p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

triangular and parametric approximations of fuzzy numbers inadvertences and corrections recent papers were dedicated to approximate fuzzy numbers by triangular trapezoidal or parametric fuzzy numbers with or without additional conditions unfortunately the result of approximation is not always a fuzzy number sometimes it is not a fuzzy set we point out the wrongs and inadvertences in some recent papers then we correct the results

aperiodic interactions on hierarchical lattices an exact criterion for the potts ferromagnet criticality we discuss the critical behaviour of the q state potts model on a diamond like hierarchical lattice with ferromagnetic interactions according to an aperiodic two letter substitutional sequence we show that the geometric deterministic fluctuations become relevant for where is the wandering exponent of the sequence d is the fractal dimension of the lattice and is the critical exponent associated with the specific heat of the uniform model we also point out that the criteria for analysing the relevance of deterministic and random fluctuations are generically different

study of the iterations of a mapping associated to a spin glass model we study the iterations of the mapping mathcal n f s frac f s 2 f 0 2 s f 0 2 with the constraintsf 1 1 f s ansn an 0 and find that except iff s s n f s approaches either 0 or 1 for s 1 ask

high order infeasible interior point methods for solving sufficient linear complementarity problems in this paper we develop systematically infeasible interior point methods of arbitrarily high order for solving horizontal linear complementarity problems that are sufficient in the sense ofcottle pang and venkateswaran 1989 the results apply to degenerate problems and problem shaving no strictly complementary solution variants of these methods are described that eventually avoid recentering steps and for which all components of the approximate solutions converge superlinearly at a high order and other variants which even terminate with a solution of the complementarity problem after finitely many steps

on the uniqueness of solutions to linear complementarity problems this paper characterizes the classu of all realn n matricesm for which the linear complementarity problem q m has a unique solution for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all it is shown that restricting the uniqueness property to the interior ofk m is necessary because whenu the problem q m has infinitely many solutions ifq belongs to the boundary of intk m it is shown thatm must have nonnegative principal minors whenu andk m is convex finally it is shown that whenm has nonnegative principal minors only one of which is 0 andk m r n thenu andk m is a closed half space

basis and partition identification for quadratic programming and linear complementarity problems optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions maximally complementary solutions can be characterized by optimal partitions on the other hand the solutions provided by simplex based pivot algorithms are given in terms of complementary bases a basis identification algorithm is an algorithm which generates a complementary basis starting from any complementary solution a partition identification algorithm is an algorithm which generates a maximally complementary solution and its corresponding partition starting from any complementary solution in linear programming such algorithms were respectively proposed by megiddo in 1991 and balinski and tucker in 1969 in this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices the presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus

real vector space of lr fuzzy intervals with respect to the shape preserving t norm based addition the construction of the membership function of fuzzy intervals is an important problem in vagueness modeling the preservation of the shape of fuzzy sets during the addition is a natural requirement in practical computation the lr fuzzy intervals introduced by dubois and prade satisfy this requirement if the addition is based on the nilpotent t norm generated by l or r shape functions the shortcoming that not any lr fuzzy interval has an opposite inverse related to shape preserving t norm based addition can be solved if the set of lr fuzzy intervals is isomorphically included in an extended set and this extended set forms a group with respect to shape preserving t norm based addition in this paper we construct the extended set of these lr fuzzy intervals we also show that the extended set is a real vector space with scalar product and the modal intervals can be considered as the elements of this extended set finally we present the algebraic form of lr fuzzy intervals and the associated application

a class of linear complementarity problems solvable in polynomial time we describe a condition number for the linear complementarity problem lcp which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used consequently we develop a class of lcps solvable in polynomial time the result suggests that the convexity or positive semidefiniteness of the lcp may not be the basic issue that separates lcps solvable and not solvable in polynomial time

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming this paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming sdp under the assumptions that the semidefinite program has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero the interior point algorithm considered here closely resembles the mizuno todd ye predictor corrector method for linear programming which is known to be quadratically convergent it is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 1 2 r the proof relies on a careful analysis of the central path for sdp it is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

differential migdal kadanoff renormalization group for disordered systems phase transitions in quenched random systems are studied within the migdal kadanoff approximation exact differential recursion relations for the distribution of coupling are derived the q state potts model with random ferromagnetic couplings is studied in two and three dimensions in three dimensions there is a non perturbative random fixed point which exists when the specific heat exponent is either positive or slightly negative if the specific heat exponent is slightly negative a multicritical point separates strong and weak disorder regimes the analytic results are confirmed by numerical simulations of the recursion relations

analysis of infeasible interior point paths arising with semidefinite linear complementarity problems we consider semidefinite monotone linear complementarity problems sdlcp in the space equation here n of real symmetric n n matrices equipped with the cone equation here n of all symmetric positive semidefinite matrices one may define weighted using any m equation here n as weight infeasible interior point paths by replacing the standard condition xy ri r 0 that defines the usual central path by xy yx 2 rm under some mild assumptions the most stringent is the existence of some strictly complementary solution of sdlcp these paths have a limit as r 0 and they depend analytically on all path parameters such as r and m even at the limit point r 0

a projection algorithm for general variational inequalities with perturbed constraint sets we consider the problem of general variational inequalities gvi with nonmonotone operator in a finite dimensional space we propose a method to solve gvi that at each iteration considers only one projection on an easy approximation of the constraint set which is important from a practical point of view we analyse the convergence of the algorithm under a weak cocoercivity condition using variational metric analysis computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case

criteria for sufficient matrices abstract column sufficient row sufficient and sufficient matrices have recently arisen in connection with the linear complementarity problem we review and supplement the basic theory of these matrix classes propose new criteria for identifying them and compare these criteria with the existing ones our main mathematical tool is principal pivoting

new characterizations of row sufficient matrices in dealing with a linear complementarity problem much depends on knowing that the matrix through which the particular lcp is defined belongs to a suitable matrix class two such classes are su the so called sufficient matrices and l which were introduced in r w cottle j s pang v venkateswaran sufficient matrices and the linear complementarity problem linear algebra appl 114 115 1989 231 249 b c eaves the linear complementarity problem manage sci 17 1971 612 634 respectively in an earlier article i adler r w cottle s verma sufficient matrices belong to l math prog 106 2006 391 401 the authors proved that su is a subclass of l by definition the class su is the intersection of two distinct classes rsu the row sufficient matrices and csu the column sufficient matrices in the present work we strengthen the aforementioned inclusion by showing that all row sufficient matrices belong to l using what we call structural properties of certain matrix classes we add to the existing characterizations of rsu in r w cottle s m guu two characterizations of sufficient matrices linear algebra appl 170 1992 65 74 s m guu r w cottle on a subclass of p0 linear algebra appl 223 224 1995 325 335 h valiaho criteria for sufficient matrices linear algebra appl 233 1996 109 129 this line of inquiry was inspired by asking what must be true of a row sufficient l matrix we establish three new characterizations of rsu in terms of the matrix classes l e0 and q0 and the structural properties of sign change invariance completeness and fullness the new characterizations of rsu provide new characterizations of su by adjoining a fourth structural property we call reflectiveness

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

a constructive characterization of q 0 matrices with nonnegative principal minors in 2 we characterized the class of matrices with nonnegative principla minors for which the linear complementarity problem always has a solution that class is contained in the one we study here our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable in short we constructively characterize the matrix class known asqo po

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

on a generalization of a normal map and equation the class of normal maps was recently investigated by robinson and ralph in connection with the study of a variational inequality defined on a polyhedral set in this paper a generalization of such a map is considered and the associated generalized normal equation is studied the latter provides a unified formulation of several generalized variational inequality and complementarity problems using degree theory some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed specializations of the results to various applications are discussed

sufficient matrices belong to l in this paper we establish a significant matrix class inclusion that seems to have been overlooked in the literature of the linear complementarity problem we show that p the class of sufficient matrices is a subclass of l in the course of demonstrating this inclusion we introduce other new matrix classes that forge interesting new connections between known matrix classes

complementarity properties of singular m matrices abstract for a matrix a whose off diagonal entries are nonpositive its nonnegative invertibility namely that a is an invertible m matrix is equivalent to a being a p matrix which is necessary and sufficient for the unique solvability of the linear complementarity problem defined by a this in turn is equivalent to the statement that a is strictly semimonotone in this paper an analogue of this result is proved for singular symmetric z matrices this is achieved by replacing the inverse of a by the group generalized inverse and by introducing the matrix classes of strictly range semimonotonicity and range column sufficiency a recently proposed idea of p matrices plays a pivotal role some interconnections between these matrix classes are also obtained

existence uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition abstract the main aim of this paper is to characterize the set of real parameters associated to a fuzzy number represented in a general form which include the most important characteristics with the following property for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter the uniqueness of the nearest trapezoidal fuzzy number with this property is proved the average euclidean distance being considered as an important property each resulting trapezoidal approximation operator is continuous the main results are illustrated by examples

a solution condition for complementarity problems with an application to spatial price equilibrium a sufficient condition is developed for the existence of solutions to complementarity problems which is applicable to all continuous functions in addition it is shown to imply a number of simpler conditions including the well known coerciveness condition finally this condition is applied to an existence problem in spatial price equilibrium where it is shown to yield not only a more general result but also a simpler method of proof

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

approximations by lr type fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated furthermore approximations in euclidean distance class can be divided into two subclasses one is to study approximations of fuzzy numbers without constraints the other one is to study approximations preserving some attributes in this paper we use lr type fuzzy numbers to approximate fuzzy numbers the proposed approximations will generalize all recent approximations without constraints in euclidean class also an efficient formula is provided

on singular n0 matrices and the class q abstract the class of n and n 0 matrices arises in the theory of global univalence of functions multivariate analysis and in complementarity theory in this paper we study singular n 0 matrices with q property

on hidden z matrices and the linear complementarity problem abstract in this paper we explore various matrix theoretic aspects of the hidden z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results in fact we revisit various results obtained for hidden z class by mangasarian cottle and pang in context of solving linear complementarity problems as linear programs we identify hidden z matrices of special category and discuss the number of solutions of the associated linear complementarity problems we also present game theoretic interpretation of various results related to hidden z class and obtain proofs following the game theoretic approach of raghavan for a subclass of z matrices

approximation of fuzzy numbers using the convolution method in this study we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number we show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non differentiable points in previous studies this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers the possible non differentiable points of which were the two endpoints of the 1 cut the construction of smoothers is a key step in the process for producing approximations we also show that if appropriate smoothers are selected then we can use the convolution method to provide approximations that are differentiable lipschitz and that also preserve the core

trapezoidal and triangular approximations preserving the expected interval in this paper an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied which is proposed by grzegorzewski and mrowka a formula for computing the improved approximation is presented moreover the nearest triangular approximation operator preserving the expected interval is also investigated

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

on the transpose of a pseudomonotone matrix and the lcp abstract in the study of linear complementary problem it is known that pseudomonotone matrices belong to the class p 0 q 0 in this note we show that under certain conditions such as invertibility or normality the transpose of a pseudomonotone matrix belongs to the class q 0

d orientation sequences for continuous functions and nonlinear complementarity problems we introduce the new concept of d orientation sequence for continuous functions it is shown that if there does not exist a d orientation sequence for a continuous function then the corresponding complementarity problem cp has a solution we believe that such a result characterizes an intrinsic property of cps as the concept of exceptional family of elements the notion of d orientation sequence of a function is also a powerful tool for investigating the existence theorems of cps we use this new tool to establish among other things a new existence result for a class of p mapping cps

on the complexity of computing the handicap of a sufficient matrix the class of sufficient matrices is important in the study of the linear complementarity problem lcp some interior point methods ipm s for lcp s with sufficient data matrices have complexity polynomial in the bit size of the matrix and its handicap in this paper we show that the handicap of a sufficient matrix may be exponential in its bit size implying that the known complexity bounds of interior point methods are not polynomial in the input size of the lcp problem we also introduce a semidefinite programming based heuristic that provides a finite upper bond on the handicap for the sub class of mathcal p matrices where all principal minors are positive

determining the handicap of a sufficient matrix abstract any linear complementarity problem with a sufficient matrix can be solved by means of the unified interior point method the complexity bound of the method is the better the smaller the so called handicap of the matrix is we propose a method for determining the handicap of a sufficient matrix and show that a sufficient matrix and its transpose have the same handicap

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

the principal pivoting method revisited the principal pivoting method ppm for the linear complementarity problem lcp is shown to be applicable to the class of lcps involving the newly identified class of sufficient matrices

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

more on positive subdefinite matrices and the linear complementarity problem abstract in this article we consider positive subdefinite matrices psbd recently studied by j p crouzeix et al siam j matrix anal appl 22 2000 66 and show that linear complementarity problems with psbd matrices of rank 2 are processable by lemke s algorithm and that a psbd matrix of rank 2 belongs to the class of sufficient matrices introduced by r w cottle et al linear algebra appl 114 115 1989 231 we also show that if a matrix a is a sum of a merely positive subdefinite copositive plus matrix and a copositive matrix and a feasibility condition is satisfied then lemke s algorithm solves lcp q a this supplements the results of jones and evers

trapezoidal approximations of fuzzy numbers preserving the expected interval algorithms and properties fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed it is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number like its ambiguity width its value and weighted expected value

best lipschitz constant of the trapezoidal approximation operator preserving the expected interval in this paper we prove new distance properties between a fuzzy number and its trapezoidal approximation preserving the expected interval then we find the best lipschitz constant of the trapezoidal approximation operator preserving the expected interval finally we use this result in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the expected interval in the case when the direct formula leads to complex calculations

two characterizations of sufficient matrices abstract two characterizations are given for the class of sufficient matrices defined by cottle pang and venkateswaran the first is a direct translation of the definition into linear programming terms the second can be thought of as a generalization of a theorem of t d parsons on p matrices

metrics and orders in space of fuzzy numbers abstract two classes of metrics p q and p where 1 p 0 q 1 in space of fuzzy numbers are introduced then a method of ranking fuzzy numbers based on these metrics is proposed and investigated not only fuzzy numbers but also the left sided and right sided fuzzy numbers are considered applications in making decisions and statistics are suggested

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

discontinuity of the trapezoidal fuzzy number valued operators preserving core we prove that any trapezoidal fuzzy number valued operator preserving core is discontinuous with respect to any weighted metric on the space of fuzzy numbers as an application we obtain the discontinuity of the weighted trapezoidal approximation operator preserving core

on the harris criterion for hierarchical lattices the authors discuss the condition necessary for the harris criteria to be valid on hierarchical lattices they prove that disorder is always relevant when the specific heat exponent alpha p is positive and show how to construct lattices for which disorder becomes relevant while alpha p is still negative

a fully polynomial time approximation algorithm for computing a stationary point of the general linear complementarity problem we apply a potential reduction algorithm to solve the general linear complementarity problem glcp minimize xty subject to ax by cz q and x y z 0 we show that the algorithm is a fully polynomial time approximation scheme fptas for computing an e approximate stationary point of the glcp note that there are some glcps in which every stationary point is a solution xty 0 these include the lcps with row sufficient matrices we also show that the algorithm is a polynomial time algorithm for a special class of glcps

existence of interval triangular and trapezoidal approximations of fuzzy numbers under a general condition abstract recently many scholars studied approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators in fact these approximations may not exist for some linear operators the purpose of this paper is to study necessary and sufficient conditions of linear operators which are preserved by interval triangular symmetric triangular trapezoidal or symmetric trapezoidal approximations of fuzzy numbers in addition an effective method for solving such problems is proposed

on the equivalence of nonlinear complementarity problems and least element problems strictly pseudomonotonez maps operating on banach lattices are considered equivalence of complementarity problems and least element problems is established under certain regularity and growth conditions this extends a recent result by riddell 1981 for strictly monotonez maps to the pseudomonotone case some other problems equivalent to the above are discussed as well

notes on sufficient matrices abstract we give some constructive characterizations for sufficient matrices and present some matrix classes whose elements are identified as column sufficient matrices in particular we give the structure of column row sufficient matrices which leads to efficient testing methods for column row sufficiency

on a subclass of p0 abstract in 1966 fiedler and ptak wrote the first systematic investigation of the matrix class p 0 consisting of all real square matrices with nonnegative principal minors in this note we focus on a particular subclass of p 0 that has arisen within the field of linear complementarity theory namely the class su of sufficient matrices our principal result is that su contains another class p we conjecture that the latter two classes are in fact the same

total dual integrality and integral solutions of the linear complementarity problem abstract this paper deals with the problem of finding an integer solution to a linear complementarity problem lcp chandrasekaran et al 1 introduced the class i of integral matrices for which the corresponding lcp has an integer solution for every integral vector q for which it has a solution and proved that for some well known matrix classes principal unimodularity forms a necessary and sufficient condition for inclusion in the class i in this paper we identify some more well known matrix classes for which principal unimodularity forms a necessary and sufficient condition for inclusion in the class i the concept of total dual integrality is utilized to obtain a necessary and sufficient condition for existence of an integer solution to lcp with a hidden k matrix we interconnect the concept of hilbert basis with principal unimodularity of a matrix and the corresponding complementary cones a necessary and sufficient condition is given for the existence of an integer solution of a linear fractional programming problem by using its lcp formulation

on the classes of fully copositive and fully semimonotone matrices abstract in this paper we consider the class c 0 f of fully copositive and the class e 0 f of fully semimonotone matrices we show that c 0 f matrices with positive diagonal entries are column sufficient we settle a conjecture made by murthy and parthasarathy to the effect that a c 0 f q 0 matrix is positive semidefinite by providing a counterexample we finally consider e 0 f matrices introduced by cottle and stone math program 27 1983 191 213 and partially address stone s conjecture to the effect that e 0 f q 0 p 0 by showing that e 0 f d c matrices are p 0 where d c is the doverspike class of matrices

nearest piecewise linear approximation of fuzzy numbers abstract the problem of the nearest approximation of fuzzy numbers by piecewise linear 1 knot fuzzy numbers is discussed by using 1 knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study they might be also perceived as a generalization of the trapezoidal approximations moreover these approximations possess some desirable properties apart from theoretical considerations approximation algorithms that can be applied in practice are also given

weighted trapezoidal approximation preserving cores of a fuzzy number recently various researchers have proved that approximations of fuzzy numbers may fail to be fuzzy numbers in this contribution we suggest a new weighted trapezoidal approximation of an arbitrary fuzzy number which preserves its cores we prove that the weighted trapezoidal approximation preserving core is always a fuzzy number we then discuss properties of the approximation strategy including translation invariance scale invariance identity and continuity the advantage is that the method presented here is simpler than other methods computationally

approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value value and ambiguity are two parameters which were introduced to represent fuzzy numbers in this paper we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number with respect to the average euclidean distance preserving the value and ambiguity to avoid the laborious calculus associated with the karush kuhn tucker theorem the working tool in some recent papers a less sophisticated method is proposed algorithms for computing the approximations many examples proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given

pc matrices and the linear complementarity problem abstract we introduce a new matrix class pc which consists of those matrices m for which the solution set of the corresponding linear complementarity problem is connected for every q rn we consider lemke s pivotal method from the perspective of piecewise linear homotopies and normal maps and show that lemke s method processes all matrices in pc q0 we further investigate the relationship of the class pc to other known matrix classes and show that column sufficient matrices are a subclass of pc as are 2 2 p0 matrices

on the nearest parametric approximation of a fuzzy number many nearest parametric approximation methods of fuzzy sets are proposed in the literature it is clear that the specific approximations may lead to the loss of information about fuzziness to overcome this problem most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation but these approximations mostly are not flexible to the decision maker s choice hence in this paper we offer a parametric fuzzy approximation method based on the decision maker s strategy as an extension of trapezoidal approximation of a fuzzy number this method comprises the selection of the form of the parametric membership function and its evaluation

the nearest approximation of a fuzzy quantity in parametric form applications of fuzzy logic and fuzzy mathematics are increasing widely all around the world thus working with fuzzy numbers are very important in many applications of fuzzy mathematics we need or it is better to work with the same fuzzy numbers in this work we approximate parametric fuzzy numbers with polynomial parametric fuzzy numbers

the linear complimentarity problem sufficient matrices and the criss cross method specially structured linear complementarity problems lcps and their solution by the criss cross method are examined the criss cross method is known to be finite for lcps with positive semidefinite bisymmetric matrices and with p matrices it is also a simple finite algorithm for oriented matroid programming problems recently cottle pang and venkateswaran identified the class of column row sufficient matrices they showed that sufficient matrices are a common generalization of p and psd matrices cottle also showed that the principal pivoting method with a clever modification can be applied to row sufficient lcps in this paper the finiteness of the criss cross method for sufficient lcps is proved further it is shown that a matrix is sufficient if and only if the criss cross method processes all the lcps defined by this matrix and all the lcps defined by the transpose of this matrix and any parameter vector

weighted semi trapezoidal approximations of fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated in the present paper we study a special class of non linear approximations with respect to a weighted euclidean distance we call it weighted semi trapezoidal approximations the proposed approximations generalize all recent approximations in euclidean distance class first we embed fuzzy numbers into a hilbert space then compute weighted semi trapezoidal approximations by means of best approximations in a closed convex subset of the hilbert space finally we propose formulas of matrix type which is more clear than the previous contributions

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

exceptional families and finite dimensional variational inequalities over polyhedral convex sets a new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper it generalizes the concepts for complementarity problem introduced by smith and isac we applied the new analytical tool to the study of existence problem for variational inequalities it is shown that our existence condition is weaker than most of the sufficient conditions which have been known

the nearest trapezoidal fuzzy number to a fuzzy quantity in this paper we introduce a fuzzy trapezoidal approximation using the metric distance between two fuzzy numbers an application of this approximation is also investigated

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

on the nearest parametric approximation of a fuzzy number revisited the problem of finding the nearest parametric approximation of a fuzzy number with respect to the average euclidean distance is completely solved properties of translation invariance scale invariance additivity preservation of expected value value and ambiguity of this new approximation operator are studied and an algorithm for computing it is provided

lipschitz functions and fuzzy number approximations we prove that some important properties of convex subsets of vector topological spaces remain valid in the space of fuzzy numbers endowed with the euclidean distance we use these results to obtain a characterization of fuzzy number valued lipschitz functions this fact helps us to find the best lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity introduced in a recent paper finally applications in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the value and ambiguity in the case when the direct formula is not applicable and an estimation for the defect of additivity of the trapezoidal approximation preserving the value and ambiguity are given

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

note on trapezoidal approximation of fuzzy numbers we show by two examples that trapezoidal approximation of fuzzy number a in grzegorzewski mrowka fuzzy sets and systems 153 2005 115 135 are not always trapezoidal

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers