a bundle bregman proximal method for convex nondifferentiable minimization by taking xk to be an approximate minimizer of where is a piecewise linear model of f constructed from accumulated subgradient linearizations of f dh is the d function of a generalized bregman function h and tk 0 convergence under implementable criteria is established by extending our recent framework of bregman proximal minimization which is of independent interest e g for nonquadratic multiplier methods for constrained minimization in particular we provide new insights into the convergence properties of bundle methods based on h 2

generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

primal and dual bregman methods with application to optical nanoscopy measurements in nanoscopic imaging suffer from blurring effects modeled with different point spread functions psf some apparatus even have psfs that are locally dependent on phase shifts additionally raw data are affected by poisson noise resulting from laser sampling and photon counts in fluorescence microscopy in these applications standard reconstruction methods em filtered backprojection deliver unsatisfactory and noisy results starting from a statistical modeling in terms of a map likelihood estimation we combine the iterative em algorithm with total variation tv regularization techniques to make an efficient use of a priori information typically tv based methods deliver reconstructed cartoon images suffering from contrast reduction we propose extensions to em tv based on bregman iterations and primal and dual inverse scale space methods in order to obtain improved imaging results by simultaneous contrast enhancement besides further generalizations of the primal and dual scale space methods in terms of general convex variational regularization methods we provide error estimates and convergence rates for exact and noisy data we illustrate the performance of our techniques on synthetic and experimental biological data

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

bundle level type methods uniformly optimal for smooth and nonsmooth convex optimization the main goal of this paper is to develop uniformly optimal first order methods for convex programming cp by uniform optimality we mean that the first order methods themselves do not require the input of any problem parameters but can still achieve the best possible iteration complexity bounds by incorporating a multi step acceleration scheme into the well known bundle level method we develop an accelerated bundle level method and show that it can achieve the optimal complexity for solving a general class of black box cp problems without requiring the input of any smoothness information such as whether the problem is smooth nonsmooth or weakly smooth as well as the specific values of lipschitz constant and smoothness level we then develop a more practical restricted memory version of this method namely the accelerated prox level apl method we investigate the generalization of the apl method for solving certain composite cp problems and an important class of saddle point problems recently studied by nesterov math program 103 127 152 2005 we present promising numerical results for these new bundle level methods applied to solve certain classes of semidefinite programming and stochastic programming problems

scaling generator coordinate method and the eigensolutions of the schrodinger equation an extremely simple approximate variational method for the study of the eigensolutions of the one dimensional schrodinger equation is introduced based upon the idea of employing scaled eigenstates of the harmonic oscillator as generating functions for the generator coordinate method the approach is successfully applied to a variety of situations

approximate generalized proximal type method for convex vector optimization problem in banach spaces in this paper we consider a convex vector optimization problem of finding weak pareto optimal solutions for an extended vector valued map from a uniformly convex and uniformly smooth banach space to a real banach space with respect to the partial order induced by a closed convex and pointed cone with a nonempty interior we propose an inexact vector valued proximal type point algorithm based on a lyapunov functional when the iterates are computed approximately and prove the sequence generated by the algorithm weakly converges to a weak pareto optimal solution of the vector optimization problem under some mild conditions our results improve and generalize some known results

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

global minimization for continuous multiphase partitioning problems using a dual approach this paper is devoted to the optimization problem of continuous multi partitioning or multi labeling which is based on a convex relaxation of the continuous potts model in contrast to previous efforts which are tackling the optimal labeling problem in a direct manner we first propose a novel dual model and then build up a corresponding duality based approach by analyzing the dual formulation sufficient conditions are derived which show that the relaxation is often exact i e there exists optimal solutions that are also globally optimal to the original nonconvex potts model in order to deal with the nonsmooth dual problem we develop a smoothing method based on the log sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal dual model and suggests labelings with maximum entropy such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions an expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization numerical experiments also show that our method outperforms several competitive approaches in various aspects such as lower energies and better visual quality

melnikov transforms bernoulli bundles and almost periodic perturbations in this paper we study nonlinear time varying perturbations of an autonomous vector field in the plane r2 we assume that the unperturbed equation i e the given vector field has a homoclinic orbit and we present a gen eralization of the melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory the key to our generalization is the concept of the melnikov transform which is a linear transformation on the space of perturbation functions the appropriate dynamical setting for studying these perturbation is the concept of a skew product flow the concept of transversality we require is best understood in this context under conditions whereby the perturbed equation admits a transversal homoclinic trajectory we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow we show the dynamics near this trajectory can have the exotic behavior of the bernoulli shift the exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle which we call the bernoulli bundle we allow all perturbations which are bounded and uniformly continuous in time thus our theory includes the classical periodic perturbations studied by melnikov quasi periodic and almost periodic perturbations as well as toroidal perturbations which are close to quasi periodic perturbations

proximal minimization methods with generalized bregman functions we consider methods for minimizing a convex function f that generate a sequence xk by taking xk 1 to be an approximate minimizer of f x dh x xk ck where ck 0 and dh is the d function of a bregman function h extensions are made to b functions that generalize bregman functions and cover more applications convergence is established under criteria amenable to implementation applications are made to nonquadratic multiplier methods for nonlinear programs

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

interior gradient and epsilon subgradient descent methods for constrained convex minimization we extend epsilon subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets in particular over r p this is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic quadratic distancelike function recently introduced by the authors we then obtain interior e subgradient descent methods which allow us to provide a natural extension of bundle methods and polyak s subgradient projection methods for nonsmooth convex minimization furthermore similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem s data

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

convergence of proximal like algorithms we analyze proximal methods based on entropy like distances for the minimization of convex functions subject to nonnegativity constraints we prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator we also consider linearly constrained convex problems and establish a quadratic convergence rate result for linear programs our analysis allows us to simplify and extend the available convergence results for these methods

a practical general approximation criterion for methods of multipliers based on bregman distances this paper demonstrates that for generalized methods of multipliers for convex programming based on bregman distance kernels including the classical quadratic method of multipliers the minimization of the augmented lagrangian can be truncated using a simple generally implementable stopping criterion based only on the norms of the primal iterate and the gradient or a subgradient of the augmented lagrangian at that iterate previous results in this and related areas have required conditions that are much harder to verify such as e optimality with respect to the augmented lagrangian or strong conditions on the convex program to be solved here only existence of a kkt pair is required and the convergence properties of the exact form of the method are preserved the key new element in the analysis is the use of a full conjugate duality framework as opposed to mainly examining the action of the method on the standard dual function of the convex program an existence result for the iterates stronger than those possible for the exact form of the algorithm is also included

on finite convergence of proximal point algorithms for variational inequalities in this paper we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem vip where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations by using this result we obtain that the well known proximal point algorithm possesses finite convergence if the solution set of vip is weakly sharp as an extension we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g sharpness

iterative regularization and nonlinear inverse scale space applied to wavelet based denoising in this paper we generalize the iterative regularization method and the inverse scale space method recently developed for total variation tv based image restoration to wavelet based image restoration this continues our earlier joint work with others where we applied these techniques to variational based image restoration obtaining significant improvement over the rudin osher fatemi tv based restoration here we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b that these procedures enhance the noise removal capability both theoretically in the sense of generalized bregman distance and for some examples experimentally in terms of the signal to noise ratio leaving less signal in the residual

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

an analysis of the em algorithm and entropy like proximal point methods the em algorithm is a popular method for maximum likelihood estimation from incomplete data this method may be viewed as a proximal point method for maximizing the log likelihood function using an integral form of the kullback leibler distance function motivated by this interpretation we consider a proximal point method using an integral form of entropy like distance function we give a convergence analysis of the resulting proximal point method in the case where the cluster points lie in the interior of the objective function domain this result is applied to a normal independent example and a gaussian mixture example to establish convergence of the em algorithm on these examples further convergence analysis of the method for maximization over an orthant is given in low dimensions sublinear convergence and schemes for accelerating convergence are also discussed

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i a generic algorithmic framework in this paper we present a generic algorithmic framework namely the accelerated stochastic approximation ac sa algorithm for solving strongly convex stochastic composite optimization sco problems while the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems the ac sa algorithm when employed with proper stepsize policies can achieve optimal or nearly optimal rates of convergence for solving different classes of sco problems during a given number of iterations moreover we investigate these ac sa algorithms in more detail such as by establishing the large deviation results associated with the convergence rates and introducing an efficient validation procedure to check the accuracy of the generated solutions

a hybrid entropic proximal decomposition method with self adaptive strategy for solving variational inequality problems in this paper we propose a hybrid nonlinear decomposition projection method for solving a class of monotone variational inequality problems the algorithm utilizes the problems structure conductive to decomposition and a projection step to get the next iterate to make the method more practical we allow solving of the subproblems approximately and adopt the constructive accuracy criterion developed recently by solodov and svaiter for classical proximal point algorithm and by the author for generalized proximal point algorithm the fejer monotonicity to the solution set of the problem is obtained by only assuming the underlying mapping is monotone and the solution set is nonempty the parameter is allowed to vary in a larger interval than that of auslender and teboulle and we also propose some improved self adaptive strategies to choose the sequence of parameters which makes the algorithm more flexible

a proximal point method for the variational inequality problem in banach spaces in this paper we prove well definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive banach spaces the proximal version we consider makes use of bregman functions whose original definition for finite dimensional spaces has here been properly extended to our more general framework

a primal dual proximal point algorithm for constrained convex programs we present a primal dual application of the proximal point algorithm to solve convex constrained minimization problems motivated by the work of eckstein math oper res 18 1993 203 about the generalized proximal point method we propose here a mixed proximal multipliers method where we improve the result of eckstein math oper res 18 1993 203 theorem 1

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

an extension of proximal methods for quasiconvex minimization on the nonnegative orthant abstract in this paper we propose an extension of proximal methods to solve minimization problems with quasiconvex objective functions on the nonnegative orthant assuming that the function is bounded from below and lower semicontinuous and using a general proximal distance it is proved that the iterations given by our algorithm are well defined and stay in the positive orthant if the objective function is quasiconvex we obtain the convergence of the iterates to a certain set which contains the set of optimal solutions and convergence to a kkt point if the function is continuously differentiable and the proximal parameters are bounded furthermore we introduce a sufficient condition on the proximal distance such that the sequence converges to an optimal solution of the problem

a generalized proximal point algorithm for the variational inequality problem in a hilbert space we consider a generalized proximal point method for solving variational inequality problems with monotone operators in a hilbert space it differs from the classical proximal point method as discussed by rockafellar for the problem of finding zeroes of monotone operators in the use of generalized distances called bregman distances instead of the euclidean one these distances play not only a regularization role but also a penalization one forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one under appropriate assumptions on the bregman distance and the monotone operator we prove that the sequence converges weakly if and only if the problem has solutions in which case the weak limit is a solution if the problem does not have solutions then the sequence is unbounded we extend similar previous results for the proximal point method with bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators which are particular cases of variational inequality problems

interior proximal and multiplier methods based on second order homogeneous kernels we study a class of interior proximal algorithms and nonquadratic multiplier methods for solving convex programs where the usual proximal quadratic term is replaced by an homogeneous functional of order two defined in terms of a convex function we prove under mild assumptions several new convergence results in both the primal and dual framework allowing also for approximate minimization in particular we introduce a new class of interior proximal methods which is globally convergent and allows for generating c multiplier methods with bounded hessians which exhibit strong convergence properties we also consider linearly constrained convex problems and establish global quadratic convergence rates results for linear programs we then study in detail a particular realization of these algorithms leading to a new class of logarithmic quadratic interior point algorithms which are shown to enjoy several attractive properties for solving constrained convex optimization problems

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

rescaling and stepsize selection in proximal methods using separable generalized distances this paper presents a convergence proof technique for a broad class of proximal algorithms in which the perturbation term is separable and may contain barriers enforcing interval constraints there are two key ingredients in the analysis a mild regularity condition on the differential behavior of the barrier as one approaches an interval boundary and a lower stepsize limit that takes into account the curvature of the proximal term we give two applications of our approach first we prove subsequential convergence of a very broad class of proximal minimization algorithms for convex optimization where different stepsizes can be used for each coordinate applying these methods to the dual of a convex program we obtain a wide class of multiplier methods with subsequential convergence of both primal and dual iterates and independent adjustment of the penalty parameter for each constraint the adjustment rules for the penalty parameters generalize a well established scheme for the exponential method of multipliers the results may also be viewed as a generalization of recent work by ben tal and zibulevsky siam j optim 7 1997 pp 347 366 and auslender teboulle and ben tiba comput optim appl 12 1999 pp 31 40 math oper res 24 1999 pp 645 668 on methods derived from varphi divergences the second application established full convergence under a novel stepsize condition of bregman function based proximal methods for general monotone operator problems over a box prior results in this area required strong restrictive assumptions on the monotone operator

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

proximal decomposition via alternating linearization a new approximate proximal point method for minimizing the sum of two convex functions is introduced it replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models the method updates the linear models and the prox center as well as the prox coefficient it is monotone in terms of the objective values and converges to a solution of the problem if any a dual version of the method is derived and analyzed applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented

a proximal point sqp trust region method for solving some special class of nonlinear semi definite programming problems abstract in this paper we consider a nonlinear semi definite programming problem that represents the fixed order h 2 and h 2 h synthesis problems a proximal point sequential quadratic programming method that makes use of trust region is developed furthermore the constrained trust region method f leibfritz e m e mostafa trust region methods for solving the optimal output feedback design problem int j contr 76 2003 501 519 which was designed to solve a nonlinear semi definite program representing the h 2 synthesis problem is extended to solve a more general nonlinear semi definite program representing the fixed order h 2 h synthesis problem numerical results for the proposed methods are given

on dual convergence of the generalized proximal point method with bregman distances the use of generalized distances e g bregman distances instead of the euclidean one in the proximal point method for convex optimization allows for elimination of the inequality constraints from the subproblems in this paper we consider the proximal point method with bregman distances applied to linearly constrained convex optimization problems and study the behavior of the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem under rather general assumptions which cover most bregman distances of interest we obtain an ergodic convergence result namely that a sequence of weighted averages of the dual sequence converges to a specific point of the dual optimal set as an intermediate result we prove under the same assumptions that the dual central path generated by a large class of barriers including the generalized bregman distances converges to the same point

an inexact proximal method for quasiconvex minimization abstract in this paper we propose an inexact proximal point method to solve constrained minimization problems with locally lipschitz quasiconvex objective functions assuming that the function is also bounded from below lower semicontinuous and using proximal distances we show that the sequence generated for the method converges to a stationary point of the problem

convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii abstract we investigate the convergence rates for total variation regularization of the problem of identifying i the coefficient q in the neumann problem for the elliptic equation div q u f in q u n g on ii the coefficient a in the neumann problem for the elliptic equation u a u f in u n g on r d d 1 when u is imprecisely given by z h 1 u z h 1 0 we regularize these problems by correspondingly minimizing the strictly convex functionals 1 2 q u q z 2 d x 1 2 q l 2 2 q and 1 2 u a z 2 d x 1 2 a u a z 2 d x 1 2 a l 2 2 a over admissible sets where u q u a is the solution of the first second neumann boundary value problem 0 is the regularization parameter taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems we obtain the convergence rates of them to the solution of the inverse problem in the sense of the bregman distance and in the l 2 norm under relatively simple source conditions without the smallness requirement on the source functions