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

full lipschitzian and holderian stability in optimization with applications to mathematical programming and optimal control the paper concerns a systematic study of full stability in general optimization models including its conventional lipschitzian version as well as the new holderian one we derive various characteri

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

on chebyshev functions and klee functions abstract associated to a lower semicontinuous function one can define its proximal mapping and farthest mapping the function is called chebyshev klee if its proximal mapping farthest mapping is single valued everywhere we show that the function f is 1 hypoconvex if its proximal mapping p f is single valued when the function f is bounded below and p f is single valued for every 0 the function must be convex similarly we show that the function f is 1 strongly convex if the farthest mapping q f is single valued when the function is the indicator function of a set this recovers the well known chebyshev problem and klee problem in r n we also give an example illustrating that a continuous proximal mapping farthest mapping needs not be locally lipschitz which answers one open question by hare and poliquin

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

local monotonicity and full stability for parametric variational systems this paper introduces and characterizes new notions of lipschitzian and holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of prox regular functions acting in finite dimensional and hilbert spaces these notions which postulate certain quantitative properties of single valued localizations of solution maps are closely related to local strong maximal monotonicity of associated set valued mappings based on advanced tools of variational analysis and generalized differentiation we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive definiteness conditions involving second order constructions of variational analysis

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

computing maximal polymatroidal network flows in the classical network flow model flows are constrained by the capacities of individual arcs in the polymatroidal network flow model introduced in this paper flows are constrained by the capacities of sets of arcs yet the essential features of the classical model are retained the augmenting path theorem the integral flow theorem and the max flow min cut theorem arc all shown to yield to straightforward generalization we describe a maximal flow algorithm which finds augmenting paths by labeling arcs instead of nodes as in the case of the classical model as a counterpart of a known result for the classical model we prove that the number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network provided each successive augmentation is made along a shortest augmenting path with ties between shortest paths broken by lexicography

two algorithms for maximizing a separable concave function over a polymatroid feasible region abstract in this paper we present two new algorithms for maximizing a separable concave function on a polymatroid both algorithms apply to the discrete as well as the continuous version of the problem the application of these algorithms to several types of polymatroids is discussed and we show that the decomposition algorithm runs in polynomial time in the discrete version for network and generalized symmetric polymatroids and the bottom up algorithm in the discrete version runs in polynomial time when the polymatroid is given as an explicit list of constraints

minimizing periodic orbits with regularizable collisions in the n body problem in the newtonian n body problem there are various subsystems with two degrees of freedom such as the collinear three body problem and the isosceles three body problem after we determine a normal form of the lagrangians of these subsystems we prove the existence of periodic solutions with regularizable collisions for these systems our result includes several examples such as schubart s orbit with or without equal masses among others

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

a proximal iterative approach to a non convex optimization problem abstract we consider a variable krasnosel skii mann algorithm for approximating critical points of a prox regular function or equivalently for finding fixed points of its proximal mapping p r o x f the novelty of our approach is that the latter is not non expansive any longer we prove that the sequence generated by such algorithm via the formula x k 1 1 k x k k p r o x k f x k where k is a sequence in 0 1 is an approximate fixed point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition

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

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

convergence of subdifferentials and normal cones in locally uniformly convex banach space abstract in this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions mosco and attouch wets convergences our analysis is devoted to proximal frechet and mordukhovich limiting normal cones and subdifferentials the results obtained can be seen as extensions of the attouch theorem to the context of non convex functions on locally uniformly convex banach space they also generalize to sequences of subsmooth sets or functions various results in the literature

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

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

epiconvergence of convexly composite functions in banach spaces this paper gives conditions ensuring the epiconvergence or gamma convergence of sequences of convexly composite functions via the study of painleve kuratowski convergence of inverse images of sets the convergence of multipliers of optimization problems associated with convexly composite functions is also established

inverse eigenvalue problem of jacobi matrix with mixed data in this paper the inverse eigenvalue problem of reconstructing a jacobi matrix from its eigenvalues its leading principal submatrix and part of the eigenvalues of its submatrix is considered the necessary and sufficient conditions for the existence and uniqueness of the solution are derived furthermore a numerical algorithm and some numerical examples are given

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

moreau envelopes of s lower regular functions abstract the paper investigates various properties of moreau envelopes and proximal mappings of s lower regular functions in particular for such a function on uniformly convex space the moreau envelope is shown to be differentiable with its derivative locally holder continuous

approximation and decomposition properties of some classes of locally d c function we study the connexion between local and global decompositions of some important subclasses of locally d c functions functions which locally split as a difference of two convex functions then we tackle the problem of regularizing such functions by the moreau yosida process and prove in particular that the class of lower c2 functions fits well this approximation procedure

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

uniform prox regularity of functions and epigraphs in hilbert spaces abstract we study the local uniform prox regularity of functions make the connection with the lower c 2 property and essentially with the prox regularity of epigraphs

removing collision singularities from action minimizers for the n body problem with free boundaries for the planar and spatial n body problems it has been proved by marchal and chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions this important result has been extended by ferrario terracini to newtonian type problems and equivariant problems it has also been used to construct many symmetric solutions for the n body problem