global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

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

nonmonotone backtracking inexact quasi newton algorithms for solving smooth nonlinear equations a family of inexact quasi newton algorithms in association with nonmonotone backtracking line search technique is proposed for solving smooth nonlinear equations global convergence of the proposed algorithms are established under the reasonable conditions we characterize the order of local convergence based on convergence behaviour of the approximate matrix of the jacobian and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithms

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 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

new quasi newton methods for unconstrained optimization problems abstract many methods for solving minimization problems are variants of newton method which requires the specification of the hessian matrix of second derivatives quasi newton methods are intended for the situation where the hessian is expensive or difficult to calculate quasi newton methods use only first derivatives to build an approximate hessian over a number of iterations this approximation is updated each iteration by a matrix of low rank in unconstrained minimization the original quasi newton equation is bk 1sk yk where yk is the difference of the gradients at the last two iterates in this paper we first propose a new quasi newton equation b k 1 s k y k in which y k is decided by the sum of yk and aksk where ak is some matrix then we give two choices of ak which carry some second order information from the hessian of the objective function the three corresponding bfgs type algorithms are proved to possess global convergence property the superlinear convergence of the one algorithm is proved extensive numerical experiments have been conducted which show that the proposed algorithms are very encouraging

pseudo monotone mappings in sobolev orlicz spaces and nonlinear boundary value problems on unbounded domains where r is a domain not necessarily bounded in r in the case where the a have at most polynomial growth in u and its derivatives this problem can be studied in the context of the mappings of monotone type from a sobolev space into its conjugate when the functions a even the top order terms do not obey any polynomial growth it is customary to use a sobolev orlicz space rather than a sobolev space although at the cost of some degree of complexity in the arguments caused mainly by the fact that orlicz spaces are not reflexive in general we shall be dealing here with the latter case the concept of a pseudo monotone operator was introduced to take into account the difference of the properties of the highest order functions a in their top order variables from the lower order derivatives when the domain r is bounded and a polynomial growth condition is satisfied it is a classical result to derive the pseudo monotonicity of the mapping t induced by the operator a in a natural manner from a relatively simple set of hypotheses leray lions conditions on the functions a 3 131 an analogous conclusion for the case of sobolev orlicz spaces was also shown to be true 161 for unbounded domains r other methods were developed to obtain some

a new backtracking inexact bfgs method for symmetric nonlinear equations a bfgs method in association with a new backtracking line search technique is presented for solving symmetric nonlinear equations the global and superlinear convergences of the given method are established under mild conditions preliminary numerical results show that the proposed method is better than the normal technique for the given problems

particle swarm algorithm for solving systems of nonlinear equations solving systems of nonlinear equations is one of the most difficult problems in all of numerical computation and in a diverse range of engineering applications newton s method for solving systems of nonlinear equations can be highly sensitive to the initial guess of the solution in this study a new particle swarm optimization algorithm is proposed to solve systems of nonlinear equations some standard systems are presented to demonstrate the efficiency of this method

weighted lorentz spaces and the hardy operator abstract we find a new expression for the norm of a function in the weighted loreritz space with respect to the distribution funetion and obtain as a simple consequence a generalization of the classical embeddings lp 1 lp lp and a new definition of the weak space p u w we also give some applications to the boundedness of the hardy operator s x0 from p0u0 w0 into p1u1 w1 with 0

chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations abstract this study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm several benchmark studies are carried out logistic map gives the best results and is utilized in solving nonlinear equation sets nine well known problems are solved with our algorithm and results are compared with quantum behaved particle swarm optimization intelligent tuned harmony search gravitational search algorithm and literature studies comparison results reveal that the proposed algorithm can cope with the highly non linear problems and outperforms many algorithms which exist in the literature

variational inequalities in orlicz sobolev spaces on etudie l existence de solutions d inegalites variationnelles associees a des operateurs differentiels elliptiques

a lyapunov type inequality for a ψ laplacian operator abstract we prove a lyapunov type inequality for a laplacian operator where is an odd increasing function which is sub multiplicative on 0 and s 1 s is a convex function for s 0 from this inequality we easily derive some previous results on the number of zeros nodal domains and bounds on eigenvalues of nontrivial solutions for certain boundary value problems including the p laplacian our method of proof does not require the standard approach via classical jensen cauchy schwarz and holder inequalities

a three terms polak ribiere polyak conjugate gradient algorithm for large scale nonlinear equations in this paper a conjugate gradient algorithm for systems of large scale nonlinear equations is designed by the following steps i a three terms conjugate gradient direction d k is presented where the direction possesses the sufficient descent property and the trust region property independent of line search technique ii a backtracking line search technique along the direction is proposed to get the step length k and construct a point iii if the point satisfies the given condition then it is the next point otherwise the projection proximal technique is used and get the next point both the direction and the line search technique are the derivative free approaches then the large scale nonlinear equations are successfully solved 100 000 variables the global convergence of the given algorithm is established under suitable conditions numerical results show that the proposed method is efficient for large scale problems

extrapolation results for q 1 given a sublinear operator t such that t ell log ell rightarrow x is bounded it can be shown that t ell q rightarrow x is bounded with constant c 1 q for every 0 q 1 in this paper we study the converse result not only for sequence spaces but for general measure spaces proving that if t lq x is bounded with constant c 1 q for every q 0 le q 1 and x is banach then t l log 1 l x is bounded moreover this result is optimal we also show that things are quite different if the banach condition on x is dropped

a ljusternik schnirelmann type theorem for eigenvalues of a prescribed mean curvature problem abstract this paper is about the following eigenvalue problem div u 1 u 2 g x u in u 0 on by using a ljusternik schnirelmann theory for variational inequalities in a space of functions of bounded variation we show the existence of infinite sequences of eigenvalues and eigenfunctions on each level of the area functional properties and finite dimensional approximation of these eigenvalues and eigenfunctions are also described

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

bfgs trust region method for symmetric nonlinear equations in this paper we propose a bfgs trust region method for solving symmetric nonlinear equations the global convergence and the superlinear convergence of the presented method will be established under favorable conditions numerical results show that the new algorithm is effective

ljusternik schnirelmann theorem for the generalized laplacian abstract let m 0 0 be an increasing continuous function with m t 0 if and only if t 0 m t as t and r n a bounded domain in this paper we show that for every r 0 the problem formula has an infinite number of eigenfunctions on the level set m u r where m t t 0 m s ds and g r r is odd satisfying some growth condition moreover we show that the sequence of associated eigenvalues tends to infinity we emphasize that no 2 condition is needed for m or for its conjugate so the associated functionals are not continuously differentiable in general