global holderian error bound for nondegenerate polynomials let f colon mathbb r n to mathbb r be a polynomial and s x in mathbb r n f x le 0 let f x max 0 f x if there exist c 0 alpha 0 beta 0 such that d x s le c f x alpha f x beta for all x in mathbb r n where d x s denotes the distance between x and s then we say that f has a global holderian error bound we prove that if f is convenient and nondegenerate with respect to its newton boundary at infinity in the sense of kouchnirenko invent math 32 1976 pp 1 32 then it has a global holderian error bound since the nondegenerate polynomials form a zariski open subset in a variety of polynomials with a given convenient newton boundary at infinity this result shows that global holderian error bounds hold for a large class of polynomials which can be relatively easy recognized

asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

error bounds for parametric polynomial systems with applications to higher order stability analysis and convergence rates the paper addresses parametric inequality systems described by polynomial functions in finite dimensions where state dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities such systems can be viewed in particular as solution sets to problems of generalized semi infinite programming with polynomial data exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry we establish a far going extension of the ojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher order holder type local error bounds results with explicitly calculated exponents the obtained results are applied to higher order quantitative stability analysis for various classes of optimization problems including generalized semi infinite programming with polynomial data optimization of real polynomials under polynomial matrix inequality constraints and polynomial second order cone programming other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings

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

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

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

on the asymptotically well behaved functions and global error bound for convex polynomials in this paper we present some new and tractable sufficient conditions for convex asymptotically well behaved awb functions moreover we establish several lipschitz and holder type global error bound results for a single convex polynomial and for functions which can be expressed as maximum of finitely many nonnegative convex polynomials an advantage of our approach is that the corresponding holder exponent in our holder type global error bound results can be determined explicitly

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

error bounds for piecewise convex quadratic programs and applications in this paper we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program these error estimates provide a unified approach for deriving many old and new error estimates for linear programs linear complementarity problems convex quadratic programs and affine variational inequality problems the approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program in a sense even robinson s result on the upper lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program as an application we derive new global error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program

error bounds for degenerate cone inclusion problems error bounds for cone inclusion problems in banach spaces are established under conditions weaker than robinson s constraint qualification the results allow the cone to be more general than the origin and therefore also generalize a classical error bound result concerning equality constrained sets in optimization applications in finite dimensional differentiable inequalities and in tangent cones are discussed

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

almost sure stability of dynamical systems under combined harmonic and stochastic excitations abstract the almost sure asymptotic stability and the associated sample behavior of mechanical and structural systems under combined harmonic and stochastic excitations are examined the method of extended stochastic averaging developed by the author is used to obtain a lower dimensional markov processes for multi degree of freedom structures subjected to both harmonic and stochastic excitations provided that the intensities of these excitations are small the approach in this paper formulated for the study of almost sure asymptotic stability is based on the well developed concept of lyapunov exponents the largest lyapunov exponent is obtained explicitly by using the lower dimensional approximate ito equations the results from this analysis are applied to study the almost sure asymptotic stability of uncoupled flapping motion of rotor blades in forward flight under the effect of atmospheric turbulence

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

approximate solutions of analytic inequality systems we consider the stability problem for a system of equalities and inequalities we prove that when such a system is satisfied approximately at a certain point it is also satisfied exactly on a nearby point

bounded linear regularity of convex sets in banach spaces and its applications this paper deals with bounded linear regularity linear regularity and the strong conical hull intersection property chip of a collection of finitely many closed convex intersecting sets in banach spaces it is shown that as in finite dimensional space setting see 6 the standard constraint qualification implies bounded linear regularity which in turn yields the strong conical hull intersection property and that the collection of closed convex sets c1 cn is bounded linearly regular if and only if the tangent cones of c1 cn has the chip and the normal cones of c1 cn has the property g uniformly on a neighborhood in the intersection c as applications we study the global error bounds for systems of linear and convex inequalities

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

on train track splitting sequences we present a structure theorem for the subsurface projections of train track splitting sequences for the proof we introduce induced tracks efficient position and wide curves as a consequence of the structure theorem we prove that train track sliding and splitting sequences give quasi geodesics in the train track graph this generalizes a result of hamenstadt

strong chip normality and linear regularity of convex sets we extend the property n introduced by jameson for closed convex cones to the normal property for a finite collection of convex sets in a hilbert space variations of the normal property such as the weak normal property and the uniform normal property are also introduced a dual form of the normal property is derived when applied to closed convex cones the dual normal property is the property g introduced by jameson normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property strong chip and various regularity properties in particular we prove that the weak normal property is a dual characterization of the strong chip and the uniform normal property is a characterization of the linear regularity moreover the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

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

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

condition operators condition numbers and condition number theorem for the generalized eigenvalue problem abstract we define the condition operators and the condition numbers associated with a well posed generalized eigenvalue problem and we study the relationship between the condition numbers and the distance to the ill posed problems

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

multihomogeneous newton methods we study multihomogeneous analytic functions and a multihomogeneous newton s method for finding their zeros we give a convergence result for this iteration and we study two examples the evaluation map and the generalized eigenvalue problem

regularities and their relations to error bounds in this paper we mainly study various notions of regularity for a finite collection c1 cm of closed convex subsets of a banach space x and their relations with other fundamental concepts we show that a proper lower semicontinuous function f on x has a lipschitz error bound resp error bound if and only if the pair epi f x 0 of sets in the product space x is linearly regular resp regular similar results for multifunctions are also established next we prove that c1 cm is linearly regular if and only if it has the strong chip and the collection nc1 z ncm z of normal cones at z has property g for each z c i 1mci provided that c1 is a closed convex cone and that c2 y is a closed vector subspace of x we show that c1 y is linearly regular if and only if there exists 0 such that each positive relative to the order induced by c1 linear functional on y of norm one can be extended to a positive linear functional on x with norm bounded by similar characterization is given in terms of normal cones

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

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

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

optimal bounded control of first passage failure of strongly non linear oscillators under combined harmonic and white noise excitations a procedure for designing the optimal bounded control of strongly non linear oscillators under combined harmonic and white noise excitations for minimizing their first passage failure is proposed first a stochastic averaging method for strongly non linear oscillators under combined harmonic and white noise excitations using generalized harmonic functions is introduced then the dynamical programming equations and their boundary and final time conditions for the control problems of maximizing reliability and of maximizing mean first passage time are formulated from the averaged ito equations by using the dynamical programming principle the optimal control law is derived from the dynamical programming equations and control constraint finally the conditional reliability function the conditional probability density and mean of the first passage time of the optimally controlled system are obtained from solving the backward kolmogorov equation and pontryagin equation an example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation

a condition number for differentiable convex inequalities for a system of differentiable convex inequalities a new bound is given for the absolute error in an infeasible point in terms of the absolute residual by using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual

error bounds for inequality systems defining convex sets the main goal in this paper is to devise an approach to explicitly calculate the constant in the hoffman s error bound for not necessarily convex inequality systems defining convex sets we give a constructive proof of the hoffman s error bound and show that we can use our method to calculate the constant at least in simple cases

maximal lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise asymptotic expansions for the exponential growth rate known as the lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise are constructed such systems arise naturally in the investigation of the stability of steady state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems almost sure stability or instability of dynamical systems depends on the sign of the maximal lyapunov exponent stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable the results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging where approximate ito equations in amplitudes and phases are obtained in the sense of weak convergence

the geometry of the disk complex we give a distance estimate for the disk complex we use the distance estimate to prove that the disk complex is gromov hyperbolic as another application of our techniques we find an algorithm which computes the hempel distance of a heegaard splitting up to an error depending only on the genus

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

error bounds for convex differentiable inequality systems in banach spaces the paper is devoted to studying the hoffman global error bound for convex quadratic affine inequality equality systems in the context of banach spaces we prove that the global error bound holds if the hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration this result is applied to establishing the equivalence between the hoffman error bound and the abadie qualification condition as well as a general version of wang pang s result 30 on error bound of holderian type the results in the present paper generalize and unify recent works by luo luo in 17 li in 16 and wang pang in 30

computational complexity on the geometry of polynomials and a theory of cost ii this paper deals with traditional algorithms newton s method and a higher order generalization due to euler these iterations schemes and their modifications have had a great success in solving nonlinear systems of equations we give some understanding of this phenomenon by giving estimates of efficiency the problem we focus on is that of finding a zero of a complex polynomial

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

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

geometry of the mapping class groups i boundary amenability we construct a geometric model for the mapping class group mathcal m mathcal c mathcal g of a non exceptional oriented surface s of genus g with k punctures and use it to show that the action of mathcal m mathcal c mathcal g on the compact metrizable hausdorff space of complete geodesic laminations for s is topologically amenable as a consequence the novikov higher signature conjecture holds for every subgroup of mathcal m mathcal c mathcal g

almost sure asymptotic stability of a general four dimensional system driven by real noise in the first part of this paper we construct an asymptotic expansion for the maximal lyapunov exponent the exponential growth rate of solutions to a linear stochastic system and the rotation numbers for a general four dimensional dynamical system driven by a small intensity real noise process stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue this work is an extension of the work of sri namachchivaya and van roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered

largest lyapunov exponent for second order linear systems under combined harmonic and random parametric excitations the principal resonance of a second order linear stochastic oscillator to combined harmonic and random parametric excitations is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning bandwidth and magnitudes of random excitation are analyzed the method of path integration is used to obtain the steady state probability density function of the system and then the largest lyapunov exponent is calculated the almost sure stability or instability of the stochastic system depends on the sign of the largest lyapunov exponent the theoretical analyses are verified by numerical results

linear semi infinite programming theory an updated survey abstract this paper presents an state of the art survey on linear semi infinite programming theory and its extensions in particular convex semi infinite programming this review updates a previous survey semi infinite programming non convex optim appl 25 1998 of the same authors on the same topic which was published in 1998

a continuation method to solve polynomial systems and its complexity in a recent work shub found comput math 9 171 178 2009 shub obtained a new upper bound for the number of steps needed to continue a known zero 0 of a system f 0 to a zero t of an input system f t following the path of pairs f t t where f t t in 0 t is a polynomial system and f t t 0 he proved that if one can choose the step size in an optimal way then the number of steps is essentially bounded by the length of the path of f t t in the so called condition metric however the proof of that result in shub found comput math 9 171 178 2009 is not constructive we give an explicit description of an algorithm which attains that complexity bound including the choice of step size

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

set regularities and feasibility problems we synthesize and unify notions of regularity both of individual sets and of collections of sets as they appear in the convergence theory of projection methods for consistent feasibility problems several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

hyperbolicity of the genus two hatcher thurston complex for the genus 1 surface with n punctures f 1 n we show that the hatcher thurston complex fancyscript ht f 1 n is hyperbolic for the genus 2 closed surface f 2 we show that the hatcher thurston complex fancyscript ht f 2 is strongly relatively hyperbolic with respect to a family of its subspaces x where ranges over all separating curves of f 2 and each x is isometric to the product of two farey graphs

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

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

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

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

harmonic measures for distributions with finite support on the mapping class group are singular kaimanovich and masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into teichmuller space converges almost surely to a point in the space pmf of projective measured foliations on the surface this defines a harmonic measure on pmf here we show that when the initial distribution has finite support the corresponding harmonic measure is singular with respect to the natural lebesgue measure class on pmf

error bounds for nondifferentiable convex inequalities under a strong slater constraint qualification a global error bound is given on the distance between an arbitrary point in then dimensional real spacern and its projection on a nonempty convex set determined bym convex possibly nondifferentiable inequalities the bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong slater constraint qualification cq which implies the ordinary slater cq a very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary slater cq is given first and then used to derive the principal global error bound

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

weak sharp minima revisited part iii error bounds for differentiable convex inclusions the notion of weak sharp minima unifies a number of important ideas in optimization part i of this work provides the foundation for the theory of weak sharp minima in the infinite dimensional setting part ii discusses applications of these results to linear regularity and error bounds for nondifferentiable convex inequalities this work applies the results of part i to error bounds for differentiable convex inclusions a number of standard constraint qualifications for such inclusions are also examined

error bound moduli for conic convex systems on banach spaces we give two explicit formulas which express the error bound moduli for conic convex systems one in terms of directional derivative and the other in terms of the coderivative as applications we study error bounds for systems of infinitely many convex inequalities

weak sharp minima revisited part ii application to linear regularity and error bounds the notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems it has been studied extensively by several authors this paper is the second of a series on this subject where the basic results on weak sharp minima in part i are applied to a number of important problems in convex programming in part ii we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming we obtain both new results and reproduce several existing results from a fresh perspective

a comparison of eigenvalue condition numbers for matrix polynomials abstract in this paper we consider the different condition numbers for simple eigenvalues of matrix polynomials used in the literature and we compare them one of these condition numbers is a generalization of the wilkinson condition number for the standard eigenvalue problem this number has the disadvantage of only being defined for finite eigenvalues in order to give a unified approach to all the eigenvalues of a matrix polynomial both finite and infinite two homogeneous condition numbers have been defined in the literature in their definition very different approaches are used one of the main goals of this note is to show that when the matrix polynomial has a moderate degree both homogeneous condition numbers are essentially the same and one of them provides a geometric interpretation of the other we also show how the homogeneous condition numbers compare with the wilkinson like eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples

error bounds for linear matrix inequalities for iterative sequences that converge to the solution set of a linear matrix inequality we show that the distance of the iterates to the solution set is at most o epsilon 2 d the nonnegative integer d is the so called degree of singularity of the linear matrix inequality and epsilon denotes the amount of constraint violation in the iterate for infeasible linear matrix inequalities we show that the minimal norm of epsilon approximate primal solutions is at least 1 o epsilon 1 2 d 1 and the minimal norm of epsilon approximate farkas type dual solutions is at most o 1 epsilon 2 d 1 as an application of these error bounds we show that for any bounded sequence of epsilon approximate solutions to a semidefinite programming problem the distance to the optimal solution set is at most o epsilon 2 k where k is the degree of singularity of the optimal solution set