conductances a method is given for deriving a lyapunov function for a two machine power system with nonzero transfer conductances the assumptions required are only mildly restrictive and are likely to be satisfied for practical values of machine parameters the method shows promise for extension to systems of more than two machines

linear inequality scaling problems let a in r m b in r m and c in r m times n be given where a is strictly positive a c scaling of the vector a is defined to be a vector a in r n with a j a j pi k 1 m u k c kj for some strictly positive vector u in r m the problem of finding a c scaling of the vector a that satisfies the linear system cx b is called the linear equality scaling problem lesp the current paper considers the linear inequality scaling problem lisp which concerns the identification of a c scaling of a which satisfies the linear inequality system cx leqq b where it is required that u leqq 1 and that u i 1 for each i with cx i b i it is shown that lisp generalizes lesp and that it unifies a number of matrix scaling problems that have been studied recently further it is shown that lisp can be reduced to one of two convex optimization problems and these reductions are used to characterize solutions to lisp and to derive necessary and sufficient condition

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

generalized integral method to derive lyapunov functions for nonlinear systems the integral method proposed by chin 1986 is extended and generalized the generalized integral method derives suitable lyapunov functions for a general class of non linear system expressed in state variables as n first order non linear differential equations four examples including those studied by mukherjee et al 1972 and miyagi and taniguchi 1981 are considered

collision computation of moving bodies abstract in this paper an explicit mathematical representation of n dimensional bodies moving in translation along general trajectories is derived this representation is used to find out if two moving bodies are going to collide an optimization problem is developed for finding the time and location of collision we consider the special cases of linear and piece wise linear trajectories the collision in this case can be obtained by solving a linear program or a sequence of linear programs respectively the problem of finding the collision time and location of several moving bodies is cast as an integer programming problem a comprehensive simulation study shows that this approach requires much lesser computation time when compared with the current approach of finding the collision between all pairs of bodies

scalings of matrices which have prespecified row sums and column sums via optimization abstract the problem of scaling a matrix so that it has given row and column sums is transformed into a convex minimization problem in particular we use this transformation to characterize the existence of such scaling or corresponding approximations we obtain new results as well as new streamlined proofs of known results

scalings of matrices satisfying line product constraints and generalizations abstract a scaling of a nonnegative matrix a is a matrix having the form a uav where u and v are square diagonal matrices which have positive diagonal elements if the matrix a is square and v u 1 we call a a symmetric scaling of a we consider two problems the first concerns the identification of a scaling of a given nonnegative matrix with prescribed row and column products the second concerns the problem of finding a symmetric scaling of a given nonnegative square matrix whose row products equal the corresponding column products for each of these two scaling problems we characterize the solutions in terms of a nonlinear convex optimization problem we use the characterization to demonstrate that feasibility of either scaling problem is equivalent to the existence of a matrix satisfying the target property and having the same pattern as the given matrix we establish uniqueness of solutions to either scaling problem whenever it is feasible and we develop algorithms for computing the desired unique scalings in the cases where the problem are feasible

on using lp to collision detection between a manipulator arm and surrounding obstacles abstract this paper is concerned with two linear programming lp models for the detection of a collision between a manipulator arm and its surrounding obstacles on the basis of a real case study in particular the first model finds the intersections between two convex polyhedra while the second one finds the intersections between suitable representations of the objects a comparison between the two approaches is presented on the basis of experimental data 9200 test cases have been analyzed

a search algorithm for motion planning with six degrees of freedom abstract the motion planning problem is of central importance to the fields of robotics spatial planning and automated design in robotics we are interested in the automatic synthesis of robot motions given high level specifications of tasks and geometric models of the robot and obstacles the movers problem is to find a continuous collision free path for a moving object through an environment containing obstacles we present an implemented algorithm for the classical formulation of the three dimensional movers problem given an arbitrary rigid polyhedral moving object p with three translational and three rotational degrees of freedom find a continuous collision free path taking p from some initial configuration to a desired goal configuration this paper describes an implementation of a complete algorithm at a given resolution for the full six degree of freedom movers problem the algorithm transforms the six degree of freedom planning problem into a point navigation problem in a six dimensional configuration space called c space the c space obstacles which characterize the physically unachievable configurations are directly represented by six dimensional manifolds whose boundaries are five dimensional c surfaces by characterizing these surfaces and their intersections collision free paths may be found by the closure of three operators which i slide along five dimensional level c surfaces parallel to c space obstacles ii slide along one to four dimensional intersections of level c surfaces and iii jump between six dimensional obstacles these operators are employed by a best first search algorithm in c space we will discuss theoretical properties of the algorithm including completeness at a resolution this paper describes the heuristic search with particular emphasis on the heuristic strategies that evaluate local geometric information at the heart of this paper lie the design and implementation of these strategies for planning paths along level c surfaces and their intersection manifolds and for reasoning about motions with three degrees of rotational freedom the problems of controlling the interaction of these strategies and of integrating diverse local experts for geometric reasoning provide an interesting application of search to a difficult domain with significant practical implications the representations and algorithms we develop impact many geometric planning problems and extend to cartesian manipulators with six degrees of freedom

matrix scaling entropy minimization and conjugate duality i existence conditions abstract we have four primary objectives in this paper first we introduce a problem called truncated matrix scaling that generalizes two well studied matrix scaling problems diagonal similarity scaling and fixed row column equivalence scaling second we derive necessary and sufficient conditions for the attainment of the infimum in the fenchel dual of a class of convex optimization problems third we show that existence of a solution for truncated matrix scaling is equivalent to the attainment of the infimum in a corresponding dual optimization problem we thereby derive necessary and sufficient conditions for the existence of a solution for truncated matrix scaling fourth we derive known existence conditions for similarity and equivalence scaling from the conditions for truncated matrix scaling

on the complexity of nonnegative matrix scaling an n n nonnegative matrix a is said to be doubly stochastic scalable if there exist two positive diagonal matrices x and y such that xay is doubly stochastic we derive an upper bound on the norms of the scaling factors x and y and give a polynomial time complexity bound on the problem of computing the scaling factors to a prescribed accuracy

an extension of a theorem of darroch and ratcliff in loglinear models and its application to scaling multidimensional matrices abstract let c c ij be an m n matrix with real entries let b be any nonzero m vector let k c b 0 be bounded let x x 1 x 2 x n y y 1 y 2 y n be two nonnegative vectors with y k and x j 0 y j 0 for any coordinate j then it is shown that there exists a k and positive numbers z 1 z 2 z m such that j x j m i 1 z c ij i for all j th theorem slightly generalizes a theorem of darroch and ratcliff in loglinear models with a completely different proof technique the proof relies on an extension of a topological theorem of kronecker to set valued maps and the duality theorem of linear programming many theorems in scaling of matrices and multidimensional matrices are direct consequences of this theorem the main idea is to associate a suitable zero one matrix of transportation with any multidimensional matrix some motivations for scaling applications are also discussed

on the complexity of general matrix scaling and entropy minimization via the ras algorithm given an n m nonnegative matrix a a ij and positive integral vectors r e n and c e m having a common one norm h the r c scaling problem is to obtain positive diagonal matrices x and y if they exist such that xay has row and column sums equal to r and c respectively the entropy minimization problem corresponding to a is to find an n m matrix z z ij having the same zero pattern as a the sum of whose entries is a given number h its row and column sums are within given integral vectors of lower and upper bounds and such that the entropy function consisting of the sum of the terms z ij ln z ij a ij is minimized when the lower and upper bounds coincide matrix scaling and entropy minimization are closely related in this paper we present several complexity bounds for the e approximate r c scaling problem polynomial in n m h 1 e and ln v v where v and v are the largest and the smallest positive entries of a respectively these bounds although not polynomial in ln 1 1 e not only provide alternative complexities for the polynomial time algorithms but could result in better overall complexities in particular our theoretical analysis supports the practicality of the well known ras algorithm in our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities but also a complexity for square nonnegative matrices having positive permanent in particular our results extend nontrivially many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by kalantari and khachiyan to the case of general r c scaling finally we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals and the sum of all entries is also prescribed balinski and demange described an ras type algorithm for its continuous version but did not analyze its complexity we show that this algorithm produces an 1 e approximate solution within complexity polynomial in n m h v v and 1 1 e

fast collision detection in four dimensional space abstract in this paper we consider the collision detection problem for general objects a four dimensional approach is proposed for this problem which detects exactly and in one step when and where the earliest collison will occur between the objects this is done by using four dimensional sets to represent the objects in both space and time the problem is then posed as a nonlinear programming problem the algorithm can handle the case of a rigid body moving on a general path in r 2 or r 3 with simultaneous translation and rotation simulation results on some example problems are given and show that the algorithm is superior to those available in the literature

a theorem of the alternative for multihomogeneous functions and its relationship to diagonal scaling of matrices abstract we prove that given a multihomogeneous function satisfying some initial conditions either it has a certain nonnegative zero over a given subspace or an associated logarithmic barrier function has a constrained stationary point under convexity precisely one of these conditions is satisfied the main ingredients of the proof are the derivation of significant properties of the constrained stationary points of the logarithmic barrier function and their relationship to corresponding points of an associated karmarkar potential function corollaries of the theorem include a duality for karmarkar s canonical linear program which happens to be a stronger version of gordan s theorem a more general duality than an existing one concerning the diagonal scaling of symmetric matrices also shown to be a stronger version of gordan s theorem and a diagonal scalability result for a class of multihomogeneous polynomials which is more general than a previously known result on the scalability of positive multidimensional matrices we also give an algorithmic proof of the theorem of the alternative through a projective algorithm which is a generalization of a modified karmarkar linear programming algorithm and in the context of nonnegative matrix scaling becomes a variant of the well known ras algorithm

scalar lyapunov function method for single and multimachine problems in power systems abstract a systematic method is proposed to derive appropriate lyapunov functions for single and multimachine problems in power systems that will enable suitable stability region estimates to be determined initially the lyapunov function for a simplified second order system is derived this function is then modified to form a suitable lyapunov function for the actual non linear system under study six examples for single and multimachine problems are considered

a recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes for a given pair of finite point setsp andq in some euclidean space we consider two problems problem 1 of finding the minimum euclidean norm point in the convex hull ofp and problem 2 of finding a minimum euclidean distance pair of points in the convex hulls ofp andq we propose a finite recursive algorithm for these problems the algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations

relaxation methods for network flow problems with convex arc costs we consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs for the case where all arc costs are strictly convex we study the convergence of a dual gauss seide type relaxation method that is well suited for parallel computation we then extend this method to the case where some of the arc costs are linear as a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in bertsekas 1 and bertsekas and tseng 2

on the scaling of multidimensional matrices abstract elementary proofs are given for theorems of bapat and raghavan on the scaling of nonnegative multidimensional matrices theorems of sinkhorn and of brualdi parter and schneider are derived as corollaries for positive two dimensional matrices hilbert s projective metric and a theorem of g birkhoff are used to prove that sinkhorn s original iterative procedure converges geometrically the ratio of convergence is estimated from the given data

on the fine grain decomposition of multicommodity transportation problems a simple algorithm for nonlinear optimization problems with multicommodity transportation constraints is developed the algorithm is of the row action type and when properly applied decomposes the underlying graph alternatingly by nodes and arcs hence a fine grain decomposition scheme is developed that is suitable for massively parallel computer architectures of the simd i e single instruction stream multiple data stream class data structures are developed for the implementation of both dense and sparse problems and details of an implementation on a connection machine cm 2 with 32k processing elements are given the dense implementation achieves computing rate of up to three gflops several aspects of the algorithm are investigated empirically computational results are reported for the solution of quadratic programs with approximately 10 million columns and 100 thousand rows

semidefinite programming and matrix scaling over the semidefinite cone abstract let e be the hilbert space of real symmetric matrices with block diagonal form diag a m where a is n n and m is an l l diagonal matrix with the inner product x y trace xy we assume n l 1 i e allow n 0 or l 0 given x e we write x 0 x 0 if it is positive semidefinite positive definite let q e e be a symmetric positive semidefinite linear operator and min x 0 5 trace xqx x 1 x 0 the problem of testing if 0 is a significant problem called homogeneous programming on the one hand the feasibility problem in semidefinite programming sdp can be formulated as a homogeneous programming problem on the other hand it is related to the generalization of the classic problem of matrix scaling let 0 1 be a given accuracy u qe e e the identity matrix in e and n n l we describe a path following algorithm that in case 0 in o n ln n u newton iterations produces d 0 d 1 such that d if 0 in o n ln n u ln ln 1 newton iterations the algorithm produces d 0 such that dqde e where d is the operator that maps w e to d1 2wd1 2 moreover we use the algorithm to prove 0 if and only if there exists d 0 such that dqde e if and only if there exists d 0 such that qd 0 thus via this duality the matrix scaling problem is a natural dual to the feasibility problem in sdp this duality also implies that in blum et al bull ams 21 1989 1 real number model of computation the decision problem of testing the solvability of matrix scaling is both in np and co np although the above complexities can be deduced from our path following algorithm for general self concordant homogeneous programming and for matrix scaling obtained in scaling dualities and self concordant homogeneous programming in finite dimensional spaces technical report lcsr tr 359 department of computer science rutgers university new brunswick nj 1999 for the problems considered here the present analysis is quite elementary short and complete this simplicity is mainly due to a new inequality derived in this paper that relates the norm of scaled quantities at two successive newton iterations and implies a quadratic rate of convergence the present algorithm is not only a simple path following algorithm for testing the solvability of feasibility problem in sdp but is also capable of testing solvability of the matrix scaling problem when n 0 the algorithm reduces to the diagonal matrix scaling linear programming algorithm of khachiyan and kalantari siam j optim 4 1992 668 as in the case of lp the algorithm of this paper can be used to solve the general sdp problem

matrix scaling entropy minimization and conjugate duality ii the dual problem we present a matrix scaling problem calledtruncated scaling and describe applications arising in economics urban planning and statistics we associate a dual pair of convex optimization problems to the scaling problem and prove that the existence of a solution for the truncated scaling problem is characterized by the attainment of the infimum in the dual optimization problem we show that optimization problems used by bacharach 1970 bachem and korte 1979 eaves et al 1985 marshall and olkin 1968 and rothblum and schneider 1989 to study scaling problems can be derived as special cases of the dual problem for truncated scaling we present computational results for solving truncated scaling problems using dual coordinate descent thereby showing that truncated scaling provides a framework for modeling and solving large scale matrix scaling problems

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

convex hull representations of models for computing collisions between multiple bodies abstract in this paper we consider a collision detection problem that frequently arises in the field of robotics given a set of bodies with their initial positions and trajectories we wish to identify the first collision that occurs between any two bodies or to determine that none exists for the case of bodies having linear trajectories we construct a convex hull representation of the integer programming model of s z selim and h a almohamad european journal of operational research 119 1 1999 121 129 and compare the relative effectiveness in solving this problem via the resultant linear program we also extend this analysis to model a situation in which bodies move along piecewise linear trajectories possibly rotating at the end of each linear segment for this case we again compare an integer programming approach with its linear programming convex hull representation and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once we provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios as well as the times at which the first collision occurs

collision detection of a moving polygon in the presence of polygonal obstacles in the plane this paper presents a new approach for the following collision detection problem in the plane let a simple polygon p rotate at a center spl ogr with constant angular velocity spl omega and translate towards a set of polygonal obstacles s with constant velocity spl nu given p and s as well as their initial positions and given also the velocities of p determine whether or not p will collide with any element of s and report the collided elements of s if collisions occurred an o mn worst case optimal algorithm is proposed to solve this problem where n is the number of vertices of p and m is the number of vertices of the obstacles in s

hamiltonian type lyapunov functions a flexible and systematic way for the construction of lyapunov functions which lends itself to the use of nonlinear programming techniques is presented the class of lyapunov functions obtained is shown to be energy functions hamiltonians for an equivalent system

dual ascent methods for problems with strictly convex costs and linear constraints a unified approach consider problems of the form text p qquad min f x ex geqq b where f is a strictly convex possibly nondifferentiable function and e and b are respectively a matrix and a vector a popular method for solving special cases of p e g network flow entropy maximization quadratic program is to dualize the constraints ex geqq b to obtain a differentiable maximization problem and then apply an iterative ascent method to solve it this method is simple and can exploit sparsity thus making it ideal for large scale optimization and in certain cases for parallel computation despite its simplicity however convergence of this method has been shown only under certain very restrictive conditions and only for certain special cases of p in this paper a block coordinate ascent method is presented for solving p that contains as special cases both dual coordinate ascent methods and dual gradient methods it is shown under certain mild assumptions on f and p that this method

distance estimation and collision prediction for on line robotic motion planning abstract an efficient method for computing the minimum distance and predicting collisions between moving objects is presented this problem has been incorporated in the framework of an in line motion planning algorithm to satisfy collision avoidance between a robot and moving objects modelled as convex polyhedra in the beginning the deterministic problem where the information about the objects is assumed to be certain is examined if instead of the euclidean norm l 1 or l norms are used to represent distance the problem becomes a linear programming problem the stochastic problem is formulated where the uncertainty is induced by sensing and the unknown dynamics of the moving obstacles two problems are considered first filtering of the distance between the robot and the moving object at the present time second prediction of the minimum distance in the future in order to predict the collision time

interval constrained matrix balancing abstract we define two interval constrained matrix balancing problems the models encompass several known formulations of problems that appear in regional input output analysis estimation of traffic over transportation and telecommunications networks and estimation of social accounting matrices we develop primal dual row action algorithms for the solution of these models and establish their convergence both algorithms generalize existing scaling algorithms for equality constrained matrix balancing problems one of the algorithms is a generalization of the well known procedure for matrix balancing called ras that can handle the range constraints and is thus called range ras the structure of the problem makes the algorithm suitable for implementation on massively parallel computers details of our implementation on a connection machine cm 2 are given numerical results for the solution of problems of size up to 500 500 are given and the performance of the algorithm is compared with that of ras