on existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite dimensional space abstract this paper is devoted to existence of trajectories to differential equations and inclusions remaining in a given closed or compact set topological properties of the boundary of this set enables us to conclude the existence of such trajectories in many cases

alternating projections on manifolds we prove that if two smooth manifolds intersect transversally then the method of alternating projections converges locally at a linear rate we bound the speed of convergence in terms of the angle between the manifolds which in turn we relate to the modulus of metric regularity for the intersection problem a natural measure of conditioning we discuss a variety of problem classes where the projections are computationally tractable and we illustrate the method numerically on a problem of finding a low rank solution of a matrix equation

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

fast bit parallel binary multipliers based on type i pentanomials in this paper a fast implementation of bit parallel polynomial basis pb multipliers over the binary extension field gf 2 m generated by type i irreducible pentanomials is presented explicit expressions for the coordinates of the multipliers and a detailed example are given complexity analysis shows that the multipliers here presented have the lowest delay in comparison to similar bit parallel pb multipliers found in the literature based on this class of irreducible pentanomials in order to prove the theoretical complexities hardware implementations over xilinx fpgas have also been performed experimental results show that the approach here presented exhibits the lowest delay with a balanced area times time complexity when it is compared with similar multipliers

semiconcave solutions of partial differential inclusions abstract we obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions these results will be used to study the behavior of singularities of the value function for a reflected control problem

the clarke generalized gradient for functions whose epigraph has positive reach we consider the class of continuous functions that map an open set rn to r with an epigraph having locally positive reach with an additional property this class contains all finite convex and c1 1 functions but also ones that are not necessarily lipschitz continuous we provide a representation formula for the clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points thus offering an alternative to the well known proximal normal formula by replacing a pointedness assumption by one of positive reach our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory and is based on an induction argument as an application we prove for a particular class of hamilton jacobi equations that an a e solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution

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

new metric properties for prox regular sets in this paper we present diverse new metric properties that prox regular sets shared with convex ones at the heart of our work lie the legendre fenchel transform and complements of balls first we show that a connected prox regular set is completely determined by the legendre fenchel transform of a suitable perturbation of its indicator function then we prove that such a function is also the right tool to extend to the context of prox regular sets the famous connection between the distance function and the support function of a convex set on the other hand given a prox regular set we examine the intersection of complements of open balls containing the set we establish that the distance of a point to a prox regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point this is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes to the best of our knowledge these results are new in the literature and show that the class of prox regular sets have good properties known in convex analysis mathematics subject classification 2010 49j52 49j53

using a signed distance function for the simulation of metal forming processes formulation of the contact condition and mesh adaptation from a lagrangian approach to an eulerian approach this paper proposes to use the metric properties of the distance function between two bodies in contact or gap function in simulations involving contact problems first the normal vectors which are involved in the formulation of the contact condition are defined through the gradient of this distance function this definition avoids to deal with the numerical penetration parameter which is generally introduced otherwise furthermore it allows the contact problem to be extended in a simple way to an eulerian formulation second this paper investigates two mesh adaptation strategies based on the properties of the distance function the first strategy consists in building a size map according to the values of this function in order to refine locally the mesh and consequently to improve the description of the contact surface the second strategy consists in adapting locally the mesh to the geometry of the contact surface this anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient a lot of elements are saved when compared with the isotropic case throughout this paper many numerical simulations are presented in the context of the forging process the deformable material is pressed between two rigid tools furthermore the algorithm used to calculate the signed distance to a surface mesh is detailed in appendix of this paper

invariant sets and lyapunov pairs for differential inclusions with maximal monotone operators we give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on hilbert spaces which is subject to a lipschitz continuous perturbation depending on the state these sets are not necessarily weakly closed as in 3 4 while the invariance criteria are still written by using only the data of the system so no need to the explicit knowledge of neither the solution of this differential inclusion nor the semi group generated by the maximal monotone operator these invariant viability results are next applied to derive explicit criteria for a lyapunov pairs of lower semi continuous not necessarily weakly lsc functions associated to these differential inclusions the lack of differentiability of the candidate lyapunov functions and the consideration of general invariant sets possibly not convex or smooth are carried out by using techniques from nonsmooth analysis

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

estimating tangent and normal cones without calculus we study simple estimates of the tangent and normal cones to a set with a nonsmooth boundary we seek estimates that are robust relative to the base point and that depend only on collections of nearby points known to lie inside and outside of the set

super efficiency of vector optimization in banach spaces using variational analysis in terms of the clarke normal cone we consider super efficiency of vector optimization in banach spaces we establish some characterizations for super efficiency in particular dropping the assumption that the ordering cone has a bounded base we extend a result in borwein and zhuang j m borwein d zhuang super efficiency in vector optimization trans amer math soc 338 1993 105 122 to the nonconvex setting 2006 elsevier inc all rights reserved

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

horizontal parabolic equation solution of radiowave propagation problems on large domains the modeling of radiowave propagation in the troposphere often involves field strength calculations in very large domains for which the usual parabolic equation techniques become prohibitively expensive the horizontal parabolic equation method hpe provides a rigorous and efficient full wave solution for this type of problem taking advantage of the fact that refractive index perturbations in the troposphere are usually confined to a limited portion of the integration domain above the height where the medium becomes standard fourier and laplace transform techniques are used to compute the field on a given horizontal with convolution kernels based on exponential or airy functions the field on the initial horizontal may be obtained by any available method the environment below the initial height may be range dependent and include terrain variations substantial computing gains in both memory requirements and execution time are obtained the method has been validated by comparison to the standard vertical pe

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

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

mixed semicontinuous perturbations of nonconvex sweeping processes in this paper we prove a theorem on the existence of a global solution of a differential inclusion governed by a class of nonconvex sweeping processes with unbounded pertubations the perturbations are not required to be convex valued

linear regularity for a collection of subsmooth sets in banach spaces using variational analysis we study the linear regularity for a collection of finitely many closed sets in particular we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

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

on the asymptotics of constrained local m estimators we discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and more generally m estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set s we give a counterexample showing that regularity of s in the sense of clarke is not sufficient for asymptotic equivalence of n consistent locally optimal m estimators we argue further that stronger properties such as so called near convexity or prox regularity of s are required in order to ensure that any two n consistent locally optimal m estimators have the same asymptotics

on reflecting boundary problem for optimal control this paper deals with mayer s problem for controlled systems with reflection on the boundary of a closed subset k the main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of hamilton jacobi bellman type

generalised prox regularity in reflexive smooth banach spaces with smooth dual norm abstract in this paper we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth banach spaces with smooth dual norm started by the first author in 9 we define a new class of nonconvex sets in banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm

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

some new regularity properties for the minimal time function a minimal time problem with linear dynamics and convex target is considered it is shown essentially that the epigraph of the minimal time function t cdot is varphi convex i e it satisfies a kind of exterior sphere condition with locally uniform radius provided t cdot is continuous several regularity properties are derived from results in g colombo and a marigonda calc var partial differential equations 25 2005 pp 1 31 including twice a e differentiability of t cdot and local estimates on the total variation of dt

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

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

letter to the editor extragradient methods for solving nonconvex variational inequalities in this paper we introduce and consider a new class of variational inequalities which are called the nonconvex variational inequalities using the projection technique we suggest and analyze an extragradient method for solving the nonconvex variational inequalities we show that the extragradient method is equivalent to an implicit iterative method the convergence of which requires only pseudo monotonicity a weaker condition than monotonicity this clearly improves on the previously known result our method of proof is very simple as compared with other techniques

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

second order sweeping process with a lipschitz perturbation abstract in this paper we prove a theorem concerning the existence of solutions for a second order sweeping process with a lipschitz single valued perturbation in hilbert space

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

the problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures abstract we obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems in order to describe the value functions we employ an infinite dimensional dual formulation instead of using the characterization via hamilton jacobi partial differential equations in this paper we consider control problems with both infinite and finite horizons the reflection is given by the normal cone to a proximal retract set

realistic surface modeling for a finite difference time domain wave propagator a new time domain wave propagator tdwp based on the two dimensional finite difference time domain fdtd technique for original paper see ibid vol v 14 p 302 307 1966 that was introduced in may 2000 issue has been augmented in this paper so that it deals with impedance boundary condition or varying terrain heights and an example of each with comparisons to other methods is presented

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

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

a novel finite difference time domain wave propagator a novel time domain wave propagator is introduced a two dimensional 2 d finite difference time domain fdtd algorithm is used to analyze ground wave propagation characteristics assuming an azimuthal symmetry surface and or elevated ducts are represented via transverse and or longitudinal refractivity and boundary perturbations in 2 d space the 2 d fdtd space extends from x 0 bottom to x spl rarr spl infin top vertically and from z spl rarr spl infin left to z spl rarr spl infin right horizontally perfectly matched layer pml blocks on the left right and top terminate the fdtd computation space to simulate a semi open propagation region the ground at the bottom is simulated either as a perfectly electrical conductor pec or as a lossy second medium a desired initial vertical field profile which has a pulse character in time is injected into the fdtd computation space the pml blocks absorb field components that propagate towards left and top the ground wave components i e the direct ground reflected and surface waves are traced longitudinally toward the right the longitudinal propagation region is covered by a finite sized fdtd computation space as if the space slides from left to right until the pulse propagates to a desired range transverse or longitudinal field profiles are obtained by accumulating the time domain response at each altitude of range and by applying the discrete fourier transformation dft at various frequencies

hypographs satisfying an external sphere condition and the regularity of the minimum time function abstract we prove that if the hypograph of a continuous function f admits at every boundary point a supporting ball then it has essentially positive reach i e the hypograph of the restriction of f outside a closed set of zero measure has locally positive reach hence such a function enjoys some properties of a concave function in particular a e twice differentiability we apply this result to a minimum time problem in the case of a nonlinear smooth dynamics and a target satisfying internal sphere condition