improving linguistic pairwise comparison consistency via linguistic discrete regions linguistic pairwise comparison matrices are widely used in decision making procedures however the matrices often give conflicting results when there are multiple criteria under consideration despite intensive research achieving consistency of such matrices remains a daunting task in this paper a novel approach based on linguistic discrete region is proposed to address the challenge unlike existing methods that require a single value for each comparison our approach allows a comparison to be expressed by a discrete region with multiple linguistic terms such front end gives users more freedom to express their opinions in the back end we propose an iterative searching algorithm that is able to achieve approximate optimal consistency for the comparison matrices with discrete region values the final results are single value matrices that not only guarantee approximate optimal consistency but comply with evaluators intentions a well as our approach does not modify any linguistic values like many existing methods we have conducted extensive evaluations and our empirical study confirms that the linguistic discrete region based approach significantly improves the consistency of linguistic pairwise comparison matrices

conceptual model of coalescence and break up in the presence of external agitation abstract a markovian probabilistic cellular automaton with the capability to capture the essential phenomenology of coalescence and break up processes in the presence of external agitation is introduced the existence of homogeneous stationary states of the model which admit large cluster formation for a range of agitation speeds is analytically predicted by mean field calculations through mean field analysis it is possible to obtain formulas that link experimental and model parameters on the base of simple measurable quantities in this way the experimental conditions for which a desirable stationary particle size distribution should be expected can be derived

a capable neural network model for solving the maximum flow problem this paper presents an optimization technique for solving a maximum flow problem arising in widespread applications in a variety of settings on the basis of the karush kuhn tucker kkt optimality conditions a neural network model is constructed the equilibrium point of the proposed neural network is then proved to be equivalent to the optimal solution of the original problem it is also shown that the proposed neural network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the maximum flow problem several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

the distance between two convex sets abstract in this paper we explore the duality relations that characterize least norm problems the paper starts by presenting a new minimum norm duality mnd theorem one that considers the distance between two convex sets roughly speaking the new theorem says that the shortest distance between the two sets is equal to the maximal separation between the sets where the term separation refers to the distance between a pair of parallel hyperplanes that separates the two sets the second part of the paper brings several examples of applications the examples teach valuable lessons about the role of duality in least norm problems and reveal new features of these problems one lesson exposes the polar decomposition which characterizes the solution of an inconsistent system of linear inequalities another lesson reveals the close links between the mnd theorem theorems of the alternatives steepest descent directions and constructive optimality conditions

riemann and weierstrass walks revisited abstract the weierstrass and riemann walks are non trivial discrete random processes to model and characterize the underlying noise in the dynamics of fluctuations for out of equilibrium systems and in more general contexts to simulate complex dynamics like order disorder phase transitions and anomalous diffusion properties in physical biological and financial systems in this work simple algorithms implemented in gnu r for both riemann and weierstrass discrete processes are presented explicit formulas for the probability distributions of n steps are obtained finally a way to connect both random processes is commented

on spherical convergence convexity and block iterative projection algorithms in hilbert space abstract we call a sequence x n in hilbert space spherical if there exists u such that lim x n u exists and is finite if moreover u is a weak accumulation point of the sequence we call the sequence spherically convergent we demonstrate that for large classes of nonexpansive possibly nonstationary discrete time processes in hilbert space the iterates are spherically convergent basic identities and orthogonality relations pertinent to this type of convergence are exhibited sufficient conditions for weak and spherical convergence in terms of the fullness of the set of fixed points of the process are established and compared spherical convergence of the general block iterative projection scheme in hilbert space is established

entropic perturbation method for solving a system of linear inequalities the problem of finding an x rn such that ax b and x 0 arises in numerous contexts we propose a new optimization method for solving this feasibility problem after converting ax b into a system of equations by introducing a slack variable for each of the linear inequalities the method imposes an entropy function over both the original and the slack variables as the objective function the resulting entropy optimization problem is convex and has an unconstrained convex dual if the system is consistent and has an interior solution then a closed form formula converts the dual optimal solution to the primal optimal solution which is a feasible solution for the original system of linear inequalities an algorithm based on the newton method is proposed for solving the unconstrained dual problem the proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence however if the system is inconsistent the unconstrained dual is shown to be unbounded moreover the same algorithm can detect possible inconsistency of the system our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region the performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by yang and murty our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region

a novel recurrent neural network with finite time convergence for linear programming in this letter a novel recurrent neural network based on the gradient method is proposed for solving linear programming problems finite time convergence of the proposed neural network is proved by using the lyapunov method compared with the existing neural networks for linear programming the proposed neural network is globally convergent to exact optimal solutions in finite time which is remarkable and rare in the literature of neural networks for optimization some numerical examples are given to show the effectiveness and excellent performance of the new recurrent neural network

a goal programming method for obtaining interval weights from an interval comparison matrix crisp comparison matrices lead to crisp weight vectors being generated accordingly an interval comparison matrix should give an interval weight estimate in this paper a goal programming gp method is proposed to obtain interval weights from an interval comparison matrix which can be either consistent or inconsistent the interval weights are assumed to be normalized and can be derived from a gp model at a time the proposed gp method is also applicable to crisp comparison matrices comparisons with an interval regression analysis method are also made three numerical examples including a multiple criteria decision making mcdm problem with a hierarchical structure are examined to show the potential applications of the proposed gp method

dominance degrees for intervals and their application in multiple attribute decision making abstract since intervals are used to express decision information in various decision making problems it is necessary to give a feasible method to rank them the most commonly used method is to employ the possibility degree to compare intervals in pairs then construct a fuzzy preference relation generally the constructed fuzzy preference relation does not keep additive multiplicative consistency in order to ensure the consistency of decision information optimization models are constructed by using the logical relationship between the elements of the consistent fuzzy preference relation and their weights then by solving the optimization models consistent fuzzy preference relations are obtained the consistent fuzzy preference relations are the proxy of the original inconsistent fuzzy preference relation constructed by the possibility degree however such treatment is flawed the inconsistent fuzzy preference relation generated from the possibility degree can t guarantee the reliability of the result while the proxy consistent fuzzy preference relation would cause bias to solve this problem the additive dominance degree and multiplicative dominance degree for intervals are proposed as a result of indirect comparison the fuzzy preference relations are constructed from the dominance degrees of intervals to ensure the consistency of decision information in the process of ranking intervals this can avoid decision making bias furthermore a new approach to interval multiple attribute decision making is proposed based on the dominance degrees

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

an automatic relaxation method for solving interval linear inequalities abstract the problem of solving iteratively a large and possibly sparse system of interval linear inequalities ax is considered an algorithm of the row action type is proposed which in each iterative step effectively takes account of a pair of inequalities describing a single interval inequality the algorithm realizes in an automatic manner a relaxation principle proposed by goffin but also allows further external relaxation parameters

a recurrent neural network for real time semidefinite programming the semidefinite programming problem is an important optimization problem that has been extensively investigated a real time solution method for solving such a problem however is still not yet available the paper proposes a recurrent neural network for this purpose first an auxiliary cost function is introduced to minimize the duality gap between the admissible points of the primal problem and the corresponding dual problem then a dynamical system is constructed to drive the duality gap to zero exponentially along any trajectory by modifying the gradient of the auxiliary cost function furthermore a subsystem is developed to circumvent the computation of matrix inverse so that the resulting overall dynamical system can be realized using a recurrent neural network the architecture of the resulting neural network is discussed the operating characteristics and performance of the proposed approach are demonstrated by means of simulation results

interval regression by tolerance analysis approach in interval linear regression analysis we are given crisp or interval data and we are to determine appropriate interval regression parameters there are various methods for interval regression many of them possess the property that while some of the resulting interval regression parameters are very wide the other parameters are crisp this drawback is the main limiting factor for such methods and much effort has been devoted to overcoming it we propose a method motivated by tolerance analysis in linear systems our method yields intervals for regression parameters the widths of which are proportional to an in advance given vector of parameters moreover the method is computationally very cheap and provides a natural measure of quality of a model first we formulate the method for the basic model of crisp input crisp output data and then extend it to crisp input interval output and interval input interval output models for the interval valued cases we study several formulations of the solution concept possibility strong possibility weak possibility necessity here strong possibility is a new concept proposed as a natural counterpart to the remaining ones we prove that the method provides optimal interval parameters meeting centrality and proportionality requirements we also show that the method provides interval regression parameters satisfying various versions of tanaka lee s inclusion property we also derive a form of a complementarity theorem for the weak possibility and necessity solution concepts since practical problems may be affected by outlying observations we show that our approach is easily adapted to deal with them we illustrate the theory by examples

possibilistic linear regression analysis for fuzzy data abstract fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined recently it has become important to deal with fuzzy data in connection with expert knowledge three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data since our formulations can be reduced to linear programming problems the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters this approach can be regarded as a fuzzy interval analysis in a fuzzy environment

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

a one layer recurrent neural network with a discontinuous activation function for linear programming a one layer recurrent neural network with a discontinuous activation function is proposed for linear programming the number of neurons in the neural network is equal to that of decision variables in the linear programming problem it is proven that the neural network with a sufficiently high gain is globally convergent to the optimal solution its application to linear assignment is discussed to demonstrate the utility of the neural network several simulation examples are given to show the effectiveness and characteristics of the neural network

on the possibilistic approach to linear regression models involving uncertain indeterminate or interval data abstract we consider linear regression models where both input data the observations of independent variables and output data the observations of the dependent variable are affected by loss of information caused by uncertainty indeterminacy rounding or censoring instead of real valued crisp data only intervals are available we study a possibilistic generalization of the least squares estimator so called ols set for the interval model investigation of the ols set allows us to quantify whether the replacement of real valued crisp data by interval values can have a significant impact on our knowledge of the value of the ols estimator we show that in the general case very elementary questions about properties of the ols set are computationally intractable assuming p np we also focus on restricted versions of the general interval linear regression model to the crisp input case taking the advantage of the fact that in the crisp input interval output model the ols set is a zonotope we design both exact and approximate methods for its description we also discuss special cases of the regression model e g a model with repeated observations

a high performance neural network for solving linear and quadratic programming problems two classes of high performance neural networks for solving linear and quadratic programming problems are given we prove that the new system converges globally to the solutions of the linear and quadratic programming problems in a neural network network parameters are usually not specified the proposed models can overcome numerical difficulty caused by neural networks with network parameters and obtain desired approximate solutions of the linear and quadratic programming problems

neural network for solving linear programming problems with bounded variables a new neural network for solving linear programming problems with bounded variables is presented the network is shown to be completely stable and globally convergent to the solutions to the linear programming problems the proposed new network is capable of achieving the exact solutions in contrast to existing optimization neural networks which need a suitable choice of the network parameters and thus can obtain only approximate solutions furthermore both the primal problems and their dual problems are solved simultaneously by the new network

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

iterative algorithm for solving a class of convex feasibility problem abstract in this paper we consider a class of convex feasibility problem which arises in quantum computation based on the matrix equation theory the feasible sets are characterized by exploiting the special structure of the linear constraints and its analytic expression is given by making use of the nice structure properties and the kkt condition we derive the projection formulas of a matrix onto the feasible sets the relaxed alternating projection method is designed to solve the convex feasibility problem numerical experiments show that the new method is feasible and effective

link function selection in stochastic multicriteria decision making models a stochastic formulation of the analytic hierarchy process ahp using an approach based on bayesian categorical data models has been developed however in categorical data models it is known that the selection of the link function may have an impact on the model estimates in particular the selection of the probit link implies an assumption that model error terms are normally distributed and this normality assumption is regularly utilized in other related methods such as the multiplicative ahp we examine model performance with respect to the choice of two model link functions with regard to point estimates it is found that the logit formulation is better able to replicate the estimates obtained by the eigenvector decomposition associated with the original formulation of the ahp by contrast the probit link produces priorities which are consistently more moderate than those of the ahp the results suggest that the logit formulation will be preferred by decision makers who wish to replicate the ahp priorities as closely as possible the results also suggest that the unexamined use of the normality assumption in other stochastic ahp methods may have an impact on priority estimates and thus is worthy of further attention

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

a neural network model for monotone linear asymmetric variational inequalities a linear variational inequality is a uniform approach for some important problems in optimization and equilibrium problems we give a neural network model for solving asymmetric linear variational inequalities the model is based on a simple projection and contraction method computer simulation is performed for linear programming lp and linear complementarity problems lcp the test results for the lp problem demonstrate that our model converges significantly faster than the three existing neural network models examined in a comparative study paper

a parallel projection method for solving generalized linear least squares problems a parallel projection algorithm is proposed to solve the generalized linear least squares problem find a vector to minimize the 2 norm distance from its image under an affine mapping to a closed convex cone in each iteration of the algorithm the problem is decomposed into several independent small problems of finding projections onto subspaces which are simple and can be tackled parallelly the algorithm can be viewed as a dual version of the algorithm proposed by han and lou 8 for the special problem under consideration stronger convergence results are established the algorithm is also related to the block iterative methods of elfving 6 dennis and steihaug 5 and the primal dual method of springarn 14

incorporating the uncertainty of decision judgements in the analytic hierarchy process abstract the uncertainty in the relative weights of a pairwise comparison matrix in the analytic hierarchy process ahp is caused by the uncertainty in our decision judgements and in many cases can not be avoided in this paper it is explicitly shown how such uncertainties can be incorporated within the framework of ahp and how the resulting uncertainties in the relative priorities of the decision alternatives can be computed the required algorithm and the computational procedures are also developed and illustrated with examples uncertainty is introduced as a fundamental concept independent of the concept of consistency with a view to extend the ahp as a decision analysis procedure

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

logarithmic least squares prioritization and completion methods for interval fuzzy preference relations based on geometric transitivity abstract this article introduces the notion of geometric transitivity which may be used to define consistent interval fuzzy preference relations some useful properties are presented for consistent interval fuzzy preference relations a close investigation reveals that existing methods for determining the analogous property of consistency are not robust to permutations of the pairwise judgments and not able to reflect the hesitancy in the decision maker s preference a parametric transformation formula is put forward to convert a normalized interval weight vector into a consistent interval fuzzy preference relation by minimizing the squared difference between the logarithm of the ratio of the original judgment and the logarithm of the ratio of the converted consistent one a logarithmic least squares model is developed to derive interval weights from any interval fuzzy preference relation based on geometric transitivity a logarithmic least squares model is established to rectify inconsistency for complete and inconsistent interval fuzzy preference relations and a logarithmic least squares completion approach is further developed to estimate missing values for incomplete interval fuzzy preference relations numerical examples are furnished to show the validity of the proposed models and compare with other existing methods

global exponential stability of recurrent neural networks for solving optimization and related problems global exponential stability is a desirable property for dynamic systems the paper studies the global exponential stability of several existing recurrent neural networks for solving linear programming problems convex programming problems with interval constraints convex programming problems with nonlinear constraints and monotone variational inequalities in contrast to the existing results on global exponential stability the present results do not require additional conditions on the weight matrices of recurrent neural networks and improve some existing conditions for global exponential stability therefore the stability results in the paper further demonstrate the superior convergence properties of the existing neural networks for optimization

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

decision making with interval probabilities handling uncertainty by interval probabilities is recently receiving considerable attention by researchers interval probabilities are used when it is difficult to characterize the uncertainty by point valued probabilities due to partially known information most of researches related to interval probabilities such as combination marginalization condition bayesian inferences and decision assume that interval probabilities are known how to elicit interval probabilities from subjective judgment is a basic and important problem for the applications of interval probability theory and till now a computational challenge in this work the models for estimating and combining interval probabilities are proposed as linear and quadratic programming problems which can be easily solved the concepts including interval probabilities interval entropy interval expectation interval variance interval moment and the decision criteria with interval probabilities are addressed a numerical example of newsvendor problem is employed to illustrate our approach the analysis results show that the proposed methods provide a novel and effective alternative for decision making when point valued subjective probabilities are inapplicable due to partially known information

znn for solving online time varying linear matrix vector inequality via equality conversion in this paper a special recurrent neural network termed zhang neural network znn is proposed and investigated for solving online time varying linear matrix vector inequality lmvi via equality conversion that is by introducing a time varying vector of which each element is great than or equal to zero such a time varying linear inequality can be converted to a time varying matrix vector equation then the znn model is developed and investigated for solving online the time varying matrix vector equation as well as the time varying lmvi by employing the znn design formula the resultant znn model exploits the time derivative information of time varying coefficients computer simulation results further demonstrate the efficacy and superiority of the proposed znn model for solving online the time varying lmvi and the converted time varying matrix vector equation

convergence results for an accelerated nonlinear cimmino algorithm we present an accelerated version of cimmino s algorithm for solving the convex feasibility problem in finite dimension the algorithm is similar to that given by censor and elfving for linear inequalities we show that the nonlinear version converges locally to a weighted least squares solution in the general case and globally to a feasible solution in the consistent case applications to the linear problem are suggested

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

classification methods for random utility models with i i d disturbances under the most probable alternative rule in this paper random utility maximization based on maximization of correct classification of the choice decisions over a given data set is considered it is shown that if the disturbance vector in the random utility model is independent and identically distributed then preference determination based on the most probable alternative reduces to deterministic utility maximization as a consequence of the above equivalence the form of the error distribution normal weibull uniform etc plays no role in the determination of the preferred alternative parameter estimation under the most probable alternative rule is carried out using two methods the first is based on the solution of an appropriately defined system of linear inequalities and the second one is based on the function optimization of a newly proposed function whose optimum is achieved when the number of correctly classified individuals is maximized the ability to use these algorithms in the framework of pattern recognition and machine learning is pointed out simulations and a real case study involving intercity travel behavior are employed to assess the proposed methods

monotone gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems abstract the relaxation method for linear inequalities iterates by projecting the current point onto the most violated constraint accelerated methods project onto the intersection of several halfspaces or onto a surrogate halfspace corresponding to a nonnegative combination of constraints we extend todd s conditions for finding best surrogate inequalities via the solution of systems of linear equations our techniques may be used for accelerating various methods for convex feasibility and optimization problems

axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations abstract this paper focuses on obtaining an interval extension of saaty s consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations imrprs as well as checking acceptability of imrprs four axiomatic properties are presented to characterize multiplicative consistency of imrprs the paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties a constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an imrpr based on the constructed interval matrix an ordinary interval multiplication based transitivity equation is developed to define consistency of imrprs a logarithmic least square model is established to find normalized interval weights from imrprs and its analytic solution is obtained by the lagrangian multiplier method the paper introduces an index to measure uniformity of row uncertainties in an imrpr and proposes a geometric consistency index to measure inconsistency of imrprs a novel method is put forward to determine acceptability of imrprs by checking acceptable consistency and acceptable uncertainty the proposed models are illustrated by eight numerical examples and a hierarchical multi criteria decision making problem dealing with recommending undergraduate students to graduate admission

an accelerated successive orthogonal projections method for solving large scale linear feasibility problems abstract we consider linear feasibility problems in the standard form ax b 1 x u the successive orthogonal projections method may be used for solving this problem using sparse orthogonal factorizations techniques for computing the projections on ax b we introduce an acceleration technique in order to speed up the generally slow convergence of the method we present some numerical experiments

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

global strict solutions to continuous coagulation fragmentation equations with strong fragmentation in this paper we give an elementary proof of the unique global in time solvability of the coagulation multiple fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate the proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument

new contraction methods for linear inequalities abstract a system of linear inequalities can be translated to an equivalent unconstrained smooth convex minimization problem based on the idea of a contraction method we present a new iterative contraction algorithm of uzawa s type for solving linear inequalities and prove its global linear convergence under the assumption that the solution set is nonempty our new step length rule allows us to take the step up to 4 times as long as that of the original uzawa s method and the resulting algorithm terminates for systems which have full dimensional solution set because the method requires only two matrix vector multiplications at each iteration and performs no transformation on the matrix elements it may be acceptable for large sparse problems and favorable for parallel computation the numerical results indicate that the new method converges much faster than the original uzawa s method

on the convergence of han s method for convex programming with quadratic objective we present an iterative method for minimizing strictly convex quadratic functions over the intersection of a finite number of convex sets the method consists in computing projections onto the individual sets simultaneously and the new iterate is a convex combination of those projections we give convergence proofs even for the inconsistent case i e when the intersection of the sets is empty

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

uncertainty index based interval assignment by interval ahp abstract in a multi attribute decision making problem indigenous values are assigned to attributes based on a decision maker s subjective judgments the given judgments are often uncertain because of the uncertainty of situations and intuitiveness of human judgments in order to reflect the uncertainty in the assigned values they are denoted as intervals whose widths represent the possibilities of attributes since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes he she gives a pairwise comparison matrix by comparing two attributes at one occasion the given matrix contains two kinds of uncertainty one is inconsistency among comparisons and the other is incompleteness of comparisons this paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix at first the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability sum or maximum of widths or ignorance then considering that too uncertain information is not useful the intervals of attributes are obtained by minimizing their uncertainty indexes

block iterative surrogate projection methods for convex feasibility problems abstract a unified framework is presented for studying the convergence of projection methods for finding a common point of finitely many closed convex sets in r n every iteration approximates each set by a half space given either by an approximate projection of the current iterate or by an aggregate inequality derived from the convex inequalities describing this set the next iterate is found by projecting the current one on a surrogate half space formed by taking a convex combination of the half space inequalities convergence to a solution is established under weak conditions that allow various acceleration techniques and choices of aggregating weights the resulting methods are block iterative and hence lend themselves to parallel implementation we show that the idea of forming cut maps via surrogate inequalities encompasses many classical as well as recently proposed methods for set intersection problems and convex feasibility problems with nondifferentiable inequalities and linear equations and inequalities

examining alternatives in the interval analytic hierarchy process using complete enumeration the analytic hierarchy process as an aid to decision makers has gained in popularity over the years recently the procedure has been generalised into the interval analytic hierarchy process rather than select a precise ratio for the comparisons between the alternatives offered to the decision maker an interval is proposed one popular strategy in examining this problem is to adopt a simulation technique here two simulation methods are considered and it is demonstrated that total enumeration is a viable alternative

eliciting dual interval probabilities from interval comparison matrices in recent years dealing with uncertainty using interval probabilities such as combination marginalization condition bayesian inferences has received considerable attention however how to elicit interval probabilities from subjective judgment is still a fundamental problem in applications of interval probabilities in this paper interval valued pairwise comparisons of events are used to characterize the judgment of a person about which one is more likely to occur between each pair of events from the interval comparison matrix we elicit the dual interval probabilities by linear programming problems the ratios of the estimated interval probabilities are used to approximate the interval comparison values from the outside and the inside of them in other words the inclusion relations between the interval comparison values and the ratios of the obtained interval probabilities hold a numerical example concerning a new product development is presented to illustrate how to estimate dual interval probabilities and use interval probabilities to make a decision

a multi step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices abstract this article presents a goal programming framework to solve group decision making problems where decision makers judgments are provided as incomplete interval additive reciprocal comparison matrices iarcms new properties of multiplicative consistent iarcms are put forward and used to define consistent incomplete iarcms a two step goal programming method is developed to estimate missing values for an incomplete iarcm the first step minimizes the inconsistency of the completed iarcms and controls uncertainty ratios of the estimated judgments within an acceptable threshold and the second step finds the most appropriate estimated missing values among the optimal solutions obtained from the previous step a weighted geometric mean approach is proposed to aggregate individual iarcms into a group iarcm by employing the lower bounds of the interval additive reciprocal judgments a two step procedure consisting of two goal programming models is established to derive interval weights from the group iarcm the first model is devised to minimize the absolute difference between the logarithm of the group preference and that of the constructed multiplicative consistent judgment the second model is developed to generate an interval valued priority vector by maximizing the uncertainty ratio of the constructed consistent iarcm and incorporating the optimal objective value of the first model as a constraint two numerical examples are furnished to demonstrate validity and applicability of the proposed approach

pairwise comparison based interval analysis for group decision aiding with multiple criteria interval ahp analytic hierarchy process was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria in this paper interval ahp is applied to group decision problems interval ahp is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker under individual interval weight vectors obtained from individual pairwise comparison matrices three approaches to obtaining a consensus interval weight vector are proposed one is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors by this approach we can count out indubitably inferior alternatives the second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when all individual opinions are similar the third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when individual opinions are not similar the usefulness of the proposed three approaches is demonstrated by simple numerical examples

on solving systems of linear inequalities with artificial neural networks the implementation of the relaxation projection algorithm by artificial neural networks to solve sets of linear inequalities is examined the different versions of this algorithm are described and theoretical convergence results are given the best known analog optimization solvers are shown to use the simultaneous projection version of it neural networks that implement each version are described the results of tests made with simulated realizations of these networks are reported these tests consisted in having all networks solve some sample problems the results obtained help determine good values for the step size parameters and point out the relative merits of the different networks

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

neurocomputing with time delay analysis for solving convex quadratic programming problems this paper presents a neural network computational scheme with time delay consideration for solving convex quadratic programming problems based on some known results a delay margin is explicitly determined for the stability of the neural dynamics under which the states of the neural network does not oscillate the configuration of the proposed neural network is provided operational characteristics of the neural network are demonstrated via numerical examples

solving convex programming problems with equality constraints by neural networks abstract this paper presents a neural network approach for solving convex programming problems with equality constraints after defining the energy function and neural dynamics of the proposed neural network we show the existence of an equilibrium point at which the neural dynamics becomes asymptotically stable it is shown that under proper conditions an optimal solution of the underlying convex programming problems is an equilibrium point of the neural dynamics and vise versa the configuration of the proposed neural network with an exact layout is provided for solving linear programming problems the operational characteristics of the neural network are demonstrated by numerical examples

new methods for linear inequalities abstract the iterative method of cimmino for solving linear equations is generalized to linear inequalities we also present a richardson type iterative method for solving the inequality problem which includes the generalized cimmino scheme convergence proofs are provided

a goal programming approach to deriving interval weights in analytic form from interval fuzzy preference relations based on multiplicative consistency abstract this paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations ifprs and obtain approximate solution based interval weights in analytic form from inconsistent ifprs the paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing 0 1 valued interval weights obtained from ifprs by examining equivalency of 0 1 valued interval weight vectors a novel framework of multiplicatively normalized interval fuzzy weights mnifws is then proposed and used to define multiplicatively consistent ifprs the paper presents significant properties for multiplicatively consistent ifprs and their associated mnifws these properties are subsequently used to establish two goal programming gp models for obtaining optimal mnifws from consistent ifprs by the lagrangian multiplier method analytic solutions of the two gp models are found for consistent ifprs the paper further devises a two step procedure for deriving approximate solution based mnifws in analytic form from inconsistent ifprs two visualized computation formulas are developed to determine the left and right bounds of approximate solution based mnifws of any ifpr the paper shows that this approximate solution is an optimal solution if an ifpr is multiplicatively consistent three numerical examples including three ifprs and comparative analyses are offered to demonstrate rationality and validity of the developed model

a simultaneous iterative method for computing projections on polyhedra a simultaneous version of hildreth s iterative algorithm for norm minimization over linear inequalities is presented proofs are given showing that the algorithm converges from any starting point to its projection on the linear constraints set in the feasible case and to the nearest least squares solution in the general case

interval multiplicative pairwise comparison matrix consistency indeterminacy and normality abstract to manifest human judgments a long established method called pairwise comparison pc has been successfully applied in the analytic hierarchy process ahp in practice human judgments are often made with uncertainty and can be characterized by an interval multiplicative pairwise comparison matrix impcm since consistency is a key issue that has plagued decision makers and researchers for a long time it is useful to propose a transformation that can effectively convert an inconsistent impcm into a consistent one especially in group decision making however a consistent impcm is not sufficient to be acceptable indeterminacy should also be considered moreover the interval priority weights should be normalized to consider consistency indeterminacy and normality simultaneously we put forward a new definition of acceptable impcm to obtain such an acceptable impcm we propose a theorem of consistency a consistent transformation and a normalized prioritization scheme as a result the proposed methods guarantee an inconsistent impcm can be directly converted into an acceptable impcm five theorems are proved to corroborate the proposed methods a numerical example is presented to illustrate the validity and superiority of the proposed methods finally discussion and conclusions are given

a simultaneous projections method for linear inequalities abstract an iterative method for solving general systems of linear inequalities is considered the method a relaxed generalization of cimmino s scheme for solving linear systems was first suggested by censor and elfving each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities the algorithm is particularly suitable for implementation on computers with parallel processors we prove convergence from any starting point for both consistent and nonconsistent systems to a feasible point in the first case and to a weighted least squares type solutions in the second

a chi square method for obtaining a priority vector from multiplicative and fuzzy preference relations abstract decision makers dms preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations this paper proposes a chi square method csm for obtaining a priority vector from multiplicative and fuzzy preference relations the proposed csm can be used to obtain a priority vector from either a multiplicative preference relation i e a pairwise comparison matrix or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures theorems and algorithm about the csm are developed three numerical examples are examined to illustrate the applications of the csm and its advantages

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

parallel projection methods and the resolution of ill posed problems abstract in this paper we consider a modification of the parallel projection method for solving overdetermined nonlinear systems of equations introduced recently by diniz ehrhardt and martinez 1 this method is based on the classical cimmino s algorithm for solving linear systems the components of the function are divided into small blocks as an attempt to correct the intrinsic ill conditioning of the system and the new iteration is a convex combination of the projections onto the linear manifolds defined by different blocks the modification suggested here was motivated by the application of the method to the resolution of a nonlinear fredholm first kind integral equation we prove convergence results and we report numerical experiments

on the asymtotic behavior of some alternate smoothing series expansion iterative methods abstract we study the asymptotic behavior of some iterative methods used in image reconstruction from projections these methods combine in each step a major iteration of a standard series expansion algorithm like art followed by the application of a smoothing matrix we prove convergence for a whole family of these methods when the matrix is symmetric and we give counterexamples for other cases

ranking trapezoidal fuzzy numbers using a parametric relation pair abstract huynh et al introduced a probability based fuzzy relation for comparing fuzzy numbers see v huynh y nakamori and j lawry 2008 40 but they did not detail how to compute it here we will consider this fuzzy relation as a probability based preference intensity index and present closed formulas for the integrals needed to compute this index for fuzzy sets that have trapezoidal membership functions also we will propose an algorithm to compute this index and a numerical method to approximate it the comparison of two fuzzy numbers should also be able to capture the situation where the order of the fuzzy numbers cannot be judged and so their order may be considered as being indifferent here using the probability based preference intensity index we will introduce two crisp relations which have a common parameter over a collection of fuzzy sets that have trapezoidal membership functions next we will show that depending on the parameter value one of them is a strict order relation and the other one may be interpreted as a relation that expresses the order indifference of fuzzy numbers we will call this latter one the order indifference relation lastly we will demonstrate how these two relations can be utilized to rank a collection of fuzzy sets that have trapezoidal membership functions

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

a stochastic ahp decision making methodology for imprecise preferences abstract existing decision making methodologies like the analytic hierarchy process ahp address imprecise pairwise comparisons by modeling crisp pairwise comparisons as fuzzy sets or a single type of probability distribution e g uniform triangular however one common issue faced by decision makers dms is bounded rationality that is dms have limited cognitive powers in specifying their preferences over multiple pairwise comparisons this result to crisp as well as imprecise pairwise comparisons furthermore given the ultimate goal of imprecise ahp is to make the decision computing weights for the criteria and the alternatives from the imprecise preferences is a must hence these various types of pairwise comparisons must be modeled using a single probability distribution for easy computation of the weights in this research a beta distribution is proposed to model the varying stochastic preferences of the dm the method of moments methodology is used to fit the varying stochastic preferences of the dm into beta stochastic pairwise comparisons since it can represent a wide variety of probability distributions additionally a non linear programming model is then developed that calculates weights which maximize the preferences of the dm while maintaining a level of consistency comparison experiments are conducted using datasets collected from literature to validate the proposed methodology

linear and quadratic programming neural network analysis neural networks for linear and quadratic programming are analyzed the network proposed by m p kennedy and l o chua ieee trans circuits syst vol 35 pp 554 562 may 1988 is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems such as the least squares problem for quadratic programming the network converges either to an equilibrium or to an exact solution depending on whether the problem has constraints or not the results also suggest an analytical approach to solve the linear system bx b without calculating the matrix inverse the results are directly applicable to optimization problems with c sup 2 convex objective functions and linear constraints the dynamics and applicability of the networks are demonstrated by simulation the distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter

a parallel subgradient projections method for the convex feasibility problem abstract an iterative method to solve the convex feasibility problem for a finite family of convex sets is presented the algorithm consists in the application of a generalization of an acceleration procedure introduced by de pierro in a parallel version of the subgradient projections method proposed by censor and lent the generated sequence is shown to converge for any starting point some numerical results are presented

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

a new variant of the zhang neural network for solving online time varying linear inequalities since march 2001 a special class of recurrent neural networks termed the zhang neural network znn has been proposed by zhang and co workers for solving online a rich repertoire of time varying problems by extending zhang et al s design formula or say the znn design formula a new variant of the znn design formula is proposed and investigated in this paper which is also based on a matrix vector valued indefinite error function in addition by employing such a novel design formula a new variant of the znn nvznn is proposed developed and investigated for online solution of time varying linear inequalities lis the resultant nvznn models are depicted in implicit dynamics and methodologically exploit the time derivative information of time varying coefficients computer simulation results further demonstrate the novelty efficacy and superiority of the proposed nvznn models for solving online time varying lis

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

models of mathematical programming for intuitionistic multiplicative preference relations in order to capture model uncertainty associated with imprecision or vagueness a decision maker may express her his judgments in terms of intuitionistic multiplicative preference relation impr two important research topics with this regard are studied in the paper 1 checking consistency of impr and 2 generating weights on the basis of this relation a new definition of consistent impr is proposed in light of this new definition the properties of consistent impr are studied in detail a transformation formula is proposed to construct a consistent impr from a given normalized intuitionistic fuzzy weight vector by minimizing the differences between the constructed consistent impr and the given impr some fractional programming models are developed to derive the intuitionistic fuzzy weight vector moreover the models are also extended to group decision making finally two numerical examples are provided to illustrate the effectiveness and practical relevance of the proposed models

rbf nets for approximating an object s boundary by image random sampling abstract our aim is to discuss the problem of approximating the boundary of an object in a binary image in contrast to earlier papers on similar topics we avoid finding contours and then their approximations instead we sample an image in a relatively small number of points 1 3 of pixels at random the collected data is used for imposing constraints on parameters of a radial basis functions rbf neural net it is proved that if the rbf net structure is sufficiently rich to approximate an object boundary then estimates of rbf net parameters tend to their true values as the number of samples approaches to infinity then a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved its implementation which is based on an iterative use of the linear programming is briefly discussed and the results of its testing are shortly reported

a conjugate gradient algorithm for sparse linear inequalities abstract this paper presents a conjugate gradient method for solving systems of linear inequalities the method is of dual optimization type and consists of two phases which can be implemented in a common framework phase 1 either finds the minimum norm solution of the system or detects the inconsistency of the system in the latter event the method proceeds to phase 2 in which an approximate least squares solution to the system is obtained the method is particularly suitable to large scale problems because it preserves the sparsity structure of the problem its efficiency is shown by computational comparisons with an sor type method

a discrete time neural network for optimization problems with hybrid constraints recurrent neural networks have become a prominent tool for optimizations including linear or nonlinear variational inequalities and programming due to its regular mathematical properties and well defined parallel structure this brief presents a general discrete time recurrent network for linear variational inequalities and related optimization problems with hybrid constraints in contrary to the existing discrete time networks this general model can operate not only on bound constraints but also on hybrid constraints comprised of inequality equality and bound constraints the model has dynamical properties of global convergence asymptotical and exponential convergences under some weaker conditions numerical examples demonstrate its efficacy and performance