the convergence of subspace trust region methods the trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence however the subproblem in the trust region method is difficult or time consuming to solve in practical computation especially in large scale problems in this paper we consider a new class of trust region methods specifically subspace trust region methods the subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace it is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially we investigate the global convergence and convergence rate of these methods

periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

a descent algorithm without line search for unconstrained optimization in this paper a new descent algorithm for solving unconstrained optimization problem is presented its search direction is descent and line search procedure can be avoided except for the first iteration it is globally convergent under mild conditions the search direction of the new algorithm is generalized and convergence of corresponding algorithm is also proved numerical results show that the algorithm is efficient for given test problems

on the exact separation of mixed integer knapsack cuts during the last decades much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets which we call knapsack cuts bixby et al the sharpest cut the impact of manfred padberg and his work mps siam series on optimization pp 309 326 2004 empirically observe that within the context of branch and cut algorithms to solve mixed integer programming problems the most important inequalities are knapsack cuts derived by the mixed integer rounding mir procedure in this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set the main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships the exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them using these benchmarks on miplib 3 0 and miplib 2003 instances we analyze the performance of mir inequalities our computations which are performed in exact arithmetic are surprising in the vast majority of the instances in which knapsack cuts yield bound improvements mir cuts alone achieve over 87 of the observed gain

kms states and conformal measures from a non constant holomorphic map on a connected riemann surface we construct an etale second countable locally compact hausdorff groupoid whose associated groupoid c algebra admits a one parameter group of automorphisms with the property that its kms states correspond to conformal measures in the sense of sullivan in this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

separation algorithms for 0 1 knapsack polytopes valid inequalities for 0 1 knapsack polytopes often prove useful when tackling hard 0 1 linear programming problems to generate such inequalities one needs separation algorithms for them i e routines for detecting when they are violated we present new exact and heuristic separation algorithms for several classes of inequalities namely lifted cover extended cover weight and lifted pack inequalities moreover we show how to improve a recent separation algorithm for the 0 1 knapsack polytope itself extensive computational results on miplib and or library instances show the strengths and limitations of the inequalities and algorithms considered

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

a majorize minimize strategy for subspace optimization applied to image restoration this paper proposes accelerated subspace optimization methods in the context of image restoration subspace optimization methods belong to the class of iterative descent algorithms for unconstrained optimization at each iteration of such methods a stepsize vector allowing the best combination of several search directions is computed through a multidimensional search it is usually obtained by an inner iterative second order method ruled by a stopping criterion that guarantees the convergence of the outer algorithm as an alternative we propose an original multidimensional search strategy based on the majorize minimize principle it leads to a closed form stepsize formula that ensures the convergence of the subspace algorithm whatever the number of inner iterations the practical efficiency of the proposed scheme is illustrated in the context of edge preserving image restoration

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

convergence of quasi newton method with new inexact line search quasi newton method is a well known effective method for solving optimization problems since it is a line search method which needs a line search procedure after determining a search direction at each iteration we must decide a line search rule to choose a step size along a search direction in this paper we propose a new inexact line search rule for quasi newton method and establish some global convergent results of this method these results are useful in designing new quasi newton methods moreover we analyze the convergence rate of quasi newton method with the new line search rule

simulating memory effects with discrete dynamical systems abstract we propose using discrete dynamical systems to model non markovian processes this implies a whole hierarchy of dynamical systems with dimensionality increasing proportional to the memory specific long range memory effects are investigated for a quadratic a cubic and a quartic map markovian processes transform the quadratic map into a normal form of the logistic equation the henon map corresponds to non markovian first generation memory effects higher memories imply new high dimensional systems non markovian processes imply new memory routes to chaos for the three polynomial maps discussed here it is possible to define critical memory lengths above which the systems essentially lose memory

a reduced space line search method for unconstrained optimization via random descent directions abstract in this paper we propose an iterative method based on reduced space approximations for unconstrained optimization problems the method works as follows among iterations samples are taken about the current solution by using for instance a normal distribution for all samples gradients are computed approximated in order to build reduced spaces onto which descent directions of cost functions are estimated by using such directions intermediate solutions are updated the overall process is repeated until some stopping criterion is satisfied the convergence of the proposed method is theoretically proven by using classic assumptions in the line search context experimental tests are performed by using well known benchmark optimization problems and a non linear data assimilation problem the results reveal that as the number of sample points increase gradient norms go faster towards zero and even more in the data assimilation context error norms are decreased by several order of magnitudes with regard to prior errors when the assimilation step is performed by means of the proposed formulation

the 0 1 knapsack problem with a single continuous variable constraints arising in practice often contain many 0 1 variables and one or a small number of continuous variables existing knapsack separation routines cannot be used on such constraints here we study such constraint sets and derive valid inequalities that can be used as cats for such sets as well for more general mixed 0 1 constraints specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable called the mixed 0 1 knapsack problem first different classes of facet defining inequalities are derived based on restriction and lifting the order of lifting particularly of the continuous variable plays an important role secondly we show that the flow cover inequalities derived for the single node flow set consisting of are flows into and out of a single node with binary variable lower and upper bounds on each are can be obtained from valid inequalities for the mixed 0 1 knapsack problem thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0 1 constraints initial computational results on a variety of problems are presented

strange attractors with one direction of instability we give simple conditions that guarantee for strongly dissipative maps the existence of strange attractors with a single direction of instability and certain controlled behaviors only the d 2 case is treated in this paper although our approach is by no means limited to two phase dimensions we develop a dynamical picture for the attractors in this class proving they have many of the statistical properties associated with chaos positive lyapunov exponents existence of srb measures and exponential decay of correlations other results include the geometry of fractal critical sets nonuniform hyperbolic behavior symbolic coding of orbits and formulas for topological entropy

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

fenchel cutting planes for integer programs a technique for generating cutting planes for integer programs is introduced that is based on the ability to optimize a linear function on a polyhedron rather than explicit knowledge of the underlying polyhedral structure of the integer program the theoretical properties of the cuts and their relationship to lagrangian relaxation are discussed the cut generation procedure is described and computational results are presented

the steepest descent algorithm without line search for p laplacian in this paper the steepest descent algorithm without line search is proposed for p laplacian its search direction is the weighted preconditioned steepest descent one and step length is estimated by a formula except the first iteration continuation method is applied for solving the p laplacian with very large p lots of numerical experiments are carried out on these algorithms all numerical results show the algorithm without line search can cut down some computational time fast convergence of these new algorithms is displayed by their step length figures these figures show that if search direction is the steepest descent one exact step lengths can be substituted properly with step lengths obtained by the formula

on the convergence of fenchel cutting planes in mixed integer programming fenchel cutting planes are based on the dual relationship between separation and optimization and can be applied in many instances where alternative cutting planes cannot they are deep in the sense of providing the maximum separation between a point hat x and a polyhedron p as measured by an arbitrary norm which is specified in the process of generating a fenchel cut this paper demonstrates a number of fundamental convergence properties of fenchel cuts and addresses the question of which norms lead to the most desirable fenchel cuts the strengths and weaknesses of the related class of 1 polar cuts are also examined

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

border collision bifurcations an explanation for observed bifurcation phenomena recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena these phenomena are part of a rich new class of bifurcations which we call border collision bifurcations a general criterion for the occurrence of border collision bifurcations is given illustrative numerical results including transitions to chaotic attractors are presented these border collision bifurcations are found in a variety of physical experiments

convergence rates of a global optimization algorithm this paper presents a best and worst case analysis of convergence rates for a deterministic global optimization algorithm superlinear convergence is proved for lipschitz functions which are convex in the direction of the global maximum concave in the direction of the global minimum computer results are given which confirm the theoretical convergence rates

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

dynamics of horizontal like maps in higher dimension we study the regularity of the green currents and of the equilibrium measure associated to a horizontal like map in under a natural assumption on the dynamical degrees we estimate the speed of convergence towards the green currents the decay of correlations for the equilibrium measure and the lyapounov exponents we show in particular that the equilibrium measure is hyperbolic we also show that the green currents are the unique invariant vertical and horizontal positive closed currents the results apply in particular to henon like maps to regular polynomial automorphisms of and to their small perturbations

gradient convergence in gradient methods with errors we consider the gradient method x t 1 x t g t s t w t where s t is a descent direction of a function f rn to re and w t is a deterministic or stochastic error we assume that gr f is lipschitz continuous that the stepsize g t diminishes to 0 and that s t and w t satisfy standard conditions we show that either f x t to infty or f x t converges to a finite value and gr f x t to0 with probability 1 in the stochastic case and in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of gr f x t or boundedness of xt

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

a new unconstrained optimization method for imprecise function and gradient values a new algorithm for unconstrained optimization is presented which is based on a modified one dimensional bisection method the algorithm actually uses only the signs of function and gradient values thus it can be applied to problems with imprecise function and gradient values it converges in one iteration on quadratic functions of n variables it rapidly minimizes general functions and it does not require evaluation or estimation of the matrix of second partial derivatives the algorithm has been implemented and tested performance information for wellknown test functions is reported q 1996 academic press inc

unfolding a chaotic bifurcation the differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased by introducing a small constant bias in the forcing to break the symmetry two discontinuous bifurcations unfold from the single merging event considering both forcing amplitude and bias together as controls a codimension two bifurcation of chaotic attractors is defined whose geometric structure in control phase space is closely related to the elementary cusp catastrophe the chaotic bifurcations are explained in terms of homoclinic structures smale cycles in the poincare map

convergence properties of nonlinear conjugate gradient methods recently important contributions on convergence studies of conjugate gradient methods were made by gilbert and nocedal siam j optim 2 1992 pp 21 42 they introduce a sufficient descent condition to establish global convergence results although this condition is not needed in the convergence analyses of newton and quasi newton methods gilbert and nocedal hint that the sufficient descent condition which was enforced by their two stage line search algorithm may be crucial for ensuring the global convergence of conjugate gradient methods this paper shows that the sufficient descent condition is actually not needed in the convergence analyses of conjugate gradient methods consequently convergence results on the fletcher reeves and polak ribiere type methods are established in the absence of the sufficient descent condition to show the differences between the convergence properties of fletcher reeves and polak ribiere type methods two examples are constructed showing that neither the boundedness of the level set nor the restriction beta k geq 0 can be relaxed for the polak ribiere type methods

statistical properties of the universal limit map of grazing bifurcations in this paper the statistical properties of an interval map having a square root singular point which characterizes grazing bifurcations of impact oscillators are studied firstly we show that in some parameter regions the map admits an induced markov structure with an exponential decay tail of the return times then we prove that the map has a unique mixing absolutely continuous invariant probability measure finally by applying the markov tower method we prove that exponential decay of correlations and the central limit theorem hold for holder continuous observations

global and superlinear convergence of an algorithm for one dimensional minimization of convex functions this paper studies an algorithm for minimizing a convex function based upon a combination of polyhedral and quadratic approximation the method was given earlier but without a good specification for updating the algorithm s curvature matrix here for the case of onedimensional minimization we provide a specification that insures convergence even in cases where the curvature scalar tends to zero or infinity under mild additional assumptions we show that the convergence is superlinear

heteroclinic tangles in time periodic equations abstract in this paper we prove that when a heteroclinic loop is periodically perturbed three types of heteroclinic tangles are created and they compete in the space of where is a parameter representing the magnitude of the perturbations the three types are a transient heteroclinic tangles containing no gibbs measures b heteroclinic tangles dominated by sinks representing stable dynamical behavior and c heteroclinic tangles with strange attractors admitting srb measures representing chaos we also prove that as 0 the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections the theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical birkhoff melnikov smale method

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

post critical set and non existence of preserved meromorphic 2 forms we present a family of birational transformations in cp2 depending on two or three parameters which do not generically preserve meromorphic 2 forms with the introduction of the orbit of the critical set vanishing condition of the jacobian also called the post critical set we get some new structures some non analytic 2 form which reduce to meromorphic 2 forms for particular subvarieties in the parameter space on these subvarieties the iterates of the critical set have a polynomial growth in the degrees of the parameters while one has an exponential growth out of these subspaces the analysis of our birational transformation in cp2 is first carried out using the diller favre criterion in order to find the complexity reduction of the mapping the integrable cases are found the identification between the complexity growth and the topological entropy is once again verified we perform plots of the post critical set as well as calculations of lyapunov exponents for many orbits confirming that generically no meromorphic 2 form can be preserved for this mapping these birational transformations in cp2 which generically do not preserve any meromorphic 2 form are extremely similar to other birational transformations we previously studied which do preserve meromorphic 2 forms we note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned but drastically different results as far as a more probabilistic approach of dynamical systems is concerned lyapunov exponents with these examples we see that the existence of a preserved meromorphic 2 form explains most of the numerical discrepancies between the topological and probabilistic approaches of dynamical systems

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

multivariate spectral gradient method for unconstrained optimization multivariate spectral gradient method is proposed for solving unconstrained optimization problems combined with some quasi newton property multivariate spectral gradient method allows an individual adaptive stepsize along each coordinate direction which guarantees that the method is finitely convergent for positive definite quadratics especially it converges no more than two steps for positive definite quadratics with diagonal hessian and quadratically for objective functions with positive definite diagonal hessian moreover based on a nonmonotone line search global convergence is established for multivariate spectral gradient algorithm at last numerical results are reported which show that this method is promising and deserves further discussing

a class of gradient unconstrained minimization algorithms with adaptive stepsize in this paper the development convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive stepsize are presented the proposed class comprises four algorithms the first two incorporate techniques for the adaptation of a common stepsize for all coordinate directions and the other two allow an individual adaptive stepsize along each coordinate direction all the algorithms are computationally efficient and possess interesting convergence properties utilizing estimates of the lipschitz constant that are obtained without additional function or gradient evaluations the algorithms have been implemented and tested on some well known test cases as well as on real life artificial neural network applications and the results have been very satisfactory

solving 0 1 integer programming problems arising from large scale planning models we present methods that are useful in solving some large scale hierarchical planning models involving 0 1 variables these 0 1 programming problems initially could not be solved with any standard techniques we employed several approaches to take advantage of the hierarchical structure of variables ordered by importance and other structures present in the models critical but not sufficient for success was a strong linear programming formulation we describe methods for strengthening the linear programs as well as other techniques necessary for a commercial branch and bound code to be successful in solving these problems

optimization on the hierarchical tucker manifold applications to tensor completion in this work we develop an optimization framework for problems whose solutions are well approximated by hierarchical tucker ht tensors an efficient structured tensor format based on recursive subspace factorizations by exploiting the smooth manifold structure of these tensors we construct standard optimization algorithms such as steepest descent and conjugate gradient for completing tensors from missing entries our algorithmic framework is fast and scalable to large problem sizes as we do not require svds on the ambient tensor space as required by other methods moreover we exploit the structure of the gramian matrices associated with the ht format to regularize our problem reducing overfitting for high subsampling ratios we also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries these samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios using these algorithms we successfully interpolate large scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow

convergence of prp method with new nonmonotone line search abstract in this paper we develop a new nonmonotone line search for prp conjugate gradient method polak ribiere polyak for minimizing functions having lipschitz continuous partial derivatives the nonmonotone line search can guarantee the global convergence of original prp method under some mild conditions numerical experiments show that prp method with the new nonmonotone line search is available and efficient in practical computation

on step size estimation of line search methods this paper concerns the step size estimation in line search method for unconstrained optimization problems some new approaches for estimating the step size are proposed and the global convergence of related line search methods are analyzed finally we discuss the choice of the initial step size in armijo line search rule

convergence of nonmonotone line search method in this paper we develop a new nonmonotone line search for general line search method and establish some global convergence theorems the new nonmonotone line search is a novel form of the nonmonotone armijo line search and allows one to choose a larger step size at each iteration which is available in constructing new line search methods and possibly reduces the function evaluations at each iteration moreover we analyze the convergence rate of some special line search methods with the new line search preliminary numerical results show that some line search methods with the new nonmonotone line search are available and efficient in practical computation

efficient maximum entropy reconstruction of nuclear magnetic resonance t1 t2 spectra this paper deals with the reconstruction of t1 t2 correlation spectra in nuclear magnetic resonance relaxometry the ill posed character and the large size of this inverse problem are the main difficulties to tackle while maximum entropy is retained as an adequate regularization approach the choice of an efficient optimization algorithm remains a challenging task our proposal is to apply a truncated newton algorithm with two original features first a theoretically sound line search strategy suitable for the entropy function is applied to ensure the convergence of the algorithm second an appropriate preconditioning structure based on a singular value decomposition of the forward model matrix is used to speed up the algorithm convergence furthermore we exploit the specific structures of the observation model and the hessian of the criterion to reduce the computation cost of the algorithm the performances of the proposed strategy are illustrated by means of synthetic and real data processing

convergence of descent method without line search line search method is sometimes a useful and efficient technique for solving unconstrained optimization problems especially for solving small and middle scale problems however a line search procedure is necessary at each iteration leading to a significant amount of computation in order to reduce the evaluations of objective functions and gradients the line search procedure should be avoided in algorithm design in this paper we propose a new descent method without line search for unconstrained optimization problems the algorithm is required to estimate some parameters at each iteration we analyze theoretically the global convergence of the new algorithm under mild conditions these theoretical conclusions can help us in designing new efficient methods for optimization problems

cutting planes for mixed integer knapsack polyhedra an algorithm for generating cutting planes for mixed integer knapsack polyhedra is described the algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper computational results are presented 1998 the mathematical programming society inc published by elsevier science b v

an accelerated double step size model in unconstrained optimization abstract this work presents a double step size algorithm with accelerated property for solving nonlinear unconstrained optimization problems using the inexact line search technique as well as the approximation of the hessian by an adequate diagonal matrix an efficient accelerated gradient descent method is developed the proposed method is proven to be linearly convergent for uniformly convex functions and also under some specific conditions linearly convergent for strictly convex quadratic functions numerical testings and comparisons show that constructed scheme exceeds some known iterations for unconstrained optimization with respect to all three tested properties number of iterations cpu time and number of function evaluations

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

a dimension reducing method for unconstrained optimization abstract a new method for unconstrained optimization in r n is presented this method reduces the dimension of the problem in such a way that it can lead to an iterative approximate formula for the computation of n 1 components of the optimum while its remaining component is computed separately using the final approximations of the other components it converges quadratically to a local optimum and it requires storage of order n 1 n 1 besides it does not require a good initial guess for one component of the optimum and it does not directly perform gradient evaluations thus it can be applied to problems with imprecise gradient values moreover a procedure for transforming the matrix formed by our method into a symmetric as well as into a diagonal one is presented furthermore the proposed dimension reducing scheme using finite difference gradient and hessian is presented the methods have been implemented and tested performance information for well known test functions is reported

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

universal limit mapping in grazing bifurcations grazing bifurcations are nonsmooth bifurcations that occur in impacting mechanical oscillators a sign of the lack of smoothness is a square root expression that arises in mappings describing the dynamics certain types of grazing bifurcations involve chaotic bands or period adding cascades with or without chaotic bands there is also a characteristic scaling behavior present here this type of bifurcation is investigated and the self similarity under scaling is used to derive a renormalized limit mapping a study of the dynamics of the limit mapping identifying all attractors for all parameter values is also presented

combining search directions using gradient flows the efficient combination of directions is a significant problem in line search methods that either use negative curvature or wish to include additional information such as the gradient or different approximations to the newton direction

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

a new trust region method for unconstrained optimization in this paper we propose a new trust region method for unconstrained optimization problems the new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions we also analyze the global linear convergence local superlinear and quadratic convergence rate of the new method numerical results show that the new trust region method is available and efficient in practical computation

a conjugate gradient method with descent direction for unconstrained optimization a modified conjugate gradient method is presented for solving unconstrained optimization problems which possesses the following properties i the sufficient descent property is satisfied without any line search ii the search direction will be in a trust region automatically iii the zoutendijk condition holds for the wolfe powell line search technique iv this method inherits an important property of the well known polak ribiere polyak prp method the tendency to turn towards the steepest descent direction if a small step is generated away from the solution preventing a sequence of tiny steps from happening the global convergence and the linearly convergent rate of the given method are established numerical results show that this method is interesting