nonexpansive retractions in hyperconvex spaces this paper is primarily concerned with the study of conditions on a hyperconvex subset d of a hyperconvex metric space m which assure that there exists a nonexpansive retraction r of m d onto d which has the property that r m d d a related question we take up is when is such a retraction r proximinal that is when does r have the propertydx rx distx dfor each x m among other things we show that if a subset d of a hyperconvex metric space m has nonempty interior and is externally hyperconvex relative to m in a very weak sense then there always exists a nonexpansive retraction of m onto d which maps m d onto d we also show that any compact weakly externally hyperconvex subset of a hyperconvex m is a proximinal nonexpansive retract of m

an empirical analysis of scenario generation methods for stochastic optimization this work presents an empirical analysis of popular scenario generation methods for stochastic optimization including quasi monte carlo moment matching and methods based on probability metrics as well as a new method referred to as voronoi cell sampling solution quality is assessed by measuring the error that arises from using scenarios to solve a multi dimensional newsvendor problem for which analytical solutions are available in addition to the expected value the work also studies scenario quality when minimizing the expected shortfall using the conditional value at risk to quickly solve problems with millions of random parameters a reformulation of the risk averse newsvendor problem is proposed which can be solved via benders decomposition the empirical analysis identifies voronoi cell sampling as the method that provides the lowest errors with particularly good results for heavy tailed distributions a controversial finding concerns evidence for the ineffectiveness of widely used methods based on minimizing probability metrics under high dimensional randomness

addendum to on the stability of functional equations on square symmetric groupoid abstract let x be a square symmetric groupoid and y d a complete metric divisible square symmetric groupoid in this paper we obtain the hyers ulam stability problem of functional inequality d g x y g x g y e x y for approximate mapping g x y of functional equation f x y f x f y differing by e x 2 0 in particular we have investigated the case of f x f y h f x 1 t f y 1 t on some set y

on the fixed point set of nonexpansive order preserving maps for every pair x and y in c a point z in c is a fixed point for t provided tz z nonexpansive maps have been studied extensively for over two decades with the objectives motivated by applications of proving the existence of fixed points under appropriate conditions and finding iteration methods leading to fixed points in particular finding conditions for the convergence of the iterates themselves we refer the reader to the survey articles of bruck b2 and kirk ki most of the results of the general theory assume convexity conditions on the banach space x or some geometrical or topological conditions on t and its iterates thus the spaces l1 and l o are usually excluded from general studies investigations in loo and other c e spaces were mode by the second author and others e g is so crandall and tartar ct gave conditions under which a nonexpansive map t on l t or on l is also order preserving that is

diversities hyperconvexity and fixed points abstract diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated in this work we take up a number of questions about hyperconvexity diversities and fixed points of nonexpansive mappings most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers examples are given for instance showing that such a metric space need not be hyperconvex but still we prove as our main result that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space

one local retracts and banach operator pairs in metric spaces abstract in this paper we first introduce the concept of nr map and then use this concept to establish the existence of common fixed points for banach operator pairs in the context of uniformly convex geodesic metric spaces new proofs of main results theorems 2 1 and 3 5 of chen and li j chen z li banach operator pair and common fixed points for nonexpansive maps nonlinear anal 74 2011 3086 3090 are presented further we prove de marr s theorem for the family of symmetric banach operator pairs in metric spaces and r trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity

scenario tree generation for multiperiod financial optimization by optimal discretization multiperiod financial optimization is usually based on a stochastic model for the possible market situations there is a rich literature about modeling and estimation of continuous state financial processes but little attention has been paid how to approximate such a process by a discrete state scenario process and how to measure the pertaining approximation error in this paper we show how a scenario tree may be constructed in an optimal manner on the basis of a simulation model of the underlying financial process by using a stochastic approximation technique consistency relations for the tree may also be taken into account

set valued contractions and fixed points abstract a common fixed point theorem is proved for a family of set valued contraction mappings in gauge spaces this result is related to a recent result of frigon for generalized contractions and it includes a method for approximating the fixed point the remainder of the paper is devoted to results for families of set valued contraction mappings in hyperconvex spaces it is proved for example that if m is a hyperconvex metric space and f is a family of set valued contractions indexed over a directed set and taking values in the space of all nonempty admissible subsets of m endowed with the hausdorff metric then the condition f x f x for all x m and implies that the set of points x m for which x f x is nonempty and hyperconvex

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

a selection theorem for quasi lower semicontinuous mappings in hyperconvex spaces we prove a continuous selection theorem for quasi lower semicontinuous mappings with values that are closed sub admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces

best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces abstract in hyperconvex metric spaces we consider best approximation invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive

hyperconvexity and nonexpansive multifunctions it is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection this implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps the result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex

the space of psi gamma additive mappings on semigroups in this paper we introduce the concept of pseudoadditive mappings from a semigroup into a banach space and we provide a generalized solution of ulam s problem for approximately additive mappings

best approximation and fixed point theorems in hyperconvex metric spaces abstract we develop best approximation and fixed point theorems in a hyperconvex metric space for multivalued mappings f that are nonexpansive or continuous in the hausdorff metric the best approximation results assume either admissible or externally hyperconvex sets as the domain x or value of the mapping the fixed point results assume either admissible sub admissible or externally hyperconvex sets as the domain or value of the mapping and that f x

generating moment matching scenarios using optimization techniques an optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming the main advantage of the method is its flexibility it can generate scenarios matching any prescribed set of moments of the underlying distribution rather than matching all moments up to a certain order and the distribution can be defined over an arbitrary set this allows for a reduction in the number of scenarios and allows the scenarios to be better tailored to the problem at hand the method is based on a semi infinite linear programming formulation of the problem that is shown to be solvable with polynomial iteration complexity a practical column generation method is implemented the column generation subproblems are polynomial optimization problems however they need not be solved to optimality it is found that the columns in the column generation approach can be efficiently generated by random sampling the number of scenarios generated matches a lower

banach operator pairs and common fixed points in hyperconvex metric spaces the purpose of this paper is to establish demarr s well known theorem for an arbitrary family of symmetric banach operator pairs in hyperconvex metric spaces without the compactness assumption we also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings as an application several results on the invariant best approximation are proved

hopfs ergodic theorem for nonlinear operators 1 t f 1 n l i 0 are a e convergent in this paper we will give a generalization of hopfs theorem to a wide class of nonlinear operators in the nonlinear situation instead of being a contraction one usually assumes that t is nonexpansive or at least norm decreasing on l i or l the definition 1 can of course be literally generalized to nonlinear mappings and the partial sums can be also defined inductively by sof f and s 1 f f s tf from the point of view of the inductive definition it is equally well justified to define inductively sof f and s i f f t s f and we will adopt throughout this paper always the latter definition and define the 1 averages a f snf there are several reasons for doing this firstly for the

on a direct method for proving the hyers ulam stability of functional equations abstract we study the stability of an equation in a single variable of the form f x a f h x b f h x with some conditions imposed on constants a b and function h the results are later applied for proving the stability by use of a direct method of a couple of functional equations in several variables they let us not only shorten various approaches obtained during the last years but also improve the approximating constants

a fixed point approach to the stability of isometries in this paper we apply a fixed point theorem to the proof of hyers ulam rassias stability property for isometries from a normed space into a banach space in which the parallelogram law holds

retractions on the fixed point set of semigroups of nonexpansive maps in hyperconvex spaces the term hyperconvex is due to aronszajn and panitchpakdi 3 for other papers dealing with hyperconvexity see 4 9 10 19 231 order intervals in l are hyperconvex but weak compact convex sets of l need not be moreover hyperconvex sets of l need not be convex a map t on a metric space for which d tx ty s d x y holds for all pairs of points is called nonexpansive there is now a considerable literature on various properties of such maps addressing the questions of existence of fixed points structure of the fixed point set and related matters among several recent surveys are 5 lo and i 11 as a general source of information on nonexpansive maps retractions and asymptotic centers we mention the book s in this paper we examine such questions for families of nonexpansive maps of hyperconvex spaces the recent papers 13 and 12 contain some related results

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

nonlinear models of diffusion on a finite space the basic convergence theorems for finite state markov chains are extended to the nonlinear case an operatort inl1 of a finite space with counting measure is called nonexpansive if tf tg 1 f g 1 holds for allf g it is shown that for anyf there exists an integer p 1 such thattpnf converges sufficient conditions forp 1 are given in the case of continuous parameter nonexpansive semigrous tt t 0 ttf converges fort

general stability of functional equations of linear type making use of a dynamical systems notion called shadowing we prove a stability result for linear functional equations in metric groups as a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2 divisibility condition

comments on the core of the direct method for proving hyers ulam stability of functional equations in this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form

scenario reduction revisited fundamental limits and guarantees the goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms we distinguish continuous scenario reduction where the new atoms may be chosen freely and discrete scenario reduction where the new atoms must be chosen from among the existing ones using the wasserstein distance as measure of proximity between distributions we identify those n point distributions on the unit ball that are least susceptible to scenario reduction i e that have maximum wasserstein distance to their closest m point distributions for some prescribed m n we also provide sharp bounds on the added benefit of continuous over discrete scenario reduction finally to our best knowledge we propose the first polynomial time constant factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential time algorithms for continuous scenario reduction

variance reduction in sample approximations of stochastic programs this paper studies the use of randomized quasi monte carlo methods rqmc in sample approximations of stochastic programs in numerical integration rqmc methods often substantially reduce the variance of sample approximations compared to monte carlo mc it seems thus natural to use rqmc methods in sample approximations of stochastic programs it is shown that rqmc methods produce epi convergent approximations of the original problem rqmc and mc methods are compared numerically in five different portfolio management models in the tests rqmc methods outperform mc sampling substantially reducing the sample variance and bias of optimal values in all the considered problems

lambda hyperconvexity in metric spaces we introduce the concept of hyperconvexity in metric spaces generalizing the classical notion of a hyperconvex metric space we show that a bounded metric space which is hyperconvex has the fixed point property for nonexpansive mappings provided 2 uniformly convex banach spaces are examples of such hyperconvex spaces for some 2 we furthermore investigate the relationship between penot s intersection property and 2hyperconvexity

importance sampling in stochastic optimization an application to intertemporal portfolio choice abstract in this paper we propose an approach to construct an analytical approximation of the zero variance importance sampling distribution we show specifically how this can be designed for the classic intertemporal portfolio choice problem with proportional transaction costs and constant relative risk aversion preferences we compare the method to standard variance reduction techniques in single period optimization and multi stage stochastic programming formulations of the problem the numerical experiments show that the method produces significant improvements in solution quality in the single period setting the number of scenarios can be reduced by a factor of 400 with maintained solution quality compared to the best standard method latin hypercube sampling using importance sampling in multi stage formulations the gaps between lower and upper bound estimates are reduced by a factor of 26 500 with maintained scenario tree size on a higher level we consider analytical approximations of the zero variance importance sampling distribution to be a promising method to improve solution quality in stochastic optimization

fixed points of a mapping and hyers ulam stability abstract we show that many general results on hyers ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem the theorem is formulated for self maps of some subsets of the space of functions from a nonempty set into the set of reals we also give some applications of that theorem e g in investigations of solutions of some difference equations and functional inequalities

on the fixed points of commuting nonexpansive maps in hyperconvex spaces abstract let t 1 t n be a finite commuting family of nonexpansive maps of a hyperconvex space such that each t i has bounded orbits we show i each point has a bounded orbit under the semigroup generated by t i ii there is a common fixed point for the family if and only if t t 1 t 2 t n has a fixed point iii for each e 0 there is a nonempty set of common e approximate fixed points for the family some additional related results are also given

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

on approximate isometries and application to stability of a functional equation abstract in this paper we show a generalized stability of isometries between banach spaces making use of this result we prove the corresponding stability of the functional equation f x y f x f y

stability of the multi jensen equation abstract in this paper we study the stability in the sense of th m rassias of multi jensen equation

the stability of a cubic type functional equation with the fixed point alternative abstract in this note we investigate the generalized hyers ulam rassias stability for the new cubic type functional equation f x y 2 z f x y 2 z f 2 x f 2 y 2 f x y 2 f x z 2 f x z 2 f y z 2 f y z by using the fixed point alternative the first systematic study of fixed point theorems in nonlinear analysis is due to g isac and th m rassias internat j math math sci 19 1996 219 228

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

approximately additive schwartz distributions generalizing the hyers ulam rassias stability theorem th m rassias on the stability of linear mapping in banach spaces proc amer math soc 72 1978 297 300 to the space of schwartz distributions we introduce a concept of approximately additive schwartz distributions and prove that every approximately additive distribution can be approximated by linear functions

invariant approximations for commuting mappings in cat 0 and hyperconvex spaces abstract in this paper for a commuting pair consisting of a point valued nonexpansive self mapping t and a set valued nonexpansive self mapping t of a hyperconvex metric space or a cat 0 space x we look for a solution to the problem of existence of z e x such that d z y d y e and z t z t z