completely bounded modules and associated extremal problems in this paper we continue our study of certain finite dimensional hilbert modules over the function algebra a cm we show that these modules are always completely bounded with the bound obtained as the matrix valued analogue of a certain scalar valued extremal problem in particular we obtain a necessary and sufficient condition for our module to be completely contractive we produce a contractive module cnm over a bm such that it is completely bounded with the complete bound equal to m that is cnm is not completely contractive

a comparison of second order epiderivatives calculus and optimality conditions abstract the first order theory of generalized derivatives is now well developed significant progress toward a second order theory has also been made two types of second order directional derivatives appear to be particularly promising for applications to optimization parabolic epiderivatives which are well suited for necessary optimality conditions and satisfy standard calculus rules for a large class of functions and the second order epiderivatives of rockafellar which lead to general sufficient optimality conditions that are close to being necessary in this paper the properties of these two types of epiderivatives are surveyed and contrasted special results for c1 1 functions are examined in detail

neural network based reliability analysis a comparative study abstract a study on the applicability of different kinds of neural networks for the probabilistic analysis of structures when the sources of randomness can be modeled as random variables is summarized the networks are employed as numerical devices for substituting the finite element code needed by monte carlo simulation the comparison comprehends two network types multi layer perceptrons and radial basis functions classifiers cost functions sum of square errors and cross entropy optimization algorithms back propagation gauss newton newton raphson sampling methods for generating the training population using uniform and actual distributions of the variables and purposes of neural network use as functional approximators and data classifiers the comparative study is performed over four examples corresponding to different types of the limit state function and structural behaviors the analysis indicates some recommended ways of employing neural networks in this field

second order characterizations of tilt stability with applications to nonlinear programming the paper is devoted to the study of tilt stable local minimizers of general optimization problems in finite dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization and this notion has been extensively studied in the literature based on advanced tools of second order variational analysis and generalized differentiation we develop a new approach to tilt stability which allows us to derive not only qualitative but also quantitative characterizations of tilt stable minimizers with calculating the corresponding moduli the implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt stable minimizers under new second order qualification and optimality conditions

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

on some relations between chaney s generalizedsecond order directional derivative and that of ben tal andzowe for a locally lipschitz real valued function f on bbb r n and x u in bbb r n our main result implies that if x is in clarke s subdifferential partial f x coming from the direction u in chaney s sense such that x u equals the directional derivative f x u then chaney s second order directional derivative f x x u when it exists coincides with the value at x of the conjugate function of the ben tal zowe second order directional derivative provided that this value is finite

reliability analysis of randomly vibrating structures with parameter uncertainties the problem of reliability analysis of randomly driven stochastically parametered vibrating structures is considered the excitation components are assumed to be jointly stationary gaussian random processes the randomness in structural parameters is modelled through a vector of mutually correlated non gaussian random variables the structure is defined to be safe if a set of response quantities remain within prescribed thresholds over a given duration of time an approximation for the failure probability conditioned on structure randomness is first formulated in terms of recently developed multivariate extreme value distributions an estimate of the unconditional failure probability is subsequently obtained using a method based on taylor series expansion this has involved the evaluation of gradients of the multivariate extreme value distribution conditioned on structure random variables with respect to the basic variables analytical and numerical techniques for computing these gradients have been discussed issues related to the accuracy of the proposed method have been examined the proposed method has applications in reliability analysis of vibrating structural series systems where different failure modes are mutually dependent illustrative examples presented include the seismic fragility analysis of a piping network in a nuclear power plant

an enhanced hybrid method for the simulation of highly skewed non gaussian stochastic fields abstract in this paper an enhanced hybrid method ehm is presented for the simulation of homogeneous non gaussian stochastic fields with prescribed target marginal distribution and spectral density function the presented methodology constitutes an efficient blending of the deodatis micaletti method with a neural network based function approximation precisely the function fitting ability of neural networks based on the resilient back propagation rprop learning algorithm is employed to approximate the unknown underlying gaussian spectrum the resulting algorithm can be successfully applied for simulating narrow banded fields with very large skewness at a fraction of the computing time required by the existing methods its computational efficiency is demonstrated in three numerical examples involving fields that follow the beta and lognormal distributions

second order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization abstract in this paper we introduce a new second order directional derivative and a second order subdifferential of hadamard type for an arbitrary nondifferentiable function we derive second order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second order in unconstrained optimization in particular we obtain two results with strongly pseudoconvex functions we also compare our conditions with the results of the recently published paper bednarik and pastor 2008 and a lot of other works published in high level journals and prove that they are particular cases of our necessary and sufficient ones we prove that the necessary optimality conditions concern more functions than the conditions in terms of lower dini directional derivative even the optimality conditions with the last derivative can be applied to a function which does not belong to some special class at last we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone constrained vector optimization

fuzzy regression analysis using neural networks abstract in this paper we propose simple but powerful methods for fuzzy regression analysis using neural networks since neural networks have high capability as an approximator of nonlinear mappings the proposed methods can be applied to more complex systems than the existing lp based methods first we propose learning algorithms of neural networks for determining a nonlinear interval model from the given input output patterns a nonlinear interval model whose outputs approximately include all the given patterns can be determined by two neural networks next we show two methods for deriving nonlinear fuzzy models from the interval model determined by the proposed algorithms nonlinear fuzzy models whose h level sets approximately include all the given patterns can be derived last we show an application of the proposed methods to a real problem

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

second order moderate derivatives f x f x it is the purpose of the present paper to introduce a notion of second order generalized derivative for arbitrary finite functions which retains the preceding properties in particular which coincides with the usual second order derivative when it exists and enjoys useful calculus rules our proposal is close to the proposal of 3 cf also 4 the variant we use is similar in spirit to the first order variant of the clarke circa derivatives introduced in l 21 here however the concept of fan plays a key role as in 3 51 let us note that the absence of built in upper semicontinuity is intimately linked with the accuracy of the concept under added assumptions of semicontinuity one recovers the notions of 3 51 our results can be found in 6

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

on generic second order gateaux differentiability let f x be a function on a banach space x we say that f is strictly gateaux differentiable if it is gateaux differentiable at every point of x and the mapping x h x h is continuous recall that a convex gateaux differentiable function is strictly gateaux differentiable in the case of a locally lipschitz function our definition coincides with more standard ones it requires that f be norm to weak star continuous

second order conditions in c 1 1 constrained vector optimization we consider the constrained vector optimization problem minc f x g x k where f n m and g n p are c1 1 functions and c m and k p are closed convex cones with nonempty interiors two type of solutions are important for our considerations namely w minimizers weakly efficient points and i minimizers isolated minimizers we formulate and prove in terms of the dini directional derivative second order necessary conditions for a point x0 to be a w minimizer and second order sufficient conditions for x0 to be an i minimizer of order two we discuss the reversal of the sufficient conditions under suitable constraint qualifications of kuhn tucker type the obtained results improve the ones in liu neittaanmaki k i ek 21

second order necessary conditions for optimality in nonsmooth nonlinear programming the notion of a second order directional derivative introduced by ben tal and zowe and a generalization of this notion by penot are used to state new second order necessary optimality conditions for a nonsmooth nonlinear programming problem with a finite number of inequality and equality constraints

the upper and lower second order directional derivatives of a sup type function the purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup type functions x sup f x t t t in terms of the first and second derivatives off x t wheret is a compact set in a metric space and we assume thatf f x and 2f x2 are continuous on n t we will give a geometrical meaning of the formula we will moreover give a sufficient condition fors x to be directionally twice differentiable

layered neural networks with gaussian hidden units as universal approximations a neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real valued maps defined on convex compact sets of rn

superlinearly convergent approximate newton methods for lc 1 optimization problems in the literature the proof of superlinear convergence of approximate newton or sqp methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions sometimes the second order derivatives of those functions are required to be lipschitzian in this paper we present approximate newton or sqp methods for solving nonlinear programming problems whose objective and constraint functions have locally lipschitzian derivatives and establishq superlinear convergence of these methods under the assumption that these derivatives are semismooth this assumption is weaker than the second order differentiability the extended linear quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives

second order necessary and sufficient conditions in nonsmooth optimization in this paper we generalize and sharpen r w chaney s results on unconstrained and constrained second order necessary and sufficient optimality conditions 5 7 for general lipschitz functions without the semismoothness assumption

strict efficiency in vector optimization the notion of strict minimum of order m for real optimization problems is extended to vector optimization its properties and characterization are studied in the case of finite dimensional spaces multiobjective problems also the notion of super strict efficiency is introduced for multiobjective problems and it is proved that in the scalar case all of them coincide necessary conditions for strict minimality and for super strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set when the objective function is frechet differentiable necessary and sufficient conditions are established for the case m 1 resulting in the situation that the strict efficiency and super strict efficiency notions coincide

on second order conditions in unconstrained optimization the main purpose of this paper is to establish the second order nonsmooth sufficient unconstrained optimality condition for so called l stable at some point functions and in this way to generalize some previous results in this direction we provide the comparisons with other results by examples

a semi analytical framework for structural reliability analysis abstract estimation of the probability of failure of structural systems can often be computationally intensive and time consuming hence alternative techniques for efficient computation of the probability of failure of complex structures while retaining the accuracy is of paramount importance to structural engineering community this paper presents a semi analytical approach for computing failure probability of structure the proposed approach utilizes i polynomial correlated function expansion a novel surrogate modelling technique to determine the moments and ii maximum entropy method to determine an analytical expression for the response probability density function pdf once the analytical expression for the pdf is obtained failure probability is calculated by numerically integrating the pdf in the failure domain as a special case when only the first two moments of response are considered an analytical formula for the probability of failure is also proposed four structural engineering problems are shown to illustrate the performance of the proposed approach it is observed that the proposed approach outperforms other existing approaches

bounded modules extremal problems and a curvature inequality abstract in this paper we study certain finite dimensional hilbert modules over the function algebra a c n these modules appear as localizations of a cowen douglas operator we show that these modules are always bounded where the bound is related to the solution of an extremal problem in particular we obtain necessary sufficient conditions for such a module to be contractive we apply the above results to produce an example of a contractive module over a b 2 which is not completely contractive

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

elimination of strict convergence in optimization we present second order optimality conditions sufficient and necessary and the characterization of convexity in terms of generalized second order derivatives some conclusions obtained by r cominetti and r correa are proved under weakened assumptions

second order variational analysis and characterizations of tilt stable optimal solutions in infinite dimensional spaces abstract the paper is devoted to developing second order tools of variational analysis and their applications to characterizing tilt stable local minimizers of constrained optimization problems infinite dimensional spaces with many results new also in finite dimensional settings the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization based on second order generalized differentiation we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second order growth the results obtained are applied to deriving new necessary and sufficient conditions for tilt stable minimizers in problems of nonlinear programming with twice continuously differentiable data in hilbert spaces

representations of function algebras abstract operator spaces and banach space geometry let g be the closed unit ball of some norm on cn and let a g be the closure of the polynomials in the sup norm we prove that if n 5 then there is a contractive representation of a g as operators on a hilbert space which is not completely contractive our technique involves introducing a numerical invariant x for a normed space x which measures the difference between the minimal operator space structure which can be assigned to x min x and the maximal structure max x we estimate x using banach space techniques we also prove that if x is any infinite dimensional subspace of the space of continuous functions on a compact hausdorff space then there exists a bounded linear map on x which is not completely bounded

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

reanalysis of linear and nonlinear structures using iterated shanks transformation reanalysis of structures is a common task in two important fields of structural engineering namely optimization and reliability it implies the solution of the kinematic equations for each new set of data in order to simplify this task several approximate methods have been developed in this paper a simple method for reanalysis applicable to either linear or nonlinear structures is proposed it is based on the improvement of the neumann resolvent series which diverges beyond a certain radius by means of the iterated shanks transformation the method is compared with two relevant proposals intended to improve the convergence of the neumann series namely the pade series and the scaled approximation methods the accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures

approximate jacobian matrices for nonsmooth continuous maps and c 1 optimization the notion of approximate jacobian matrices is introduced for a continuous vector valued map it is shown for instance that the clarke generalized jacobian is an approximate jacobian for a locally lipschitz map the approach is based on the idea of convexificators of real valued functions mean value conditions for continuous vector valued maps and taylor s expansions for continuously g teaux differentiable functions i e c1 functions are presented in terms of approximate jacobians and approximate hessians respectively second order necessary and sufficient conditions for optimality and convexity of c1 functions are also given

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

approximating a second order directional derivative for nonsmooth convex functions for a lower semicontinuous convex function f the approximate second order directional derivative d delta mapsto f varepsilon x 0 d delta is defined through the varepsilon directional derivative f varepsilon x d the function v d x mapsto v d x f varepsilon x d is for all varepsilon 0 locally lipschitz on inf domf and at those points where it is not differentiable v d admits a directional derivative v d x 0 delta for all delta which we precisely denote by f varepsilon x 0 d delta the objective of the present work is two fold to classify all the possible differentiability properties of v d according to the behavior of the function lambda mapsto f x 0 lambda d on r and to study the existence or nonexistence of the limit of f varepsilon x 0 d delta when varepsilon to 0

necessary and sufficient conditions for isolated local minima of nonsmooth functions we consider the mathematical programming problem find inf f x mid x in c where f is an arbitrary extended real valued function and c a subset of a finite dimensional space we give necessary and sufficient optimality conditions for this problem generalizing previous results of a auslendex this journal 22 1984 pp 239 254

generalized hessian for c 1 1 functions in infinite dimensional normed spaces the subject of this paper is the systematic study of second order notions concerning differentiable functions with lipschitz derivative the results and notions are motivated by recent papers of cominetti correa and hiriart urruty the first goal of this paper is the comparison of several known second order directional derivatives the second goal is the introduction of a generalized hessian which is a set of certain symmetric bilinear forms the relation of this generalized hessian to other existing second order derivatives is also described

some thoughts on ando s theorem and parrott s example in this short note we present an elementary proof of ando s theorem within a restricted class p of homomorphisms modeled after parrott s example we also show by explicit estimation that the cb norm of the contractive homomorphism of the tri disc algebra induced by the commuting triple of parrott exceeds 1 indeed we construct a polynomial p with matrix coefficients with the property p p in particular we show that there are contractive homomorphisms of the tri disc algebra which are not even 2 contractive