clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

necessary conditions for constrained optimization problems with semicontinuous and continuous data we consider nonsmooth constrained optimization problems with semicontinuous and continuous data in banach space and derive necessary conditions sl without constraint qualification in terms of smooth subderivatives and normal cones these results in different versions are set in reflexive and smooth banach spaces

variational analysis in nonreflexive spaces and applications to control problems with l 1 perturbations we provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is l 1 our underlying technique relies on a recent sequential description of both the generalized gradient of clarke and of the approximate g subdifferential of functions defined on a smooth banach space we also show that the proximal limit formula for the generalized gradient and its l p analogue are direct consequences of these sequential formulas related characterizations of lipschitzness of a function on a smooth space are given

bounded sets of lagrange multipliers for vector optimization problems in infinite dimension the aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of lagrange multipliers for vectorial optimization problems in infinite dimension in some smooth cases these conditions turn out to be necessary for the existence of multipliers as well

quadratic tests for detection of abrupt changes in multivariate signals this article considers the problem of detecting abrupt changes in the mean of a multivariate gaussian random signal a fixed sample size spl chi sup 2 test is compared against the optimum sequential tests spl chi sup 2 cusum and spl chi sup 2 glr

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

the lagrange multiplier rule for multifunctions in banach spaces we study general constrained multiobjective optimization problems with objectives being closed multifunctions in banach spaces in terms of the coderivatives and normal cones we provide generalized lagrange multiplier rules as necessary optimality conditions of the above problems in an asplund space setting sharper results are presented

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

the fermat rule for multifunctions for super efficiency abstract using variational analysis we study the vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in terms of the coderivatives and normal cones we present fermat s rules as necessary or sufficient conditions for a super efficient solution of the above problems

nondifferentiable multiplier rules for optimization and bilevel optimization problems in this paper we study optimization problems with equality and inequality constraints on a banach space where the objective function and the binding constraints are either differentiable at the optimal solution or lipschitz near the optimal solution necessary and sufficient optimality conditions and constraint qualifications in terms of the michel penot subdifferential are given and the results are applied to bilevel optimization problems

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

information bounds and quick detection of parameter changes in stochastic systems by using information theoretic bounds and sequential hypothesis testing theory this paper provides a new approach to optimal detection of abrupt changes in stochastic systems this approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems in addition it leads to detection rules which have manageable computational complexity for on line implementation and yet are nearly optimal under the different performance criteria considered

homotopy classes in sobolev spaces and the existence of energy minimizing maps consider the problem of finding maps f m n that are stationary for an energy functional such as fm idfl p where m and n are compact riemannian manifolds and p l such maps may be found by minimizing the functional but if we minimize among all maps from m to n then the minimum is 0 and is attained only by constant maps thus in order to find nontrivial stationary maps we would like to use the topology of m and n to define classes of maps from m to n in which we can minimize the functional for instance one could try to minimize among maps in a given homotopy class but this is not possible in general unless p dimm since a minimizing sequence of mappings in one homotopy class can converge in the appropriate weak topology to a map in another homotopy class however in this paper we show that it is possible to minimize among maps fwhose restrictions to a lower dimensional skeleton of a triangulation of m belong to a given homotopy class to state the results precisely we need to refer to certain sobolev spaces and norms we will assume without loss of generality that m and n are submanifolds of euclidean spaces r m and r respectively and we let lip m n denote the space of lipschitz maps from m to n we define li p m r to be the space of all functions fe lp m r n such that there exist functions

a nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions abstract fuzzy optimization conditions in terms of the frechet subdifferential for reflexive spaces were investigated by borwein treiman and zhu 1998 in 1 to achieve the nondegenerate form it is well known that some qualification conditions should be assumed in this paper we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in asplund spaces in particular reflexive spaces for optimization problems with semi continuous and continuous data the results are even new in finite dimensional frameworks

a mathematical framework for new fault detection schemes in nonlinear stochastic continuous time dynamical systems in this work a mathematical unifying framework for designing new fault detection schemes in nonlinear stochastic continuous time dynamical systems is developed these schemes are based on a stochastic process called the residual which reflects the system behavior and whose changes are to be detected a quickest detection scheme for the residual is proposed which is based on the computed likelihood ratios for time varying statistical changes in the ornstein uhlenbeck process several expressions are provided depending on a priori knowledge of the fault which can be employed in a proposed cusum type approximated scheme this general setting gathers different existing fault detection schemes within a unifying framework and allows for the definition of new ones a comparative simulation example illustrates the behavior of the proposed schemes

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

existence and characterization of efficient decisions with respect to cones existence and characterization theorems for the efficient nondominated set of decisions inr n are presented the existence is proved when the set of decisions satisfies some compactness conditions the efficient set is characterized in terms of the exposed efficient decisions when certain convexity and compactness conditions are satisfied

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints this paper discusses hamiltonian necessary conditions for a nonsmooth multiobjective optimal control problem with endpoint constraints involving a general preference the transversality condition in our necessary conditions is stated in terms of normal cones to the level sets of the preference examples involving a number of useful preferences are discussed

constrained epsilon minimax test for simultaneous detection and classification a constrained epsilon minimax test is proposed to detect and classify nonorthogonal vectors in gaussian noise with a general covariance matrix and in presence of linear interferences this test is epsilon minimax in the sense that it has a small loss of optimality with respect to the purely theoretical and incalculable constrained minimax test which minimizes the maximum classification error probability subject to a constraint on the false alarm probability this loss is even more negligible as the signal to noise ratio is large furthermore it is also an epsilon equalizer test since its classification error probabilities are equalized up to a negligible difference when the signal to noise ratio is sufficiently large an asymptotically equivalent test with a very simple form is proposed this equivalent test coincides with the generalized likelihood ratio test when the vectors to classify are strongly separated in term of euclidean distance numerical experiments on active user identification in a multiuser system confirm the theoretical findings

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

necessary conditions for constrained optimization problems in smooth banach spaces and applications we derive necessary optimality conditions for constrained minimization problems with lower semicontinuous inequality constraints continuous equality constraints and a set constraint in smooth banach spaces then we apply these necessary optimality conditions to derive subdifferential characterizations of singular normal vectors to the epigraph and the graph of a function subdifferential calculus and necessary optimality conditions for mathematical programs with equilibrium constraints

epi lipschitz like sets in banach space theorems and examples on considere un espace de banach x et on etudie quelles proprietes un ensemble c dans x doit avoir pour que le cone tangent de clarke t c x et le cone normal n c x mesurent adequatement le comportement a la frontiere de x dans c

lagrange multipliers in nonsmooth semi infinite optimization problems using the variational analysis technique in terms of the epi coderivative we provide lagrange multiplier rules for a class of semi infinite optimization problems where all functions are lower semicontinuous or locally lipschitz

second order asymptotic optimality in multisensor sequential change detection a generalized multisensor sequential change detection problem is considered in which a number of possibly correlated sensors monitor an environment in real time the joint distribution of their observations is determined by a global parameter vector and at some unknown time there is a change in an unknown subset of components of this parameter vector the goal is to detect the change as soon as possible while controlling the rate of false alarms we establish the second order asymptotic optimality with respect to lorden s criterion of various generalizations of the cusum rule that is we show that their additional expected worst case detection delay relative to the one that could be achieved if the affected subset was known remains bounded as the rate of false alarm goes to 0 for any possible subset of affected components this general framework incorporates the traditional multisensor setup in which only an unknown subset of sensors is affected by the change the latter problem has a special structure which we exploit in order to obtain feasible representations of the proposed schemes we present the results of a simulation study where we compare the proposed schemes with scalable detection rules that are only first order asymptotically optimal finally in the special case that the change affects exactly one sensor we consider the scheme that runs in parallel the local cusum rules and study the problem of specifying the local thresholds

optimum problems with nonsmooth equality constraints abstract in this paper first and higher order necessary conditions for constrained extremum problems are obtained under weak regularity assumptions concerning the functions describing the equality as well as the inequality conditions of the problem

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

necessary optimality conditions and exact penalization for non lipschitz nonlinear programs when the objective function is not locally lipschitz constraint qualifications are no longer sufficient for karush kuhn tucker kkt conditions to hold at a local minimizer let alone ensuring an exact penalization in this paper we extend quasi normality and relaxed constant positive linear dependence condition to allow the non lipschitzness of the objective function and show that they are sufficient for kkt conditions to be necessary for optimality moreover we derive exact penalization results for the following two special cases when the non lipschitz term in the objective function is the sum of a composite function of a separable lower semi continuous function with a continuous function and an indicator function of a closed subset we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function when the non lipschitz term is the sum of a continuous function and an indicator function of a closed subset we also show that our problem admits an exact penalization under an extended quasi normality involving the coderivative of the continuous function

conical regularization for multi objective optimization problems abstract this paper focuses on multi objective optimization problem with linear infinite dimensional constraints the main novelty of the contribution is that we do not assume that the ordering cone has a nonempty interior and consequently there might not exist kkt system to compute the solution numerically we devise a regularization framework which consists of replacing the ordering cone by a family of solid cones we provide general features of the original problem and the family of regularized problems using tools from the set valued analysis we prove that the weak outer limit of regularized solutions is the set of minimal elements of the original problem we scalarize the original problem as well as the regularized analogs and establish that the corresponding scalarized solutions parametrize the regularized minimal elements we demonstrate the convergence in the kuratowski mosco sense of the regularized solutions to the efficient set of the original problem we prove an a priori holder continuity estimate for the regularized efficient set valued map when the original problem is regular and a verifiable norm boundedness property of the multipliers holds we present detailed numerical examples to show the efficacy of the developed framework

topology and sobolev spaces abstract consider the sobolev class w 1 p m n where m and n are compact manifolds we present some sufficient conditions which guarantee that w 1 p m n is path connected we also discuss cases where w 1 p m n admits more than one component there are still a number of open problems especially concerning the values of p where a change in homotopy classes occurs

a generalized change detection problem the purpose of this paper is to give a new statistical approach to the change diagnosis detection isolation problem the change detection problem has received extensive research attention however the change isolation problem has for the most part been ignored we consider a stochastic dynamical system with abrupt changes and investigate the multiple hypotheses extension of lorden s 1971 results we introduce a joint criterion of optimality for the detection isolation problem and then design a change detection isolation algorithm we also investigate the statistical properties of this algorithm we prove a lower bound for the criterion in a class of sequential change detection isolation algorithms it is shown that the proposed algorithm is asymptotically optimal in this class the theoretical results are applied to the case of additive changes in linear stochastic models

ideal weakly efficient solutions for vector optimization problems we establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded the solutions to be referred are ideal strong weakly efficient weakly pareto also several necessary and or sufficient conditions for the solution set to be non empty and compact are established moreover some characterizations of the non emptiness boundedness of the convex solution set in case the solutions are searched in a subset of the real line are also given however the solution set fails to be convex in general in addition special attention is addressed when the underlying cone is the non negative orthant and when the semi strict quasiconvexity of each component of the vector valued function is assumed our approach is based on the asymptotic description of the functions and sets some examples illustrating such an approach are also exhibited

scalarization and optimality conditions for vector equilibrium problems abstract in this paper we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions henig efficient solutions and globally efficient solutions to the vector equilibrium problems without the convexity assumption using nonsmooth analysis and the scalarization results we provided the necessary conditions for weakly efficient solutions henig efficient solutions globally efficient solutions and superefficient solutions to vector equilibrium problems by the assumption of convexity we gave sufficient conditions for those solutions as applications we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems

conic positive definiteness and sharp minima of fractional orders in vector optimization problems abstract motivated by the fact that the usual positive definiteness does not work in an infinite space we introduce the concept of s positive definiteness with respect to an ordering cone in a general banach space and show that the s positive definiteness plays the same role as the usual positive definiteness in the finite dimensional case as applications we study sharp and weak sharp minima of fractional orders in vector optimization

topology of sobolev mappings ii this paper addresses some topological and analytical issues concern ing sobolev mappings between compact riemannian manifolds among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of brezis and li in particular we solved two conjectures in bl we also give a topological obstruction for the weak sequential density of smooth maps in a given sobolev mapping space finally we show a necessary and suf ficient topological condition under which the smooth maps are strongly dense in the sobolev spaces the earlier result theorem 1 of b2 was shown to be not correct

degrees of efficiency and degrees of minimality in this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure usually different scalarizing functions are used in order to obtain the various solutions of the vector problem here we consider different kinds of solutions of the same scalarized problem our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency besides the usual notions of weakly efficient and efficient points which are characterized as global and strict global solutions of the scalarized problem we also consider some restricted notions of efficiency such as strict and proper efficiency which are characterized as tikhonov well posed minima and sharp minima for the scalarized problem

nonparametric sequential change point detection by a vertically trimmed box method this paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions a nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box as such it will be coined as the vertical box control chart v box chart its implementation requires merely to count the number of data points which fall into the box attached to the last available observation no a priori knowledge of data distributions is required and proper tuning of the box size provides a quick detection technique this is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the v box chart these theoretical results are verified by simulation studies which indicate that the v box chart may provide quick detection with zero delay for jumps of moderate sizes its averaged run length to detection is more favorable than the one for the classical ewma method by comparison with the classical shewhart chart which was optimized for normal errors our method provides comparable or better performance

topology of sobolev mappings iii we establish a necessary and sufficient topological condition for maps that are in w1 p m n to be connected by continuous paths in w1 p m n to maps in w1 q m n q q 1 we also obtain a necessary and sufficient topological condition under which w1 q m n is strongly dense in w1 p m n several results concerning sequential weak density of smooth or w1 q m n maps in w1 p m n are also proven 2003 wiley periodicals inc

two strategies in the problem of change detection and isolation the comparison between optimal sequential and nonsequential fixed size sample strategies in the problem of abrupt change detection and isolation is discussed in particular we show that sometimes a simple fixed size sample algorithm is almost as efficient as an optimal sequential algorithm which leads to a burdensome number of arithmetical operations

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

necessary optimality conditions for weak sharp minima in set valued optimization abstract the aim of the present paper is to get necessary optimality conditions for a general kind of sharp efficiency for set valued mappings in infinite dimensional framework the efficiency is taken with respect to a closed convex cone and as the basis of our conditions we use the mordukhovich generalized differentiation we have divided our work into two main parts concerning on the one hand the case of a solid ordering cone and on the other hand the general case without additional assumptions on the cone in both situations we derive some scalarization procedures in order to get the main results in terms of the mordukhovich coderivative but in the general case we also carryout a reduction of the sharp efficiency to the classical pareto efficiency which in addition with a new calculus rule for frechet coderivative of a difference between two maps allows us to obtain some results in frechet form

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

on global singularities of sobolev mappings we define various invariants for sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong sobolev topology we show that these invariants classify various types of global singularities of sobolev maps these invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the sobolev space

lifting in besov spaces abstract let be a smooth bounded simply connected domain in r n and let u be a complex valued measurable function on such that u x 1 a e assume that u belongs to a besov space b p q s we investigate whether there exists a real valued function b p q s r such that u e i this complements the corresponding study in sobolev spaces due to bourgain brezis and the first author the microscopic parameter q turns out to play an important role in some limiting situations the analysis of this lifting problem relies on some interesting new properties of besov spaces in particular a non restriction property when q p

on board component fault detection and isolation using the statistical local approach we describe both the key principles and real application examples of a unified theory which allows us to perform the on board incipient fault detection and isolation tasks involved in monitoring for condition based maintenance this theory is known under the name of local approach and it is especially suited to component faults it aims at transforming complex detection problems concerning a parameterized stochastic process into the universal problem of monitoring the mean of a gaussian vector based on a small deviation assumption the key tools are the first order taylor expansion and the asymptotic gaussianity of a convenient parameter estimating function ml ls iv and subspace identification methods are addressed in this perspective in the case of dynamic processes given in state space form the approach may also call for observer based state estimation or state elimination experiments concerning both simulated and real data for linear and nonlinear dynamical processes are reported how the key principles and features of the local approach compare with those of other approaches is discussed

holonomy and skyrme s model in this paper we consider two generalizations of the skyrme model one is a variational problem for maps from a compact 3 manifold to a compact lie group the other is a variational problem for flat connections we describe the path components of the configuration spaces of smooth fields for each of the variational problems we prove that the invariants separating the path components are well defined for not necessarily smooth fields with finite skyrme energy we prove that for every possible value of these invariants there exists a minimizer of the skyrme functional throughout the paper we emphasize the importance of holonomy in the skyrme model some of the results may be useful in other contexts in particular we define the holonomy of a distributionally flat l 2 loc connection the local developing maps for such connections need not be continuous

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

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

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions

first and second order sufficient conditions for strict minimality in nonsmooth vector optimization abstract in this paper we present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function with this aim the notion of support function to a vector problem is introduced in such a way that the scalar case and the multiobjective case in particular are contained the obtained results extend the multiobjective ones to this case moreover specializing to a feasible set defined by equality inequality and set constraints first and second order sufficient conditions by means of lagrange multiplier rules are established

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

metric regularity of epigraphical multivalued mappings and applications to vector optimization in this work we combine in a meaningful way two techniques of variational analysis and nonsmooth optimization on one hand we use the error bound approach to study the metric regularity of some special types of multifunctions and on the other hand we exploit the incompatibility between the metric regularity and the pareto minimality this method allows us to present some varepsilon fermat rules for set valued optimization problem in the setting of general banach spaces our results are comparable to several recent results in literature

a fuzzy necessary optimality condition for non lipschitz optimization in asplund spaces in this paper we consider a general optimization problem in the broad class of asplund spaces we derive a new necessary optimality condition in the so called lagrangian fuzzy form without standard lipschitz conditions we also give a chain rule for frechet subdifferentials and subdifferential representations of frechet normals to constrained sets

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we