subgradient based neural networks for nonsmooth nonconvex optimization problems this paper presents a subgradient based neural network to solve a nonsmooth nonconvex optimization problem with a nonsmooth nonconvex objective function a class of affine equality constraints and a class of nonsmooth convex inequality constraints the proposed neural network is modeled with a differential inclusion under a suitable assumption on the constraint set and a proper assumption on the objective function it is proved that for a sufficiently large penalty parameter there exists a unique global solution to the neural network and the trajectory of the network can reach the feasible region in finite time and stay there thereafter it is proved that the trajectory of the neural network converges to the set which consists of the equilibrium points of the neural network and coincides with the set which consists of the critical points of the objective function in the feasible region a condition is given to ensure the convergence to the equilibrium point set in finite time moreover under suitable assumptions the coincidence between the solution to the differential inclusion and the ldquoslow solutionrdquo of it is also proved furthermore three typical examples are given to present the effectiveness of the theoretic results obtained in this paper and the good performance of the proposed neural network

on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

lower bounds for parallel algebraic decision trees parallel complexity of convex hulls and related problems we show that any parallel algorithm in the fixed degree algebraic decision tree model that answers membership queries in w r n using p processors requires w n log p n rounds where w is the number of connected components of w we further prove a similar result for the average case complexity we give applications of this result to various fundamental problems in computational geometry like convex hull construction and trapezoidal decomposition and also present algorithms with matching upper bounds

complementarity problems this paper provides an introduction to complementarity problems with an emphasis on applications and solution algorithms various forms of complementarity problems are described along with a few sample applications which provide a sense of what types of problems can be addressed effectively with complementarity problems the most important algorithms are presented along with a discussion of when they can be used effectively we also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems finally we provide a brief summary of current research trends

an efficient algorithm for decomposing a polygon into star shaped polygons abstract in this paper we show how a theorem in plane geometry can be converted into a o n log n algorithm for decomposing a polygon into star shaped subsets the computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods

convergence of neural networks for programming problems via a nonsmooth lojasiewicz inequality this paper considers a class of neural networks nns for solving linear programming lp problems convex quadratic programming qp problems and nonconvex qp problems where an indefinite quadratic objective function is subject to a set of affine constraints the nns are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic which enable to implement an exact penalty method a new method is exploited to address convergence of trajectories which is based on a nonsmooth lstrokojasiewicz inequality for the generalized gradient vector field describing the nn dynamics the method permits to prove that each forward trajectory of the nn has finite length and as a consequence it converges toward a singleton furthermore by means of a quantitative evaluation of the lstrokojasiewicz exponent at the equilibrium points the following results on convergence rate of trajectories are established 1 for nonconvex qp problems each trajectory is either exponentially convergent or convergent in finite time toward a singleton belonging to the set of constrained critical points 2 for convex qp problems the same result as in 1 holds moreover the singleton belongs to the set of global minimizers and 3 for lp problems each trajectory converges in finite time to a singleton belonging to the set of global minimizers these results which improve previous results obtained via the lyapunov approach are true independently of the nature of the set of equilibrium points and in particular they hold even when the nn possesses infinitely many nonisolated equilibrium points

a linear time algorithm for obtaining the convex hull of a simple polygon in this paper a linear time algorithm is described for finding the convex hull of a simple i e non self intersecting polygon

applications of ramsey s theorem to decision tree complexity combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented problems that are defined by simple inequalities between inputs called order invariant problems are considered a decision tree is called k bounded if each query depends on at most k variables no further assumptions on the type of queries are made it is proved that one can replace the queries of any k bounded decision tree that solves an order invariant problem over a large enough input domain with k bounded queries whose outcome depends only on the relative order of the inputs as a consequence all existing lower bounds for comparison based algorithms are valid for general k bounded decision trees where k is a constant an o n log n lower bound for the element uniqueness problem and several other problems for any k bounded decision tree such that k o n c and c n 1 2 bounded decision trees of complexity o n that solve the element uniqueness problem all the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well

how to reduce the average complexity of convex hull finding algorithms abstract let x 1 x n be a sequence of independent r d valued random vectors with a common density the following class of convex hull finding algorithms is considered find the extrema in a finite number of carefully chosen directions eliminate the x i s that belong to the interior of the polyhedron formed by these extrema apply an 0 n worst case complexity algorithm to find the convex hull of the remaining points we give weak sufficient conditions that imply that the overall average complexity is 0 n we also show that for the standard normal density the average complexity is 0 n whenever n n log n

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

a new linear convex hull algorithm for simple polygons corresp a new optimal algorithm for computing the convex hull of a simple polygon in the plane along with a proof of correctness is presented the main novelty of the proposed algorithm is its use of existing concepts such as the unimodality of the area of a triangle inscribed in a convex polygon and sldansky s scan the combination of these concepts gives us new insight into the problem

parallel permutation and sorting algorithms and a new generalized connection network o k log n algorithms are obtained to permute and sort n data items on cube and perfect shuffle computers with n sup 1 1 k processing elements 1 k log n these algorithms lead directly to a generalized connection network construction having o k log n delay and o kn sup 1 1 k log n contact pairs this network has the advantage that the switches can be set in o k log n time by either a cube or perfect shuffle computer with n sup 1 1 k processing elements 23 references

exponential stability of impulsive high order hopfield type neural networks with time varying delays this paper considers the problems of global exponential stability and exponential convergence rate for impulsive high order hopfield type neural networks with time varying delays by using the method of lyapunov functions some sufficient conditions for ensuring global exponential stability of these networks are derived and the estimated exponential convergence rate is also obtained as an illustration an numerical example is worked out using the results obtained

complexity of large update interior point algorithm for p k linear complementarity problems in this paper we propose a new large update primal dual interior point algorithm for p k linear complementarity problems lcps we extend bai et al s primal dual interior point algorithm for linear optimization lo problems to p k lcps with generalized kernel functions new search directions and proximity functions are proposed based on a simple kernel function which is neither a logarithmic barrier nor a self regular we show that if a strictly feasible starting point is available then the new large update primal dual interior point algorithms for solving p k lcps have o 1 2 k nlogn m 0 e polynomial complexity which is similar to the polynomial complexity obtained for lo and give a simple complexity analysis this proximity function has not been used in the complexity analysis of interior point method ipm for p k lcps before

hadamard s theorem for locally lipschitzian maps if the derivativef p of a continuously differentiable mapffrom r into itself is an invertible linear transformation for each p then by the classical inverse function theorem f is a local homeomorphism every p has a neighborhood u mapped homeomorphically onto its image f u a natural and intriguing question under what further conditions will f be a homeomorphism of r onto itself was answered in one way by hadamard 8 in 1906 using a condition on the norms b max bh h i of the inverses b f p

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

an optimal algorithm for detecting weak visibility of a polygon notation and a theorem are presented which using a result of b chazelle and l j guibas 1985 enable the authors to design an o n log n algorithm for reporting all visibility edges of a given n vertex polygon improving on this bound to o n is presently focused upon this problem is solved for polygons with at least one given visibility edge it is assumed that both endpoints of this edge are convex vertices subsequently it is shown how to drop this restriction the general case of detecting weak edge visibility of an arbitrary simple polygon is dealt with

finding the convex hull of a simple polygon in linear time abstract though linear algorithms for finding the convex hull of a simply connected polygon have been reported not all are short and correct a compact version based on sklansky s original idea 7 and bykat s counter example 8 is given its complexity and correctness are also shown

stability analysis for neural dynamics with time varying delays by using the usual additive neural network model a delay independent stability criterion for neural dynamics with perturbations of time varying delays is derived we extend previously known results obtained by gopalsamy and he 1994 to the time varying delay case and present decay estimates of solutions of neural networks the asymptotic stability is global in the state space of neuronal activations from the techniques used in this paper it is shown that our criterion ensures stability of neural dynamics even when the delay functions vary violently with time our approach provides an effective method for the stability analysis of neural dynamics with delays

on a convex hull algorithm for polygons and its application to triangulation problems abstract a frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases due to its simplicity the algorithm is nevertheless attractive in this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons some application areas where such polygons occur are briefly discussed in addition it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0 n time

constant potential primal dual algorithms a framework we start with a study of the primal dual affine scaling algorithms for linear programs using ideas from kojima et al mizuno and nagasawa and new potential functions we establish a framework for primal dual algorithms that keep a potential function value fixed we show that if the potential function used in the algorithm is compatible with a corresponding neighborhood of the central path then the convergence proofs simplify greatly our algorithms have the property that all the iterates can be kept in a neighborhood of the central path without using any centering in the search directions

a new parameterization of digital straight lines a 1 1 correspondence is established between digital straight lines which start at a fixed point and a simple set of quadruples of integer parameters such a representation by parameters is useful for enumeration first the authors show a 1 1 correspondence between point pairs in a planar set of points and the linear dichotomies of this set then from the equivalence between digital lines and linear dichotomies of points on the digitization grid they prove a 1 1 and onto correspondence between digital straight lines starting at a fixed point and a well defined set of pairs of grid points it follows that four parameters uniquely represent any given digital line with a fixed starting point an o n algorithm is given for determining the parameters from the digital line as well as o log n algorithms for transforming between these parameters and the parameters suggested by l dorst and a w m smeulders 1984

on the conditions for success of sklansky s convex hull algorithm abstract the convex hull algorithm for simple polygons due to sklansky fails in some cases but its extreme simplicity compared to the later algorithms revived an interest in this algorithm a sufficient condition for its success was given recently by toussaint and avis they have proved that the algorithm works for polygons known as weakly externally visible polygons in this paper a new notion called external left visibility is introduced and it is shown that this is a necessary and sufficient condition for the success of sklansky s algorithm moreover algorithms testing simple polygons for external left visibility and weak external visibility are given

a note on bicriterion programming this note examines the complexity of solving the bicriterion vector maximum problem and the associated linear weighted criterion problem over a finite set of alternatives it presents an algorithm for linear weighted problems with separable objectives that is considerably more efficient than complete enumeration of all alternatives and shows that the algorithm is essentially optimal for this problem

superlinear primal dual affine scaling algorithms for lcp we describe an interior point algorithm for monotone linear complementarity problems in which primal dual affine scaling is used to generate the search directions the algorithm is shown to have global and superlinear convergence with q order up to but not including two the technique is shown to be consistent with a potential reduction algorithm yielding the first potential reduction algorithm that is both globally and superlinearly convergent

almost sure exponential stability of recurrent neural networks with markovian switching this paper presents new stability results for recurrent neural networks with markovian switching first algebraic criteria for the almost sure exponential stability of recurrent neural networks with markovian switching and without time delays are derived the results show that the almost sure exponential stability of such a neural network does not require the stability of the neural network at every individual parametric configuration next both delay dependent and delay independent criteria for the almost sure exponential stability of recurrent neural networks with time varying delays and markovian switching parameters are derived by means of a generalized stochastic halanay inequality the results herein include existing ones for recurrent neural networks without markovian switching as special cases finally simulation results in three numerical examples are discussed to illustrate the theoretical results

randomized algorithms for binary search and load balancing on fixed connection networks with geometric applications there are now a number of fundamental problems in computational geometry that have optimal algorithms on pram models this paper presents randomized parallel algorithms that execute on an n processor butterfly interconnection network in o log n time for the following problems of input size n trapezoidal decomposition visibility triangulation and two dimensional convex hull these algorithms involve tackling some of the very basic problems like binary search and load balancing that are taken for granted in pram models apart from a two dimensional convex hull algorithm these are the first nontrivial geometric algorithms that attain this performance on fixed connection networks these techniques use a number of ideas from flashsort that have to be modified to handle more difficult situations it seems likely that they will have wider applications

improved global robust stability of interval delayed neural networks via split interval generalizations abstract the problem of global robust stability of hopfield type delayed neural networks with the intervalized network parameters is revisited recently a computationally tractable i e linear matrix inequality lmi based global robust stability criterion derived from an earlier criterion based on dividing the given interval into more that two intervals has been presented in the present paper generalizations i e division of the given interval into m intervals where m is an integer greater than or equal to 2 is considered and some new lmi based global robust stability criteria are derived it is shown that in some cases m 2 may not suffice i e m 2 may be needed to realize the improvement an example showing the effectiveness of the proposed generalization is given the paper also provides a complete and systematic explanation of the split interval idea

efficient parallel convex hull algorithms parallel algorithms are presented to identify i e detect and enumerate the extreme points of the convex hull of a set of planar points using a hypercube pyramid tree mesh of trees mesh with reconfigurable bus exclusive read exclusive write parallel random access machine erew pram and modified aks network it is shown that the problem of identifying the convex hull for a set of planar points given arbitrarily cannot be solved faster than sorting

synchronization of competitive neural networks with different time scales abstract in this paper the exponential synchronization problem for a class of competitive neural networks is investigated moreover without assuming the active functions to be differentiable and bounded some exponential synchronization criteria are devised by lyapunov functionals linear matrix inequality lmi approach the leibniz newton formula the conditions are less conservative than existing ones reported in the literature for delayed neural networks the applicability of the conditions are validated by an illustrative example

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis ii in this paper we obtain new sufficient conditions ensuring existence uniqueness and global asymptotic stability gas of the equilibrium point for a general class of delayed neural networks dnns via nonsmooth analysis which makes full use of the lipschitz property of functions defining dnns based on this new tool of nonsmooth analysis we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring gas are actually unnecessary and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped finally we derive some new sufficient conditions which are easy to check comparison with some related existing results is conducted and advantages are illustrated with examples throughout our paper spectral properties of the matrix a a sup spl tau play an important role which is a distinguished feature from previous studies here a and a sup spl tau are respectively the feedback and the delayed feedback matrix defining the neural network under consideration

a new criterion for global robust stability of interval delayed neural networks a novel criterion for the global robust stability of hopfield type interval neural networks with delay is presented an example showing the effectiveness of the present criterion is given

a lower bound to finding convex hulls given a set s of n distinct points x i y i 0 leq i n the convex hull problem is to determine the vertices of the convex hull h s all the known algorithms for solving this problem have a worst case running time of cn log n or higher and employ only quadratic tests i e tests of the form f x 0 y 0 x 1 y 1 x n 1 y n 1 0 with f being any polynomial of degree not exceeding 2 in this paper we show that any algorithm in the quadratic decision tree model must make cn log n tests for some input

instance optimal geometric algorithms we prove the existence of an algorithm a for computing 2d or 3d convex hulls that is optimal for every point set in the following sense for every sequence of n points and for every algorithm a in a certain class a the running time of a on input is at most a constant factor times the running time of a on the worst possible permutation of for a in fact we can establish a stronger property for every sequence of points and every algorithm a the running time of a on is at most a constant factor times the average running time of a over all permutations of we call algorithms satisfying these properties instance optimal in the order oblivious and random order setting such instance optimal algorithms simultaneously subsume output sensitive algorithms and distribution dependent average case algorithms and all algorithms that do not take advantage of the order of the input or that assume the input are given in a random order the class a under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments to establish an instance specific lower bound we deviate from traditional ben or style proofs and adopt a new adversary argument for 2d convex hulls we prove that a version of the well known algorithm by kirkpatrick and seidel 1986 or chan snoeyink and yap 1995 already attains this lower bound for 3d convex hulls we propose a new algorithm we further obtain instance optimal results for a few other standard problems in computational geometry such as maxima in 2d and 3d orthogonal line segment intersection in 2d finding bichromatic l close pairs in 2d offline orthogonal range searching in 2d offline dominance reporting in 2d and 3d offline half space range reporting in 2d and 3d and offline point location in 2d our framework also reveals a connection to distribution sensitive data structures and yields new results as a byproduct for example on online orthogonal range searching in 2d and online half space range reporting in 2d and 3d

an analysis of global asymptotic stability of delayed cellular neural networks in this paper a new sufficient condition is given for the uniqueness and global asymptotic stability of the equilibrium point for delayed cellular neural networks dcnns this condition imposes constraints on the feedback and delayed feedback matrices of a dcnn independently of the delay parameter this result is also compared with the previous results derived in the literature

a convex hull inclusion test a new characterization of the interior of the convex hull of a finite point set is given an inclusion test based on this characterization is on average almost linear in the number of points times the dimensionality

a counter example to a convex hull algorithm for polygons abstract a linear time algorithm was recently published pattern recognition22 561 565 1989 for computing the convex hull of a simple polygon in this note we present a counter example to that algorithm

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

a filtering technique for fast convex hull construction in r2 abstract this work presents an optimization technique that reduces the computational cost for building the convex hull from a set of points the proposed method pre processes the input set filtering all points inside an eight vertex polygon in o n time and returns a reduced set of candidate points ordered and distributed across four priority queues experimental results show that for a normal distribution of points in two dimensional space the filtering approach in conjunction with the graham scan is up to 10 faster than the qhull library and between 1 7 to 10 faster than the convex hull methods available in the cgal library results on the worst case scenario when all points lie in the circumference show that a slight random radial displacement of the points make this method the fastest one moreover when increasing the magnitude of this displacement the performance of the proposed method scales at a faster rate than the other methods in terms of memory efficiency the proposed implementation manages to use from 3 to 6 less memory than the other methods the reason behind this memory improvement is because the proposed method stores indices of the input arrays avoiding duplicates of the original floating points furthermore the approach extends the problem size up to n 2 40 by employing 5 byte indices instead of 8 bytes when n 2 32 the optimization technique presented in this work has shown to be significantly useful in accelerating the computation of the convex hull and it is not limited just to the combination with the graham scan but it can also be used in conjunction with other convex hull algorithms

description of object shapes by apparent boundary and convex hull the notions of the apparent boundary and the strict apparent boundary of an object are introduced which together with the convex hull provide an automatic approach for object shape analysis various properties of the apparent boundary and the strict apparent boundary are established which show that they are good approximations of the real object boundary applications of the apparent boundary and the convex hull to the automatic shape characterization of planktonic foraminifera are presented

exponential stability of discrete time neural networks with delay and impulses abstract in this paper we investigate the exponential stability of discrete time neural networks with impulses and time varying delay the discrete time neural networks are derived by discretizing the corresponding continuous time counterparts with different discretization methods the impulses are classified into three classes input disturbances stabilizing and neutral type the impulses are neither helpful for stabilizing nor destabilizing the neural networks and then by using the excellent ideology introduced recently by chen and zheng w h chen w x zheng global exponential stability of impulsive neural networks with variable delay an lmi approach ieee trans circuits syst i 56 6 2009 1248 1259 the connections between the impulses and the utilized lyapunov function are fully explored with respect to each type of impulse novel techniques that used to realize the ideology in discrete time situation are proposed and it is shown that they are essentially different from the continuous time case several criteria for global exponential stability of the discrete time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods

on the convex layers of a planar set let s be a set of n points in the euclidean plane the convex layers of s are the convex polygons obtained by iterating on the following procedure compute the convex hull of s and remove its vertices from s this process of peeling a planar point set is central in the study of robust estimators in statistics it also provides valuable information on the morphology of a set of sites and has proven to be an efficient preconditioning for range search problems an optimal algorithm is described for computing the convex layers of s the algorithm runs in 0 n log n time and requires o n space also addressed is the problem of determining the depth of a query point within the convex layers of s i e the number of layers that enclose the query point this is essentially a planar point location problem for which optimal solutions are therefore known taking advantage of structural properties of the problem however a much simpler optimal solution is derived

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

the symmetric all furthest neighbor problem given a set p of n points on the plane a symmetric furthest neighbor sfn pair of points p q is one such that both p and q are furthest from each other among the points in p a pair of points is antipodal if it admits parallel lines of support in this paper it is shown that a sfn pair of p is both a set of extreme points of p and an antipodal pair of p it is also shown that an asymmetric furthest neighbor asfn pair is not necessarily antipodal furthermore if p is such that no two distances are equal it is shown that as many as and no more than n 2 pairs of points are sfn pairs a polygon is unimodal if for each vertex pk k 1 n the distance function defined by the euclidean distance between pk and the remaining vertices traversed in order contains only one local maximum the fastest existing algorithms for computing all the asfn or sfn pairs of either a set of points a simple polygon or a convex polygon require 0 n log n running time it is shown that the above results lead to an 0 n algorithm for computing all the sfn pairs of vertices of a unimodal polygon

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

convex hull properties and algorithms abstract convex hull ch is widely used in computer graphic image processing cad cam and pattern recognition we investigate ch properties and derive new properties 1 ch vertices coordinates monotonically increase or decrease 2 the edge slopes monotonically decrease using these properties we proposed two algorithms i e ch algorithm for planar point set and ch algorithm for two available chs the main ideas of the proposed algorithms are as follows a planar point set is divided into several subsets by the extreme points and vertices in these subsets are then separately calculated during the computation the ch properties are used to eliminate concave points this can reduce the computational cost and then improves the speed our first algorithm can extract ch with o n log n time which is the lower bound of planar ch extraction and the second algorithm can obtain ch with o m n time at the worst case

global exponential stability of multitime scale competitive neural networks with nonsmooth functions in this paper we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions which models a literally inhibited neural network with unsupervised hebbian learning the network has two types of state variables one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights based on the nonsmooth analysis techniques we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

generalized lyapunov approach for convergence of neural networks with discontinuous or non lipschitz activations the paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non lipschitz functions some tools are developed which enable us to apply a lyapunov like approach to differential equations with discontinuous right hand side modeling the neural network dynamics the tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable lyapunov function and a comparison principle for this time derivative which yields conditions for exponential convergence or convergence in finite time by means of the lyapunov like approach a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions moreover new results on global convergence in finite time are established which are applicable to neuron activations with jump discontinuities or neuron activations modeled by means of continuous non lipschitz holder functions

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

global stability for cellular neural networks with time delay a sufficient condition related to the existence of a unique equilibrium point and its global asymptotic stability for cellular network networks with delay dcnns is derived it is shown that the condition relies on the feedback matrices and is independent of the delay parameter furthermore this condition is less restrictive than that given in the literature

real time fuzzy regression analysis a convex hull approach abstract in this study we present an enhancement of fuzzy regression analysis with regard to its aspect of real time processing let us recall that fuzzy regression generalizes the concept of classical numeric regression in the sense of bringing additional capabilities that allow the model to deal with fuzzy granular data we show that a convex hull method provides a useful vehicle to reduce computing time which becomes of particular relevance in case of real time data analysis our objective is to develop an efficient real time fuzzy regression analysis based on the use of convex hull specifically a beneath beyond algorithm in this algorithm the re construction of convex hull edges depends on incoming vertices while a re computing procedure can be realized in real time we demonstrate the use of the developed enhancement to application to unit performance assessment and air pollution data an important role of convex hull is contrasted with the limitations of linear programming used in the standard regression

optimal parallel hypercube algorithms for polygon problems we present parallel techniques on hypercubes for solving optimally a class of polygon problems we thus obtain optimal o log n time n processor hypercube algorithms for the problems of computing the portions of an n vertex simple polygonal chain c that are visible from a given source point computing the convex hull of c testing an n vertex simple polygon p for monotonicity and other related problems as well previously it was not known how to achieve these complexity bounds on hypercubes one of the main difficulties being that there is no known optimal sorting hypercube algorithm that achieves these bounds in fact these are the first optimal geometric hypercube algorithms that do not assume that the input is given already sorted by x or y coordinates the hypercube model we use is the standard one with o 1 local memory per processor and with one port communication

a logarithmic time sort for linear size networks a randomized algorithm that sorts on an n node network with constant valence in o log n time is given more particularly the algorithm sorts n items on an n node cube connected cycles graph and for some constant k for all large enough it terminates within k log n time with probability at least 1 n

new lower bounds for convex hull problems in odd dimensions we show that in the worst case omega n ceil d 2 1 n log n sidedness queries are required to determine whether the convex hull of n points in real d is simplicial or to determine the number of convex hull facets this lower bound matches known upper bounds in any odd dimension our result follows from a straightforward adversary argument a key step in the proof is the construction of a quasi simplicial n vertex polytope with omega n ceil d 2 1 degenerate facets while it has been known for several years that d dimensional convex hulls can have omega n floor d 2 facets the previously best lower bound for these problems is only omega n log n using similar techniques we also obtain simple and correct proofs of erickson and seidel s lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane as a related result we show that detecting simplicial convex hulls in real d is ceil d 2 sum hard in the sense of gajentaan and overmars

mesh computer algorithms for computational geometry asymptotically optimal parallel algorithms are presented for use on a mesh computer to determine several fundamental geometric properties of figures for example given multiple figures represented by the cartesian coordinates of n or fewer planar vertices distributed one point per processor on a two dimensional mesh computer with n simple processing elements theta n sup 1 2 or time algorithms are given for identifying the convex hull and smallest enclosing box of each figure given two such figures a theta n sup 1 2 or time algorithm is given to decide if the two figures are linearly separable given n or fewer planar points theta n sup 1 2 or time algorithms are given to solve the all nearest neighbor problems for points and for sets of points given n or fewer circles convex figures hyperplanes simple polygons orthogonal polygons or iso oriented rectangles theta n sup 1 2 or time algorithms are given to solve a variety of area and intersection problems since any serial computer has worst case time of omega n when processing n points these algorithms show that the mesh computer provides significantly better solutions to these problems

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

a generalized lmi based approach to the global asymptotic stability of delayed cellular neural networks a novel linear matrix inequality lmi based criterion for the global asymptotic stability and uniqueness of the equilibrium point of a class of delayed cellular neural networks cnns is presented the criterion turns out to be a generalization and improvement over some previous criteria

common asymptotic behavior of solutions and almost periodicity for discontinuous delayed and impulsive neural networks the paper considers a general neural network model with impulses at a given sequence of instants discontinuous neuron activations delays and time varying data and inputs it is shown that when the neuron interconnections satisfy an m matrix condition or a dominance condition then the state solutions and the output solutions display a common asymptotic behavior as time t it is also shown via a new technique based on prolonging the solutions of the delayed neural network to that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network finally this paper shows that for almost periodic data and inputs the selected solution is almost periodic moreover it is robust with respect to a large class of perturbations of the data analogous results also hold for periodic data and inputs a by product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case nonstationary almost periodic solutions in an otherwise globally convergent nonimpulsive neural network to the authors knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features i e impulses discontinuous neuron activations and delays the results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given

a linear time algorithm for computing the convex hull of an ordered crossing polygon in this paper we describe a linear time algorithm for finding the convex hull of an ordered crossing polygon and prove its correctness

multilist layering complexity and applications abstract a natural combinatorial generalization of the convex layer problem termed multilist layering is introduced it is observed to be p complete in the general case when the number of lists or layer size are bounded by s n multilist layering is shown to be logspace hard for the class of problems solvable simultaneously in polynomial time and space s n on the other hand simultaneous polynomial time and o s n log n space solutions in the above cases are provided thus a natural almost complete problem for steve s classes sc 1 sc 2 4 is in particular obtained also nc algorithms for multilist layering when the number of lists or the layer size is bounded by a constant are given as a result the first nc solutions sc solutions respectively for the convex layer problem where the number of orientations or the layer size are bonded by a constant polylog bounded respectively are derived

geometric algorithms for digitized pictures on a mesh connected computer although mesh connected computers are used almost exclusively for low level local image processing they are also suitable for higher level image processing tasks we illustrate this by presenting new optimal in the o notational sense algorithms for computing several geometric properties of figures for example given a black white picture stored one pixel per processing element in an n n mesh connected computer we give n time algorithms for determining the extreme points of the convex hull of each component for deciding if the convex hull of each component contains pixels that are not members of the component for deciding if two sets of processors are linearly separable for deciding if each component is convex for determining the distance to the nearest neighboring component of each component for determining internal distances in each component for counting and marking minimal internal paths in each component for computing the external diameter of each component for solving the largest empty circle problem for determining internal diameters of components without holes and for solving the all points farthest point problem previous mesh connected computer algorithms for these problems were either nonexistent or had worst case times of n2 since any serial computer has a best case time of n2 when processing an n n image our algorithms show that the mesh connected computer provides significantly better solutions to these problems

exploring complexity of large update interior point methods for p κ linear complementarity problem based on kernel function abstract interior point methods not only are the most effective methods in practice but also have polynomial time complexity the large update interior point methods perform in practice much better than the small update methods which have the best known theoretical complexity in this paper motivated by the complexity results for linear optimization based on kernel functions we extend a generic primal dual interior point algorithm based on a new kernel function to solve p linear complementarity problems by using some elegant and simple tools and having interior point condition we show that the large update primal dual interior point methods for solving p linear complementarity problems enjoys o q 1 2 n log n q 1 q log n e iteration bound which becomes o 1 2 n log n log log n log n e with special choices of the parameter q this bound is much better than the classical primal dual interior point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization field some computational results have been provided

a self adjusting interior point algorithm for linear complementarity problems abstract based on the min max principle the standard centering equation in the interior point method is replaced by the optimality condition of a new proximity measure function thus a self adjusting mechanism is constructed in the new perturbed system the newton direction can be adjusted self adaptively according to the information of last iterates a self adjusting interior point method is given based on the new perturbed system numerical comparison is made between this algorithm and a primal dual interior point algorithm using standard perturbed system results demonstrate the efficiency and some advantages of the proposed algorithm

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

computing the convex hull of a simple polygon abstract an algorithm for computing the convex hull of a simple polygon is presented its correctness and complexity are also shown

convergence of newton s method for convex best interpolation in this paper we consider the problem of finding a convex function which interpolates given points and has a minimal l2 norm of the second derivative this problem reduces to a system of equations involving semismooth functions we study a newton type method utilizing clarke s generalized jacobian and prove that its local convergence is superlinear for a special choice of a matrix in the generalized jacobian we obtain the newton method proposed by irvine et al and settle the question of its convergence by using a line search strategy we present a global extension of the newton method considered the efficiency of the proposed global strategy is confirmed with numerical experiments

efficient operations on discrete paths we present linear time and space operations on discrete paths first we compute the outer hull of any discrete path as a consequence a linear time and space algorithm is obtained for computing the convex hull next we provide a linear algorithm computing the overlay graph of two simple closed paths from this overlay graph one can easily compute the intersection union and difference of two jordan polyominoes i e polyominoes whose boundary is a jordan curve the linear complexity is obtained by using an enriched version of a data structure introduced by brlek koskas and provencal a quadtree for representing points in the discrete plane z z augmented with neighborhood links which was introduced in particular to decide in linear time if a discrete path is self intersecting

robust synchronization of an array of coupled stochastic discrete time delayed neural networks this paper is concerned with the robust synchronization problem for an array of coupled stochastic discrete time neural networks with time varying delay the individual neural network is subject to parameter uncertainty stochastic disturbance and time varying delay where the norm bounded parameter uncertainties exist in both the state and weight matrices the stochastic disturbance is in the form of a scalar wiener process and the time delay enters into the activation function for the array of coupled neural networks the constant coupling and delayed coupling are simultaneously considered we aim to establish easy to verify conditions under which the addressed neural networks are synchronized by using the kronecker product as an effective tool a linear matrix inequality lmi approach is developed to derive several sufficient criteria ensuring the coupled delayed neural networks to be globally robustly exponentially synchronized in the mean square the lmi based conditions obtained are dependent not only on the lower bound but also on the upper bound of the time varying delay and can be solved efficiently via the matlab lmi toolbox two numerical examples are given to demonstrate the usefulness of the proposed synchronization scheme

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed