error bounds for 2 regular mappings with lipschitzian derivatives and their applications we obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature specifically we assume that the constraints mapping has a lipschitzian derivative and satisfies a certain 2 regularity condition at the point under consideration this setting directly subsumes the classical regular case and the twice differentiable 2 regular case for which error bounds are known but it is significantly richer than either of these two cases when applied to a certain equation based reformulation of the nonlinear complementarity problem our results yield an error bound under an assumption more general than b regularity the latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available we also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems

newton type methods for optimization problems without constraint qualifications we consider equality constrained optimization problems where a given solution may not satisfy any constraint qualification but satisfies the standard second order sufficient condition for optimality based on local identification of the rank of the constraints degeneracy via the singular value decomposition we derive a modified primal dual optimality system whose solution is locally unique nondegenerate and thus can be found by standard newton type techniques using identification of active constraints we further extend our approach to mixed equality and inequality constrained problems and to mathematical programs with complementarity constraints mpcc in particular for mpcc we obtain a local algorithm with quadratic convergence under the second order sufficient condition only without any constraint qualifications not even the special mpcc constraint qualifications

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

an accelerated newton method for equations with semismooth jacobians and nonlinear complementarity problems we discuss local convergence of newton s method to a singular solution x of the nonlinear equations f x 0 for f mathbb r n rightarrow mathbb r n it is shown that an existing proof of griewank concerning linear convergence to a singular solution x from a starlike domain around x for f twice lipschitz continuously differentiable and x satisfying a particular regularity condition can be adapted to the case in which f is only strongly semismooth at the solution further newton s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second newton step these results are applied to a nonlinear equations reformulation of the nonlinear complementarity problem ncp whose derivative is strongly semismooth when the function f arising in the ncp is sufficiently smooth conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinear equations reformulation of the ncp at x

a linear programming approximation to the eigenvector method in the analytic hierarchy process eigenvector method em is a well known approach to deriving priorities from pairwise comparison matrices in the analytic hierarchy process ahp which requires the solution of a set of nonlinear eigenvalue equations this paper proposes an approximate solution approach to the em to facilitate its computation we refer to the approach as a linear programming approximation to the em or lpaem for short as the name implies the lpaem simplifies the nonlinear eigenvalue equations as a linear programming for solution it produces true weights for perfectly consistent pairwise comparison matrices numerical examples are examined to show the validity and effectiveness of the proposed lpaem and its significant advantages over a recently developed linear programming method entitled lp gw ahp in rank preservation

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

analytic network process for pattern classification problems using genetic algorithms the analytic network process anp is a useful technique for multi attribute decision analysis mcda that employs a network representation to describe interrelationships between diverse attributes owing to effectiveness of the anp in allowing for complex interrelationships between attributes this paper develops an anp based classifier for pattern classification problems with interdependence or independence among attributes to deal with interdependence this study employs genetic algorithms gas to automatically determine elements in the supermatrix that are not easily user specified to find degrees of importance of respective attributes then with the relative importance for each attribute in the limiting supermatrix the current work determines the class label of a pattern by its synthetic evaluation experimental results obtained by the proposed anp based classifier are comparable to those obtained by other fuzzy or non fuzzy classification methods

groups where free subgroups are abundant given an infinite topological group g and a cardinal 0 we say that g is almost free if the set of tuples gi i g which freely generate free subgroups of g is dense in g in this note we examine groups having this property and construct examples for instance we show that if g is a non discrete hausdorff topological group that contains a dense free subgroup of rank 0 then g is almost free a consequence of this is that for any infinite set the group of all permutations of is almost 2 free we also show that an infinite topological group is almost 0 free if and only if it is almost n free for each positive integer n this generalizes the work of dixon and gartside and knight

aggregation of direct and indirect judgments in pairwise comparison matrices with a re examination of the criticisms by bana e costa and vansnick in a recent paper by bana e costa and vansnick c a bana e costa j c vansnick a critical analysis of the eigenvalue method used to derive priorities in ahp european journal of operational research 187 3 2008 1422 1428 analytic hierarchy process ahp particularly its eigenvector method em used for deriving priorities from pairwise comparison matrices was criticized for the violation of a so called condition of order preservation cop due to this violation the em was considered to have a serious fundamental weakness which makes the use of ahp as a decision support tool very problematic the consistency ratio cr index in the ahp was also criticized for its failure to act as an alert of this violation of cop in this paper we look into decision makers overall judgments which can be obtained through the aggregation of their direct and indirect judgments and then re examine bana e costa and vansnick s numerical examples with a detailed analysis to show the invalidity of their criticisms

a new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process abstract the deahp method for weight deviation and aggregation in the analytic hierarchy process ahp has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices which makes its application very restrictive this paper proposes a new data envelopment analysis dea method for priority determination in the ahp and extends it to the group ahp situation in this new dea methodology two specially constructed dea models that differ from the deahp model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent the new dea method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers dms subjective judgments for inconsistent pairwise comparison matrices in hierarchical structures the new dea method utilizes the simple additive weighting saw method for aggregation of the best local priorities without the need of normalization numerical examples are examined throughout the paper to show the advantages of the new dea methodology and its potential applications in both the ahp and group decision making

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

consistency analysis of triangular fuzzy reciprocal preference relations abstract in order to simulate the uncertainty associated with impression or vagueness a decision maker may give her his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making the study of their consistency becomes a very important aspect to avoid a misleading solution based on the reciprocity property this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations the new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by buckley 1985 the properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail in addition the shortcomings of the proof procedure of the proposition given by wang and chen 2008 are pointed out and the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations finally using the n 1 restricted comparison ratios a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed and an algorithm is shown to make a consistent decision ranking numerical results are further calculated to illustrate the new definition and the obtained algorithm

automorphisms of relatively free nilpotent groups of infinite rank let f be a free group and v a characteristic subgroup of f then the natural homomorphism from f to f v gives rise to a homomorphism 1 aut f aut f i from the automorphism group of f to the automorphism group of f v in this paper we shall prove that x is surjective provided that f has infinite rank and f v is nilpotent

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

dealing with singularities in nonlinear unconstrained optimization we propose a new trust region based optimization algorithm for solving unconstrained nonlinear problems whose second derivatives matrix is singular at a local solution we give a theoretical characterization of the singularity in this context and we propose an iterative procedure which allows to identify a singularity in the objective function during the course of the optimization algorithm and artificially adds curvature to the objective function numerical tests are performed on a set of unconstrained nonlinear problems both singular and non singular results illustrate the significant performance improvement compared to classical trust region and filter algorithms proposed in the literature the approach is also shown to be competitive with tensor methods in terms of efficiency while reaching a higher level of robustness

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

the theory of 2 regularity for mappings with lipschitzian derivatives and its applications to optimality conditions we study local structure of a nonlinear mapping near points where standard regularity and or smoothness assumptions need not be satisfied we introduce a new concept of 2 regularity a certain kind of second order regularity for a once differentiable mapping whose derivative is lipschitz continuous under this 2 regularity condition we obtain the representation theorem and the covering theorem i e stability with respect to right hand side perturbations under assumptions that are weaker than those previously employed in the literature for results of this type these results are further used to derive a constructive description of the tangent cone to a set defined by 2 regular equality constraints and optimality conditions for related optimization problems the class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation based reformulation optimality conditions for mathematical programs with equivalently reformulated complementarity constraints are also discussed

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

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

reconstructing the topology of clones function clones are sets of functions on a fixed domain that are closed under composition and contain the projections they carry a natural algebraic structure provided by the laws of composition which hold in them as well as a natural topological structure provided by the topology of pointwise convergence under which composition of functions becomes continuous inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems we study questions of the following type in which situations does the algebraic structure determine the topological structure of a functional clone we pay particular attention to function clones which contain an oligomorphic permutation group and discuss applications of this situation in model theory and theoretical computer science

the small index property for infinite dimensional classical groups we show that the countable dimensional analogues of the finite classical groups have the strong small index property that is if g is one of these groups and v is its natural module and if h is a subgroup of g of index less than 2 then there exists a finite dimensional subspace x of v such that h is sandwiched between the pointwise and setwise stabilisers in g of x