a family of quasi newton methods for nonlinear equations with direct secant updates of matrix factorizations a new family of least change secant methods for solving systems of nonlinear algebraic equations is introduced at each iteration of the methods of this family the matrix b k is represented as a product of the form a k 1 r k where the matrices a k and r k have a determined structure and a k 1 r k 1 are chosen by the use of a least change secant rule the first and second methods of broyden math comp 19 1965 pp 577 593 math comp 25 1971 pp 285 294 and the methods of schubert math comp 24 1970 pp 27 30 johnson and austria siam j numer anal 20 1983 pp 315 325 and chadee tech report sol 85 8 dept of operations research stanford university stanford ca 1985 are members of this family local and superlinear convergence results are proved and particular members of the family that seem to be of practical usefulness are introduced

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

spectral gradient projection method for solving nonlinear monotone equations an algorithm for solving nonlinear monotone equations is proposed which combines a modified spectral gradient method and projection method this method is shown to be globally convergent to a solution of the system if the nonlinear equations to be solved is monotone and lipschitz continuous an attractive property of the proposed method is that it can be applied to solving nonsmooth equations we also give some preliminary numerical results to show the efficiency of the proposed method

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

a note on extending actions of infinitesimal group schemes abstract we prove that iterative derivations on projective line cannot be expanded to iterative hasse schmidt derivations in the case when the iterativity rule is given by a non algebraic formal group

convergence properties of the bfgs algoritm the bfgs method is one of the most famous quasi newton algorithms for unconstrained optimization in 1984 powell presented an example of a function of two variables that shows that the polak ribiere polyak prp conjugate gradient method and the bfgs quasi newton method may cycle around eight nonstationary points if each line search picks a local minimum that provides a reduction in the objective function in this paper a new technique of choosing parameters is introduced and an example with only six cyclic points is provided it is also noted through the examples that the bfgs method with wolfe line searches need not converge for nonconvex objective functions

a globally convergent newton gmres subspace method for systems of nonlinear equations newton krylov methods are variants of inexact newton methods where the approximate newton direction is taken from a subspace of small dimension here we introduce a new hybrid newton gmres method where a global strategy restricted to a low dimensional subspace generated by gmres is performed the obtained process is consistent with preconditioning and with matrix free implementation computational results indicate that our proposal enhances the classical backtracking inexact method

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

a globally and superlinearly convergent gauss newton based bfgs method for symmetric nonlinear equations in this paper we present a gauss newton based bfgs method for solving symmetric nonlinear equations which contain as a special case an unconstrained optimization problem a saddle point problem and an equality constrained optimization problem a suitable line search is proposed with which the presented bfgs method exhibits an approximate norm descent property under appropriate conditions global convergence and superlinear convergence of the method are established the numerical results show that the proposed method is successful

global convergence of the nonmonotone mbfgs method for nonconvex unconstrained minimization in this paper we propose a new nonmonotone armijo type line search and prove that the mbfgs method proposed by li and fukushima with this new line search converges globally for nonconvex minimization some numerical experiments show that this nonmonotone mbfgs method is efficient for the given test problems

fundamental study the multiple facets of the canonical direct unit implicational basis the notion of dependencies between attributes arises in many areas such as relational databases data analysis data mining formal concept analysis knowledge structures formalization of dependencies leads to the notion of so called full implicational systems or full family of functional dependencies which is in one to one correspondence with the other significant notions of closure operator and of closure system an efficient generation of a full implicational system or a closure system can be performed from equivalent implicational systems and in particular from the bases for such systems for example the so called canonical basis this paper shows the equality between five other bases originating from different works and satisfying various properties in particular they are unit implicational systems the three main properties of this unique basis are the directness canonical and minimal properties whence the name canonical direct unit implicational basis given to this unit implicational system the paper also gives a nice characterization of this canonical basis and makes precise its link with the prime implicants of the horn function associated to a closure operator it concludes that it is necessary to compare more closely related works made independently and with a different terminology in order to take advantage of the really new results in these works

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

an assessment of nonmonotone linesearch techniques for unconstrained optimization the purpose of this paper is to discuss the potential of nonmonotone techniques for enforcing convergence of unconstrained minimization algorithms from starting points distant from the solution linesearch based algorithms are considered for both small and large problems and extensive numerical experiments show that this potential is sometimes considerable a new variant is introduced in order to limit some of the identified drawbacks of the existing techniques this variant is again numerically tested and appears to be competitive finally the impact of preconditioning on the considered methods is examined

on minimal sets of graded attribute implications we explore the structure of non redundant and minimal sets consisting of graded if then rules the rules serve as graded attribute implications in object attribute incidence data and as similarity based functional dependencies in a similarity based generalization of the relational model of data based on our observations we derive a polynomial time algorithm which transforms a given finite set of rules into an equivalent one which has the least size in terms of the number of rules

a modified bfgs type quasi newton method with line search for symmetric nonlinear equations problems abstract in this paper using approximate gradient of the norm square metric function we present an inexact mbfgs method with line search for solving symmetric nonlinear equations which is a generalization of the mbfgs method proposed by li and fukushima 2001 for solving smooth unconstrained optimization the used approximate gradient of the proposed method only requires computing two residuals at every iteration and is easy to obtain by utilizing symmetric structure of the system we establish global convergence of the proposed method and report some numerical results

canonical dichotomous direct bases closure systems are usually characterized in terms of implications the directness property of implicational systems is a key issue in their computational usability in this work we focus on this property studying its connection with the structure of implicational systems and the design of methods for transforming any implicational system into an equivalent direct implicational system we introduce a new paradigm based on the bipartition of the implicational sets into two components according to their behavior wrt the closure in addition we present the notions of two new direct bases named dd basis and canonical dd basis also providing two methods to compute each of them the advantages of the dichotomous approach will be shown both from the theoretical and empirical points of view

the bfgs method with exact line searches fails for non convex objective functions this work shows that the bfgs method and other methods in the broyden class with exact line searches may fail for non convex objective functions

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

a globally convergent bfgs method with nonmonotone line search for non convex minimization in this paper we propose a modified bfgs broyden fletcher goldfarb shanno method with nonmonotone line search for unconstrained optimization under some mild conditions we show that the method is globally convergent without a convexity assumption on the objective function we also report some preliminary numerical results to show the efficiency of the proposed method

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

comments on a family of derivative free conjugate gradient methods for large scale nonlinear systems of equations j comput appl math 224 2009 11 19 in this note a simple counter example shows that the proof of lemma 3 3 in 1 w cheng y xiao and q hu a family of derivative free conjugate gradient methods for large scale nonlinear systems of equations j comput appl math 224 2009 11 19 is not correct which implies that lemma 3 2 in 1 is not enough to ensure lemma 3 3 in 1 a new proof is given which leads to a stronger result than lemma 3 2 in 1 and this result not only guarantees that lemma 3 3 in 1 holds but also improves the corresponding global convergence theorem 3 1 in 1

a nonmonotone line search technique and its application to unconstrained optimization a new nonmonotone line search algorithm is proposed and analyzed in our scheme we require that an average of the successive function values decreases while the traditional nonmonotone approach of grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 requires that a maximum of recent function values decreases we prove global convergence for nonconvex smooth functions and r linear convergence for strongly convex functions for the l bfgs method and the unconstrained optimization problems in the cute library the new nonmonotone line search algorithm used fewer function and gradient evaluations on average than either the monotone or the traditional nonmonotone scheme

splittings in gbl algebras i the general case abstract we study and characterize splitting algebras in varieties of integral residuated semi lattices the main result is a complete characterization of the splitting algebras in the variety of gbl e w algebras i e integral bounded commutative and divisible residuated lattices

a family of derivative free conjugate gradient methods for large scale nonlinear systems of equations in this paper we propose a family of derivative free conjugate gradient methods for large scale nonlinear systems of equations they come from two modified conjugate gradient methods w y cheng a two term prp based descent method numer funct anal optim 28 2007 1217 1230 l zhang w j zhou d h li a descent modified polak ribiere polyak conjugate gradient method and its global convergence ima j numer anal 26 2006 629 640 recently proposed for unconstrained optimization problems under appropriate conditions the global convergence of the proposed method is established preliminary numerical results show that the proposed method is promising

a modified bfgs method and its global convergence in nonconvex minimization in this paper we propose a modification of the bfgs method for unconstrained optimization a remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function under certain conditions we also establish superlinear convergence of the method

generalized hasse schmidt varieties and their jet spaces building on the abstract notion of prolongation developed in 10 the theory of iterative hasse schmidt rings and schemes is introduced simultaneously generalising difference and hasse schmidt differential rings and schemes this work provides a unified formalism for studying difference and differential algebraic geometry as well as other related geometries as an application hasse schmidt jet spaces are constructed generally allowing the development of the theory for arbitrary systems of algebraic partial difference differential equations where constructions by earlier authors applied only to the finite dimensional case in particular it is shown that under appropriate separability assumptions a hasse schmidt variety is determined by its jet spaces at a point