kkt conditions for weak compact convex sets theorems of the alternative and optimality conditions abstract we present several equivalent conditions for the karush kuhn tucker conditions for weak compact convex sets using them we extend several existing theorems of the alternative in terms of weak compact convex sets such extensions allow us to express the kkt conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints furthermore several new equivalent optimality conditions for optimization problems with inequality constraints are obtained

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

links between functions and subdifferentials abstract a function in a class f x is said to be subdifferentially determined in f x if it is equal up to an additive constant to any function in f x with the same subdifferential a function is said to be subdifferentially representable if it can be recovered from a subdifferential we identify large classes of lower semicontinuous functions that possess these properties

unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

equivalence among various derivatives and subdifferentials of the distance function abstract for a nonempty closed set c in a normed linear space x with uniformly g teaux differentiable norm it is shown that the distance function d c is strictly differentiable at x x c iff it is regular at x iff its modified upper or lower dini subdifferential at x is a singleton iff its upper or lower dini subdifferential at x is nonempty iff its upper or lower dini derivative at x is subadditive moreover if x is a hilbert space then d c is frechet differentiable at x x c iff its frechet subdifferential at x is nonempty many characteristics of proximally smooth sets and convex closed sets in a hilbert space are also given

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

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

efficient computation of rcs from scatterers of uncertain shapes we propose a way of accounting for the lack of detailed knowledge about material shapes in computational time domain electromagnetics we use legendre gauss lobatto stroud 2 and stroud 3 quadrature formulas to solve the resulting stochastic equation and we show the efficiency of the proposed method over statistical monte carlo simulations we also show how the radar cross section rcs in scattering is affected by the uncertainty in shape of the objects and by the direction of the incident field

optimality conditions in nonconvex optimization via weak subdifferentials abstract in this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions the well known necessary and sufficient optimality condition of nonsmooth convex optimization given in the form of variational inequality is generalized to the nonconvex case by using the notion of weak subdifferentials the equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented

optimality conditions for lower semi continuous functions abstract let f be a lower semi continuous and bounded below function from a banach space x into where x is assumed to admit a lipschitz smooth bump function generalizing results of chaney we study optimality conditions for formula to be a local minimum point of f these conditions are described in terms of generalized chaney s subdifferentials and second order derivatives

quasidifferentials and maximal normal operators quasidifferentials are studied with the theory of maximal normal operators the quasidifferential of a normally quasidifferentiable function is a pair of upper and lower semicontinuous operators which are special maximal normal operators the function sum of the upper and lower semicontinuous operators is the clarke subdifferential of this function basic calculus and minimal forms of quasidifferentials are also discussed

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

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

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

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

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

topological properties of the approximate subdifferential abstract the approximate subdifferential introduced by mordukhovich has attracted much attention in recent works on nonsmooth optimization potential advantages over other concepts of subdifferentiability might be related to its nonconvexity this is motivation to study some topological properties more in detail as the main result it is shown that any weakly compact subset of any hilbert space may be obtained as the kuratowski painleve limit of approximate subdifferentials from a one parametric family of lipschitzian functions sharper characterizations are possible for strongly compact subsets as a consequence in any hilbert space the approximate subdifferential of a suitable lipschitzian function may be homeomorphic both in the strong and weak topology to the cantor set further results relate the approximate subdifferential to specific topological types to the one dimensional case which is extraordinary in some sense and to the value function of a c 1 optimization problem

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

on the multiplicity of dini subgradients in separable spaces abstract for an extended valued function on a separable banach space the dini subdifferential is either single valued or empty on a large subset of the domain because the set of points at which it contains two or more elements can be covered by a countable family of lipschitz hypersurfaces this generalizes a finite dimensional result of benoist and has applications to the hadamard differentiability of regularly locally convex functions

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

optimality conditions for an isolated minimum of order two in c1 constrained optimization abstract in this paper we obtain first and second order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint first order sufficient conditions are derived in terms of generalized convex functions in the necessary conditions we suppose that the data are continuously differentiable a notion of strongly kt invex inequality constrained problem is introduced it is shown that each kuhn tucker point is an isolated global minimizer of order two if and only if the problem is strongly kt invex the article could be considered as a continuation of i ginchev v i ivanov second order optimality conditions for problems with c 1 data j math anal appl 340 2008 646 657

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

comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule abstract a comparison principle for viscosity sub and super solutions of second order elliptic partial differential equations is derived using the fuzzy sum rule of non smooth calculus this method allows us to weaken the assumptions made on the function f when the equation f x u d u d 2 u 0 is under consideration

iteration functions in some nonsmooth optimization algorithms recently several globally convergent model algorithms based on iteration functions have been proposed for solving nonsmooth optimization problems in particular pang han and rangaraj proposed such an algorithm for minimizing a locally lipschitzian function we determine properties of iteration functions calculus existence we also identify characteristics of functions that possess iteration functions we show that a locally lipschitzian function has a pang han rangaraj iteration function only when the function is pseudo regular in the sense of borwein and that a subsmooth lower c1 function always has a pang han rangaraj iteration function

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

the maximal normal operator space and integration of subdifferentials on nonconvex functions on definit des operateurs normaux maximaux et des fonctions primales sur des ensembles normaux on trouve que la normalite maximale est conservee dans plusieurs operations et que les operateurs normaux maximaux ont de bonnes proprietes de fermeture et de limitation

epidifferentiability of the map of infima in hilbert spaces this article deals with derivatives for set valued maps that take values in ordered vector spaces in particular it concerns about the relationship between the epiderivatives of a set valued map and its associated map of infima when the image space is a real separable hilbert space ordered by an orthonormal basis by using a variational technique based on a decoupling of the ordering cone into half spaces we show that both epiderivatives coincide under certain hypothesis of compactness and stability furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set valued maps

convergence for pseudo monotone semiflows on product ordered topological spaces in this paper we consider a class of pseudo monotone semiflows which only enjoy some weak monotonicity properties and are defined on product ordered topological spaces under certain conditions several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium which improve and extend some corresponding results already known some applications to delay differential equations are presented

convergence of solutions of a class of delay differential systems abstract this paper presents a new and concise approach for investigating the convergence of the solutions of a class of delay differential systems it is shown that each solution of the delayed system is convergent to a constant vector the result obtained extends the existing ones in the literature

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

differential properties of the moreau envelope in a vector space endowed with a uniformly g teaux differentiable norm it is proved that the moreau envelope enjoys many remarkable differential properties and that its subdifferential can be completely described through a certain approximate proximal mapping this description shows in particular that the moreau envelope is essentially directionally smooth new differential properties are derived for the distance function associated with a closed set moreover the analysis when applied to the investigation of the convexity of tchebyshev sets allows us to recover several known results in the literature and to provide some new ones

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

stochastic analysis of scattered field by building facades using polynomial chaos this paper presents a statistical assessment of scattered field from a building facade having rando m physical and geometrical parameters a simple inhomogeneous model is considered for the building and the calculation met hod is based on green s functions the basis of polynomial chaos expansion method is explained and applied to estimate the sca ttered electric field from a building facade having 8 random parameters in specular and non specular scenarios uncertainty analysis and total output distribution are discussed in differen t diffraction zones of the building ireless communications particularly mobile networks experienced a great expansion worldwide these networks are very dense in urban areas and th e electromagnetic field distribution is highly depend ent on city structures it is essential to have predictive tool s to assess as accurately as possible the distribution of electrom agnetic fields in order to enable optimized implementation of base stations and respect the constraints regarding huma n exposure wave propagation simulators are based on different models of urban environment with different assumptions on the building architectural details given the adopted model the y incorporate appropriate calculation methods we can mention empirical statistical theoretical site specific models or a combination of them to generate a hybrid model 1 empirical models 2 are based on the extensive measurement campaigns and are highly related to the structure of the city statistical models 3 are also based on measurements and are u sed to give wide band multipath model of the environment theoretical physical 4 models are obtained by i mposing some general idealized conditions on the geometry they usually use physical optics with or without diffrac tion phenomena site specific models 5 depend on a lar ge number of parameters which are related to a given s ite and their computing time depending on the complexity o f the environment can be long these models usually employ ray

second order sufficient conditions in nonsmooth optimization second order conditions are given which are sufficient to guarantee that a given point be a local solution to certain types of finite dimensional nonsmooth nonlinear programming problems both unconstrained and constrained problems are considered the conditions of the sufficiency theorems presented here are expressed in terms of the generalized gradients of nonsmooth analysis and with one exception certain second order directional derivatives these sufficiency theorems are complementary to some necessity theorems which are proved elsewhere

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

convergence of bounded solutions for a class of systems of delay differential equations abstract in this paper the convergence of a class of systems of delay differential equations is considered we show that every bounded solution of such systems tends to an equilibrium under certain hypotheses our results extend some corresponding ones already known