perturbed kepler problem in quaternionic form a new technique for solving the perturbed kepler problem is presented as its approach is essentially algebraic it can be easily computerized and carried out to any perturbation order it also eliminates the notorious difficulty of small divisors and is well suited to deal with commensurable orbital periods as demonstrated by its explanation of kirkwood gaps

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

majorizing functions and convergence of the gauss newton method for convex composite optimization we introduce a notion of quasi regularity for points with respect to the inclusion f x in c where f is a nonlinear frechet differentiable function from mathbb r v to mathbb r m when c is the set of minimum points of a convex real valued function h on mathbb r m and f satisfies the l average lipschitz condition of wang we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the gauss newton method with quasi regular initial points for the convex composite function h circ f results are new even when the initial point is regular and f is lipschitz

extending the applicability of gauss newton method for convex composite optimization on riemannian manifolds we present a semi local convergence analysis of the gauss newton method for solving convex composite optimization problems in riemannian manifolds using the notion of quasi regularity for an initial point using a combination the l average lipszhitz condition and the center l 0 average lipschitz condition we introduce majorizing sequences for the gauss newton method that are more precise than in earlier studies consequently our semi local convergence analysis for the gauss newton method has the following advantages under the same computational cost weaker sufficient convergence conditions more precise estimates on the distances involved and an at least as precise information on the location of the solution

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

second order optimality conditions in nonlinear programming obtained by way of epi derivatives dedicated to the memory of robin w chaney second order optimality conditions for finite dimensional smooth and nonsmooth nonlinear programming are obtained by a new method that emphasizes a close connection with geometrical approximation of the essential objective function the approximation is secured by the use of certain epi derivatives defined by epiconvergence the optimality conditions are expressed in a form that covers general interval constraints and their possible representation through penalties or an augmented lagrangian an abstract constraint involving restriction to a convex polyhedron is incorporated

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

on newton s method under holder continuous derivative a mysovskii type theorem for newton s method under k p holder continuous derivative is considered in this paper for the application studied by others the new condition is weaker than ones in the literature also we prove that the optimal convergent order is p 1 for 0 p 1

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

equivalences of the large deviation principle for gibbs measures and critical balance in the ising model in this paper we obtain the equivalence of the large deviation principle for gibbs measures with and without an external field for the ising model the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube b h ash 0 for variousb we find a critical balance at a valueb0 ofb

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

extended newton methods for multiobjective optimization majorizing function technique and convergence analysis we consider the extended newton method for approaching a pareto optimum of a multiobjective optimization problem establish quadratic convergence criteria and estimate a radius of convergence ball

optimality conditions for non finite valued convex composite functions burke 1987 has recently developed second order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued in this note we present a technique for reducing the infinite valued case to the finite valued one we then use this technique to extend the results in burke 1987 to the case in which the convex function may take infinite values we conclude by comparing these results with those established by rockafellar 1989 for the piecewise linear quadratic case

unified approaches for solvable and unsolvable linear complementarity problems abstract in this paper general linear complementarity problems lcps are studied via global optimization problems in particular unsolvable lcps are reformulated as multicriteria optimization minimax optimization and quadratic programming problems the solvability and unsolvability of lcps are obtained via these reformulations furthermore first order and second order global optimality conditions of lcps are derived some examples are also given to demonstrate these optimality conditions

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

symplectic regularization of binary collisions in the circular n 2 sitnikov problem we present a brief overview of the regularizing transformations of the kepler problem and we relate the euler transformation with the symplectic structure of the phase space of the n body problem we show that any particular solution of the n body problem where two bodies have rectilinear dynamics can be regularized by a linear symplectic transformation and the inclusion of the euler transformation into the group of symplectic local diffeomorphisms over the phase space as an application we regularize a particular configuration of the restricted circular n 2 body problem

on the layering transition of an sos surface interacting with a wall ii the glauber dynamics we continue our study of the statistical mechanics of a 2d surface above a fixed wall and attracted towards it by means of a very weak positive magnetic fieldh in the solid on solid sos approximation when the inverse temperature is very large in particular we consider a glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions using the results proved in the first paper of this series we show that for allh h k 1 h k h k being the critical values of the magnetic field found in the previous paper the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions on the contrary forh h k and free boundary conditions we show that the gap in a cube of sidel is bounded from above and from below by a negative exponential ofl our results provide a strong indication that contrary to what happens in two dimensions for the three dimensional dynamical ising model in a finite cube at low temperature and very small positive external field with boundary conditions that are opposite to the field on one face of the cube and are absent free on the remaining faces the rate of exponential convergence to equilibrium which is positive in infinite volume may go to zero exponentially fast in the side of the cube

second order optimality conditions for problems with c1 data abstract in this paper we obtain second order optimality conditions of karush kuhn tucker type and fritz john one for a problem with inequality constraints and a set constraint in nonsmooth settings using second order directional derivatives in the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable but their gradients are not necessarily locally lipschitz in the sufficient conditions for a global minimum x we assume that the objective function is differentiable at x and second order pseudoconvex at x a notion introduced by the authors i ginchev v i ivanov higher order pseudoconvex functions in i v konnov d t luc a m rubinov eds generalized convexity and related topics in lecture notes in econom and math systems vol 583 springer 2007 pp 247 264 the constraints are both differentiable and quasiconvex at x in the sufficient conditions for an isolated local minimum of order two we suppose that the problem belongs to the class c 1 1 we show that they do not hold for c 1 problems which are not c 1 1 ones at last a new notion parabolic local minimum is defined and it is applied to extend the sufficient conditions for an isolated local minimum from problems with c 1 1 data to problems with c 1 one

convergence of gauss newton s method and uniqueness of the solution abstract in this paper we study the convergence of gauss newton s method for nonlinear least squares problems under the hypothesis that derivative satisfies some kinds of weak lipschitz condition we obtain the exact estimates of the radii of convergence ball of gauss newton s method and the uniqueness ball of the solution new results can be used to determinate approximation zero of gauss newton s method

two conditions concerning newton s method in this note we compare two conditions concerning newton s method and find out that even under some weaker condition one has the convergence of the newton s method and a better error bound than the relative strong condition

bound on the mass gap for finite volume stochastic ising models at low temperature we consider a sequence of finite volume z d d 2 reversible stochastic ising models in the low temperature regime and having invariant measures satisfying free boundary conditions we show that associated with the models are random hitting times whose expectations regarded as a function of grow exponentially in d 1 d moreover the mass gaps for the models shrink exponentially fast in d 1 d a geometrical lemma is employed in the analysis which states that if a peierls contour is sufficiently small relative to the faces of then the fraction of the contour tangent to the faces is less than a constant smaller than one

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

on convergence of the gauss newton method for convex composite optimization the local quadratic convergence of the gauss newton method for convex composite optimization f h f is established for any convex function h with the minima set c extending burke and ferris results in the case when c is a set of weak sharp minima for h

a gauss newton method for convex composite optimization an extension of the gauss newton method for nonlinear equations to convex composite optimization is described and analyzed local quadratic convergence is established for the minimization ofh f under two conditions namelyh has a set of weak sharp minima c and there is a regular point of the inclusionf x c this result extends a similar convergence result due to womersley this journal 1985 which employs the assumption of a strongly unique solution of the composite functionh f a backtracking line search is proposed as a globalization strategy for this algorithm a global convergence result is established with a quadratic rate under the regularity assumption

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

constraint qualifications for extended farkas s lemmas and lagrangian dualities in convex infinite programming for an inequality system defined by a possibly infinite family of proper functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions under the new constraint qualifications we obtain characterizations of those reverse convex inequalities which are a consequence of the constrained system and we provide necessary and or sufficient conditions for a stable farkas lemma to hold similarly we provide characterizations for constrained minimization problems to have the strong or strong stable lagrangian dualities several known results in the conic programming problem are extended and improved

epi convergent smoothing with applications to convex composite functions smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems in this note we synthesize and extend recent results due to beck and teboulle on infimal convolution smoothing for convex functions with those of x chen on gradient consistency for nonconvex functions we use epi convergence techniques to define a notion of epi smoothing that allows us to tap into the rich variational structure of the subdifferential calculus for nonsmooth nonconvex and nonfinite valued functions as an illustration of the versatility and range of epi smoothing techniques the results are applied to the general constrained optimization for which nonlinear programming is a special case

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

large deviations for the contact process and two dimensional percolation summary the following results are proved 1 for the upper invariant measure of the basic one dimensional supercritical contact process the density of l s has the usual large deviation behavior the probability of a large deviation decays exponentially with the number of sites considered 2 for supercritical two dimensional nearest neighbor site or bond percolation the density ya of sites inside a square a which belong to the infinite cluster has the following large deviation properties the probability that ya deviates from its expected value by a positive amount decays exponentially with the area of a while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of a these two problems are treated together in this paper because similar techniques renormalization are used for both

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

convergence analysis of the gauss newton method for convex inclusion and convex composite optimization problems abstract using the convex process theory we study the convergence issues of the iterative sequences generated by the gauss newton method for the convex inclusion problem defined by a cone c and a smooth function f the derivative is denoted by f the restriction in our consideration is minimal and even in the classical case the initial point x 0 is assumed to satisfy the following two conditions f is lipschitz around x 0 and the convex process t x 0 defined by t x 0 f x 0 c is surjective our results are new in giving sufficient conditions which are weaker than the known ones and have a remarkable property being affine invariant ensuring the convergence of the iterative sequence with initial point x 0 the same study is also made for the so called convex composite optimization problem with objective function given as the composite of a convex function with a smooth map

variational analysis of a composite function a formula for the lower second order epi derivative an exact formula is established for the lower second order epi derivative of a function of the form g f x where f is a smooth map from one banach space into another and g is a convex function generally not everywhere finite unconstrained minimization of such functions typically arise as an equivalent in one or another sense reduction form for many important classes of constrained optimization problems the formula is further applied to study epi differentiability of the max function x max q x q q

second order necessary and sufficient conditions for convex composite ndo in convex composite ndo one studies the problem of minimizing functions of the formf h f whereh m is a finite valued convex function andf n m is continuously differentiable this problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework e g convex inclusions minimax problems and penalty methods for constrained optimization in the present work we extend the second order theory developed by a d ioffe in 11 12 13 for the case in whichh is sublinear to arbitrary finite valued convex functionsh moreover a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature

shannon mcmillan theorems for discrete random fields along curves and lower bounds for surface order large deviations the notion of a surface order specific entropy hc p of a two dimensional discrete random field p along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c existence is shown by proving a corresponding shannon mcmillan theorem we obtain a representation of hc p as a mixture of specific entropies along the tangent lines of c as an application the specific entropy along curves is used to refine follmer and ort s lower bound for the large deviations of the empirical field of an attractive gibbs measure from its ergodic behaviour in the phase transition regime

approximate solutions for abstract inequality systems we consider conic inequality systems of the type f x ge k 0 with approximate solution x 0 associated to a parameter tau where f is a twice frechet differentiable function between hilbert spaces x and y and ge k is the partial order in y defined by a nonempty convex not necessarily closed cone k subseteq y we prove that under the suitable conditions the system f x ge k 0 is solvable and the ratio of the distance from x 0 to the solution set s over the distance from f x 0 to the cone k has an upper bound given explicitly in terms of tau and x 0 we show that the upper bound is sharp applications to analytic function inequality equality systems on euclidean spaces are given and the corresponding results of dedieu siam j optim 11 2000 pp 411 425 are extended and significantly improved

convergence of the gauss newton method for convex composite optimization under a majorant condition under the hypothesis that an initial point is a quasi regular point we use a majorant condition to present a new semilocal convergence analysis of an extension of the gauss newton method for solving convex composite optimization problems in this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where a gauss newton sequence is well behaved

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

the kustaanheimo stiefel map the hopf fibration and the square root map on r3 and r4 abstract we study the kustaanheimo stiefel map ksm from u r 4 0 to x r 3 0 and the principal circle bundle p u x s 1 that it induces we show that the ksm is the appropriate generalization of the squaring map z z 2 z c and not quaternion multiplication in that the ksm induces a principal circle bundle on s 3 s 2 namely the hopf fibration while quaternion squaring is degenerate because the dimension of the fibers is not constant we construct two square root branches from the upper and lower half of r 3 to r 3 x 1 where x 1 is the nonpositive x 1 axis in r 3 and resembles the cut used to define the standard complex square root branches z we glue these two branches together we introduce what we like to call ks cylindrical coordinates with a 2 dimensional axis of rotation we also introduce what we call ks torical and spherical coordinates we use the ks cylindrical coordinates to define the full square root map on an s 1 cover of r 3 given by r 3 s 1 where is an equivalence relation on x 1 s 1

on concentration inequalities and their applications for gibbs measures in lattice systems we consider gibbs measures on the configuration space s mathbb z d where mostly d ge 2 and s is a finite set we start by a short review on concentration inequalities for gibbs measures in the dobrushin uniqueness regime we have a gaussian concentration bound whereas in the ising model and related models at sufficiently low temperature we control all moments and have a stretched exponential concentration bound we then give several applications of these inequalities whereby we obtain various new results amongst these applications we get bounds on the speed of convergence of the empirical measure in the sense of kantorovich distance fluctuation bounds in the shannon mcmillan breiman theorem fluctuation bounds for the first occurrence of a pattern as well as almost sure central limit theorems

newton s method for solving generalized equations kantorovich s and smale s approaches abstract in this paper we study newton type methods for solving generalized equations involving set valued maps in banach spaces kantorovich type theorems both local and global versions are proved as well as the quadratic convergence of the newton sequence we also extend smale s classical theory to generalized equations these results are new and can be considered as an extension of many known ones in the literature for classical nonlinear equations our approach is based on tools from variational analysis the metric regularity concept plays an important role in our analysis

constraint qualifications for optimality conditions and total lagrange dualities in convex infinite programming abstract for an inequality system defined by an infinite family of proper convex functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications under the new constraint qualifications we provide necessary and or sufficient conditions for the kkt rules to hold similarly we provide characterizations for constrained minimization problems to have total lagrangian dualities several known results in the conic programming problem are extended and improved

lagrange multipliers and optimality lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first order optimality conditions formally as a system of equations modern applications with their emphasis on numerical methods and more complicated side conditions than equations have demanded deeper understanding of the concept and how it fits into a larger theoretical picture a major line of research has been the nonsmooth geometry of one sided tangent and normal vectors to the set of points satisfying the given constraints another has been the game theoretic role of multiplier vectors as solutions to a dual problem interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black and white constraints to be replaced by penalty expressions this paper traces su

on convergence rates of linearized proximal algorithms for convex composite optimization with applications in the present paper we investigate a linearized proximal algorithm lpa for solving a convex composite optimization problem each iteration of the lpa is a proximal minimization of the convex composite function with the inner function being linearized at the current iterate the lpa has the attractive computational advantage that the solution of each subproblem is a singleton which avoids the difficulty as in the gauss newton method gnm of finding a solution with minimum norm among the set of minima of its subproblem while still maintaining the same local convergence rate as that of the gnm under the assumptions of local weak sharp minima of order p p in 1 2 and a quasi regularity condition we establish a local superlinear convergence rate for the lpa we also propose a globalization strategy for the lpa based on a backtracking line search and an inexact version of the lpa we further apply the lpa to solve a possibly nonconvex feasibility problem as well as a sensor network localiza

local properties of algorithms for minimizing nonsmooth composite functions this paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionf x f x h c x wheref andc are smooth buth c may be nonsmooth local convergence at a second order rate is established for the generalized gauss newton method whenh is convex and globally lipschitz and the minimizer is strongly unique local convergence at a second order rate is established for a generalized newton method when the minimizer satisfies nondegeneracy strict complementarity and second order sufficiency conditions assuming the minimizer satisfies these conditions necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established necessary and sufficient conditions for a two step superlinear rate of convergence are also established when only reduced curvature information is available all these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

descent methods for composite nondifferentiable optimization problems we present a framework for the development of globally defined descent algorithms for the minimization of non differentiable objective functionsf h o f withh convex within our structure the global convergence properties of the cauchy modified newton gauss newton and variable metric methods are easily established along with that of several new approaches examples illustrating the calculational techniques are provided

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

exponential convergence under mixing we show that for a mixing stationary sequence of bounded random variables the average of the firstn variables converges exponentially fast withn to the mean value of these random variables

extended newton methods for conic inequalities approximate solutions and the extended smale α theory abstract using the convex process theory and the majorizing technique we study the convergence issues of the iterative sequences generated by the extended newton method for solving the conic inequality system f c 0 defined by a cone c and a frechet differentiable function f satisfying the extended weak condition convergence criterion of the extended newton method is presented as an application we establish when f is analytic an extended theory similar to smale s theory for nonlinear analytic equations

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

finite size scaling for first order phase transitions rigorous results abstract recent rigorous results on finite size effects for lattice models are reviewed we consider symmetric and asymmetric first order phase transitions in cubic and cylindrical geometries and discuss the determination of transition points from finite size data

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

some new results on the two dimensional kinetic ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel with open boundary conditions in the absence of an external field and at large inverse temperature we prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap as a consequence we are able to show that there exists a new time scaleteven much smaller than the global relaxation timetrel such that with large probability any initial configuration first relaxes to one of the two phases in a time scale of orderteven and only after a time scale of the order oftrel does it reach the final equilibrium by jumping via a large deviation to the opposite phase it also follows that with large probability the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time

iteration functions in some nonsmooth optimization algorithms abstract in this paper it is proved that a locally lipschitzian function has a phr iteration function or a qs iteration function if and only if it is pseudo regular and a locally lipschitzian function has a positive homogeneous phr iteration function or a positive homogeneous qs iteration function if and only if it is continuously differentiable in addition we also give the computable phr iteration functions or qs iteration functions for a piece wise smooth function and a subsmooth function respectively

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise