differentiability of the metric projection in hilbert space a study is made of differentiability of the metric projection p onto a closed convex subset k of a hilbert space h when k has nonempty interior the gateaux or fr het smoothness of its boundary can be related with some precision to gateaux or frechet differentiability properties of p for instance combining results in 3 with earlier work of r d holmes shows that k has a c2 boundary if and only if p is cl in h k and its derivative p has a certain invertibility property at each point an example in 5 shows that if the c2 condition is relaxed even slightly then p can be nondifferentiable fr het in h k

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

hermitian harmonic maps from complete hermitian manifolds to complete riemannian manifolds in this paper we study a nonlinear elliptic system of equations imposed on a map from a complete hermitian non kahler manifold to a riemannian manifold this system is more appropriate to hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are hermitian harmonic maps it was first studied by jost and yau in j y and was applied to study the rigidity of compact hermitian manifolds we extend their existence and uniqueness results to the case where both domain and target manifolds are complete hopefully the results will be useful to study corresponding rigidity of complete hermitian manifolds let m be a complex manifold with hermitian metric h and let n be a riemannian manifold with metric gij and christoffel symbols i jk a hermitian harmonic map u m n satisfies the following elliptic system

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

perturbed optimization in banach spaces iii semi infinite optimization this paper is devoted to the study of perturbed semi infinite optimization problems i e minimization over er n with an infinite number of inequality constraints we obtain the second order expansion of the optimal value function and the first order expansion of approximate optimal solutions in two cases i when the number of binding constraints is finite and ii when the inequality constraints are parametrized by a real scalar these results are partly obtained by specializing the sensitivity theory for perturbed optimization developed in part i cf siam j control optim 34 1996 pp 1151 1171 and deriving specific sharp lower estimates for the optimal value function which take into account the curvature of the positive cone in the space c omega of continuous real valued functions

conjugate points for variational problems with equality and inequality state constraints in this paper variational problems with equality and inequality state constraints are considered the theory of conjugate points for these problems is developed and necessary conditions for weak local optimality are derived in terms of this concept and the legendre condition for the case of inequality constraints the envelope like effect is taken into consideration in the accessory problem

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

uplink downlink duality via minimax duality the sum capacity of a gaussian vector broadcast channel is the saddle point of a minimax gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint this sum capacity result has been proved using two different methods one based on decision feedback equalization and the other based on a duality between uplink and downlink channels this paper illustrates the connection between the two approaches by establishing that uplink downlink duality is equivalent to lagrangian duality in minimax optimization this minimax lagrangian duality relation allows the optimal transmit covariance and the least favorable noise covariance matrices in a gaussian vector broadcast channel to be characterized in terms of the dual variables in particular it reveals that the least favorable noise is not unique further the new lagrangian interpretation of uplink downlink duality allows the duality relation to be generalized to gaussian vector broadcast channels with arbitrary linear constraints however duality depends critically on the linearity of input constraints duality breaks down when the input constraint is an arbitrary convex constraint this shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink downlink duality representation

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

perturbed optimization in banach spaces i a general theory based on a weak directional constraint qualification using a directional form of constraint qualification weaker than robinson s we derive an implicit function theorem for inclusions and use it for first and second order sensitivity analyses of the value function in perturbed constrained optimization we obtain holder and lipschitz properties and under a no gap condition first order expansions for exact and approximate solutions as an application differentiability properties of metric projections in hilbert spaces are obtained using a condition generalizing polyhedricity we also present in the appendix a short proof of a generalization of the convex duality theorem in banach spaces

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

duality and optimization for generalized multi hop mimo amplify and forward relay networks with linear constraints we consider a generalized multi hop mimo amplify and forward af relay network with multiple sources destinations and arbitrarily number of relays we establish two dualities and the corresponding dual transformations between such a network and its dual respectively under single network linear constraint and per hop linear constraint a unified optimization framework is proposed to find a stationary point for an important class of nonconvex optimization problems of af relay networks based on a local lagrange dual method where the primal algorithm only finds a stationary point for the inner loop problem of maximizing the lagrangian w r t the primal variables the input covariance matrices are shown to satisfy a polite water filling structure at a stationary point of the inner loop problem the duality and polite water filling are exploited to design fast primal algorithms compared with the existing algorithms the proposed optimization framework with duality based primal algorithms can be used to solve more general problems with lower computation cost

on sensitivity analysis of nonlinear programs in banach spaces the approach via composite unconstrained optimization this article studies local sensitivity analysis of nonlinear programming problems in banach spaces with arbitrary sets of solutions and no a priori regularity properties this is done by first developing a corresponding theory for unconstrained optimization involving simple composite functions and then applying this theory to general nonlinear programs by way of certain reduction principles

mimo b mac interference network optimization under rate constraints by polite water filling and duality we take two new approaches to design efficient algorithms for transmitter optimization under rate constraints in order to guarantee the quality of service for mimo b mac interference networks a b mac network is a generalized interference network that is a combination of multiple interfering broadcast channels bc and multiaccess channels mac two related optimization problems maximizing the minimum of weighted rates under a sum power constraint and minimizing the sum power under rate constraints are considered the first approach takes advantage of existing algorithms for sinr problems by building a bridge between rate and sinr through the design of optimal mappings between them the second approach exploits the polite water filling structure which is the network version of water filling satisfied by all the pareto optimal input of a large class of achievable regions of b mac networks it replaces most generic optimization algorithms currently used for such networks and reduces the complexity while demonstrating superior performance even in non convex cases both centralized and distributed algorithms are designed and the performance is analyzed in addition to numeric examples

differential stability of two stage stochastic programs two stage stochastic programs with random right hand side are considered optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations respectively conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces explicit formulas for the derivatives are derived special attention is paid to the role of a lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function

first order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality so called elliptic mpecs are formulated in reflexive banach spaces with the goal of deriving explicit first order optimality conditions amenable to the development of numerical procedures variational analytic concepts are both applied and further developed the paper is split into two main parts the first part concerns the derivation of conditions in which the lower level state constraints are assumed to be polyhedric sets this part is then completed by two examples the latter of which involves pointwise bilateral bounds on the gradient of the state the second part focuses on an important nonpolyhedric example namely when the lower level state constraints are presented by pointwise bounds on the euclidean norm of the gradient of the state a formula for the second order mosco epiderivative of the indicator function for this convex set is derived this result is then used to demonstrate the hadamard directional differentiability of the solution mapping of the variational inequality which then leads to the derivation of explicit strong stationarity conditions for this problem

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

legendre type optimality conditions for a variational problem with inequality state constraints they differ from the legendre clebsch condition they give information about the hesse matrix of the integrand at not only inactive points but also active points on the other hand since the inequality state constraints can be regarded as an infinite number of inequality constraints they sometimes form an envelope according to a general theory 9 one has to take the envelope into consideration when one consider second order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint however we show that we do not need to take it into account when we consider legendre type conditions finally we show that any inequality state constraint forms envelopes except two trivial cases we prove it by presenting an envelope in a visible form

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

variational analysis of a composite function a formula for the lower second order epi derivative an exact formula is established for the lower second order epi derivative of a function of the form g f x where f is a smooth map from one banach space into another and g is a convex function generally not everywhere finite unconstrained minimization of such functions typically arise as an equivalent in one or another sense reduction form for many important classes of constrained optimization problems the formula is further applied to study epi differentiability of the max function x max q x q q

max min sinr coordinated multipoint downlink transmission duality and algorithms this paper considers the max min weighted signal to interference plus noise ratio sinr problem subject to multiple weighted sum power constraints where the weights can represent relative power costs of serving different users first we study the power control problem we apply nonlinear perron frobenius theory to derive closed form expressions for the optimal value and solution and an iterative algorithm which converges geometrically fast to the optimal solution then we use the structure of the closed form solution to show that the problem can be decoupled into subproblems each involving only one power constraint next we study the multiple input single output miso transmit beamforming and power control problem we use uplink downlink duality to show that this problem can be decoupled into subproblems each involving only one power constraint we apply this decoupling result to derive an iterative subgradient projection algorithm for the problem

second order necessary and sufficient conditions for convex composite ndo in convex composite ndo one studies the problem of minimizing functions of the formf h f whereh m is a finite valued convex function andf n m is continuously differentiable this problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework e g convex inclusions minimax problems and penalty methods for constrained optimization in the present work we extend the second order theory developed by a d ioffe in 11 12 13 for the case in whichh is sublinear to arbitrary finite valued convex functionsh moreover a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

sinr constrained beamforming for a mimo multi user downlink system algorithms and convergence analysis consider a multiple input multiple output mimo downlink multi user channel a well studied problem in such a system is the design of linear beamformers for power minimization with the quality of service qos constraints the most representative algorithms for solving this class of problems are the so called minimum mean square error mmse second order cone programming socp algorithm visotksy and madhow optimum beamforming using transmit antenna arrays proc ieee veh technol conf may 1999 vol 1 pp 851 856 wong zheng and ng convergence analysis of downlink mimo antenna system using second order cone programming proc 62nd ieee veh technol conf sep 2005 pp 492 496 and the uplink downlink duality udd algorithm codreanu tolli juntti and latva aho joint design of tx rx beamformers in mimo downlink channel ieee trans signal process vol 55 no 9 pp 4639 4655 sep 2007 the former is based on alternating optimization of the transmit and receive beamformers while the latter is based on the well known uplink dowlink duality theory despite their wide applicability the convergence to karush kuhn tucker kkt solutions of both algorithms is still open in the literature in this paper we rigorously establish the convergence of these algorithms for qos constrained power minimization qcpm problem with both single stream and multiple streams per user cases key to our analysis is the development and analysis of a new mmse dual algorithm which connects the mmse socp and the udd algorithm our numerical experiments show that 1 all these algorithms can almost always reach points with the same objective value irrespective of initialization and 2 the mmse socp mmse dual algorithm works well while the udd algorithm may fail with an infeasible initialization

network duality for multiuser mimo beamforming networks and applications for any multiple input multiple output mimo network with linear beamformers it is shown that there exists a dual network that attains the same signal to interference plus noise ratio sinr performance this network duality concept is a generalization of the virtual uplink concept investigated by rashid farrokhi we first develop the dual relation using a generic duality theory in linear programming this approach naturally leads to the construction of a dual network and provides machinery for handling a generalized cost function we then consider the joint mimo beamforming and power control problem with individual sinr constraints we apply the network duality to this problem and propose a high performance iterative algorithm which compared with past centralized approaches has improved convergence behavior finally we propose a simpler distributed version of the algorithm tailored for a cellular downlink this algorithm can be readily implemented in a distributed fashion since it does not require information exchange between base stations the algorithm achieves performance close to that of centralized ones and outperforms other decentralized approaches available in the literature

perturbed optimization in banach spaces ii a theory based on a strong directional constraint qualification we study the sensitivity of the optimal value and optimal solutions of perturbed optimization problems in two cases the first one is when multipliers exist but only the weak and not the strong second order sufficient optimality condition is satisfied the second case is when no lagrange multipliers exist to deal with these pathological cases we are led to introduce a directional constraint qualification stronger than in part i of this paper which reduces to the latter in the important case of equality inequality constrained problems we give sharp upper estimates of the cost based on paths varying as the square root of the perturbation parameter and under a no gap condition obtain the first term of the expansion for the cost when multipliers exist we study the expansion of approximate solutions as well we show in the appendix that the strong directional constraint qualification is satisfied for a large class of problems including regular problems in the sense of robinson

transmit optimization with improper gaussian signaling for interference channels this paper studies the achievable rates of gaussian interference channels with additive white gaussian noise awgn when improper or circularly asymmetric complex gaussian signaling is applied for the gaussian multiple input multiple output interference channel mimo ic with the interference treated as gaussian noise we show that the user s achievable rate can be expressed as a summation of the rate achievable by the conventional proper or circularly symmetric complex gaussian signaling in terms of the users transmit covariance matrices and an additional term which is a function of both the users transmit covariance and pseudo covariance matrices the additional degrees of freedom in the pseudo covariance matrix which is conventionally set to be zero for the case of proper gaussian signaling provide an opportunity to further improve the achievable rates of gaussian mimo ics by employing improper gaussian signaling to this end this paper proposes widely linear precoding which efficiently maps proper information bearing signals to improper transmitted signals at each transmitter for any given pair of transmit covariance and pseudo covariance matrices in particular for the case of two user gaussian single input single output interference channel siso ic we propose a joint covariance and pseudo covariance optimization algorithm with improper gaussian signaling to achieve the pareto optimal rates by utilizing the separable structure of the achievable rate expression an alternative algorithm with separate covariance and pseudo covariance optimization is also proposed which guarantees the rate improvement over conventional proper gaussian signaling

optimal control of a parabolic equation with time dependent state constraints in this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain in the presence of a state constraint the latter is integral over the space and has to be satisfied each time using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations we show that both the control and multiplier are continuous in time under some natural geometric hypotheses we can prove that extended polyhedricity holds allowing us to obtain no gap second order optimality conditions that characterize quadratic growth an expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right hand side of the state equation

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

a unified analysis of max min weighted sinr for mimo downlink system this paper studies the max min weighted signal to interference plus noise ratio sinr problem in the multiple input multiple output mimo downlink where multiple users are weighted according to priority and are subject to a weighted sum power constraint first we study the multiple input single output miso and single input multiple output simo problems using nonlinear perron frobenius theory as a by product we solve the open problem of convergence for a previously proposed miso algorithm by wiesel eldar and shamai in 2006 furthermore we unify our analysis with respect to the previous alternate optimization algorithm proposed by tan chiang and srikant in 2009 by showing that our miso result can in fact be derived from their algorithm next we combine our miso and simo results into an algorithm for the mimo problem we show that our proposed algorithm is optimal when the channels are rank one or when the network is operating in the low signal to noise ratio snr region finally we prove the parametric continuity of the mimo problem in the power constraint and we use this insight to propose a heuristic initialization strategy for improving the performance of our generally suboptimal mimo algorithm the proposed initialization strategy exhibits improved performance over random initialization

on the marginal function in nonlinear programming for several types of finite or infinite dimensional optimization problems the marginal function or optimal value function is characterized by different local approximations such as generalized gradients generalized directional derivatives directional hadamard or dini derivatives we give estimates for these terms which are determined by multipliers satisfying necessary optimality conditions when the functions which define the optimization problem are more than once continuously differentiable then higher order necessary conditions are employed to obtain refined estimates for the marginal function as a by product we give a new equivalent formulation of clarke s multiplier rule for nonsmooth optimization problems this shows that the set of all multipliers satisfying these necessary conditions is the union of a finite number of closed convex cones

second order sufficiency and quadratic growth for nonisolated minima for standard nonlinear programming problems the weak second order sufficient condition is equivalent to the quadratic growth condition as far as the set of minima consists of isolated points and some qualification hypothesis holds this kind of condition is instrumental in the study of numerical algorithms and sensitivity analysis the arm of the paper is to study the relations between various types of sufficient conditions and quadratic growth in cases when the set of minima may have nonisolated points

sensitivity analysis for relaxed optimal control problems with final state constraints in this article we compute a second order expansion of the value function of a family of relaxed optimal control problems with final state constraints parameterized by a perturbation variable the sensitivity analysis is performed for controls that we call r strong solutions they are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given r and having an associated trajectory in a small neighborhood for the uniform norm in this framework relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function

second order conditions for optimization problems with constraints using a projective approach new necessary conditions and new sufficient conditions for optimization problems with explicit or implicit constraints are examined they are compared to previous ones a particular emphasis is given to mathematical programming problems with non polyhedral constraints this case occurs in particular when the constraints are defined in functional spaces

on lower bounds of the second order directional derivatives of ben tal zowe and chaney let f be a regular locally lipschitz real valued function defined on an open convex subset of a normed space we show that at any unit direction u the upper second order derivative d 2f u 0 in the sense of dem yanov and pevnyi dem yanov v f a b pevnyi 1974 expansion with respect to a parameter of the extremal values of game problems ussr computational math and math phys 14 33 45 ben tal and zowe ben tal a j zowe 1982 necessary and sufficient optimality conditions for a class of nonsmooth minimization problems math programming24 70 91 has the same lower bounds as the lower second order derivatives d 2f u 0 consequently one can characterize the convexity of f in terms of these derivatives we also obtain the corresponding results in terms of chaney s second order directional derivatives

transmit beamforming and power control for optimizing the outage probability fairness in miso networks this paper studies the joint beamforming and power control in a multiuser multi input single output network by utilizing the only statistical channel distribution information such information consists of slowly varying covariance matrices in the beamforming network that can be employed to reduce instantaneous feedback overhead in transmission utilizing solely the statistical channel information we study how to minimize the maximum outage probability under a weighted sum power constraint that guarantees max min fairness to all users this problem is however generally hard to solve due to the nonconvexity and nonlinear coupling between beamformer and power variables first assuming a fixed beamformer set we use the nonlinear perron frobenius theory to design a decentralized algorithm with provable geometrically fast convergence rate to compute the optimal power then for the general case we examine a certainty equivalent margin counterpart with outage mapped thresholds that incorporate the statistical channel information we show that a network duality for this certainty equivalent problem can be useful to decouple the coupling between the beamformer and power variables this nonlinear perron frobenius theory motivated approach yields a feasible beamformer and power allocation that are near optimal as compared to monte carlo averaging simulations

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

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

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

liouville theorems for harmonic maps we prove several liouville theorems for harmonic maps between certain classes of riemannian manifolds in particular the results can be applied to harmonic maps from the euclidean space r m g 0 to a large class of riemannian manifolds our assumptions on the harmonic maps concern the asymptotic behavior of the maps at