a comparison of second order epiderivatives calculus and optimality conditions abstract the first order theory of generalized derivatives is now well developed significant progress toward a second order theory has also been made two types of second order directional derivatives appear to be particularly promising for applications to optimization parabolic epiderivatives which are well suited for necessary optimality conditions and satisfy standard calculus rules for a large class of functions and the second order epiderivatives of rockafellar which lead to general sufficient optimality conditions that are close to being necessary in this paper the properties of these two types of epiderivatives are surveyed and contrasted special results for c1 1 functions are examined in detail

on some relations between chaney s generalizedsecond order directional derivative and that of ben tal andzowe for a locally lipschitz real valued function f on bbb r n and x u in bbb r n our main result implies that if x is in clarke s subdifferential partial f x coming from the direction u in chaney s sense such that x u equals the directional derivative f x u then chaney s second order directional derivative f x x u when it exists coincides with the value at x of the conjugate function of the ben tal zowe second order directional derivative provided that this value is finite

polynomial structures for nilpotent groups if a polycyclic by finite rank k group admits a faithful affine representation making it acting on rk properly discontinuously and with compact quotient we say that admits an affine structure in 1977 john milnor questioned the existence of affine structures for such groups very recently examples have been obtained showing that even for torsion free finitely generated nilpotent groups n affine structures do not always exist it looks natural to consider affine structures as examples of polynomial structures of degree one we introduce the concept of a canonical type polynomial structure for polycyclic by finite groups using the algebraic framework of the seifert fiber space construction and a nice cohomology vanishing theorem we prove the existence and uniqueness up to conjugation of canonical type polynomial structures for virtually finitely generated nilpotent groups applying this uniqueness to a result going back to mal cev it follows that for torsion free finitely generated nilpotent groups each canonical polynomial structure is expressed in polynomials of limited degree the minimal degree needed for obtaining a polynomial structure will determine the affine defect number we prove that the known counterexamples to milnor s question have the smallest possible affine defect i e affine defect number equal to one

polycyclic by finite groups admit a bounded degree polynomial structure for a polycyclic by finite group gamma of hirsch length h an affine resp polynomial structure is a representation of gamma into rm aff bbb r h resp rm p bbb r h the group of polynomial diffeomorphisms letting gamma act properly discontinuously on bbb r h recently it was shown by counter examples that there exist groups gamma even nilpotent ones which do not admit an affine structure thus giving a negative answer to a long standing question of john milnor we prove that every polycyclic by finite group gamma admits a polynomial structure which moreover appears to be of a special simple type called canonical and as a consequence of this consists entirely of polynomials of a bounded degree the construction of this polynomial structure is a special case of an iterated seifert fiber space construction which can be achieved here because of a very strong and surprising cohomology vanishing theorem

the uniqueness of polynomial crystallographic actions let gamma be a polycyclic by finite group it is proved in 8 that gamma admits a polynomial action of bounded degree on mathbb r n which is properly discontinuous and such that the quotient gamma backslash mathbb r n is compact we prove here that such an action is unique up to conjugation by a polynomial transformation of mathbb r n

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

second order moderate derivatives f x f x it is the purpose of the present paper to introduce a notion of second order generalized derivative for arbitrary finite functions which retains the preceding properties in particular which coincides with the usual second order derivative when it exists and enjoys useful calculus rules our proposal is close to the proposal of 3 cf also 4 the variant we use is similar in spirit to the first order variant of the clarke circa derivatives introduced in l 21 here however the concept of fan plays a key role as in 3 51 let us note that the absence of built in upper semicontinuity is intimately linked with the accuracy of the concept under added assumptions of semicontinuity one recovers the notions of 3 51 our results can be found in 6

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

aspherical manifolds with virtually 3 step nilpotent fundamental group a closed aspherical manifold of dimension 3 4 with virtually 3 step nilpotent fundamental group supports a complete affinely flat structure topological and seifert bundle classifications are given for a special case called seifert relatives of flat riemannian manifolds we use the seifert fiber space construction as a main tool 0 introduction it is well known that given a torsion free virtually poly z group ir there exists a closed smooth k r 1 manifold eg a double coset space of a lie group a question we are interested in is whether such a manifold is homeomorphic to a quotient of r by an action of a discrete group of affine motions the best known result seems to be j scheuneman s 18 stating that the answer is yes if ir is 3 step nilpotent in this paper we generalize this to virtually 3 step nilpotent groups in dimension 3 4 the main idea is to refine the seifert fiber space construction of p conner and f raymond 4 this gives rise to an embedding of such ir into the group of affine motions so that rm ir is compact then we apply t farrell and w c hsiang s theorem 6 stating that any two closed aspherical manifolds with isomorphic fundamental groups which are virtually nilpotent are homeomorphic in dimension 3 4 this paper is organized as follows in section 1 we refine the seifert fiber space construction by restricting the coefficient module to some submodules of m w rk the main result is that if the first coboundary homomorphism 6 of certain cohomology exact sequence is surjective then one can convert the algebraic object group into a geometric one group action manuscript received november 23 1981 manuscript revised may 14 1981 this is a part of the author s ph d thesis at the university of michigan 1981 the second part geometric realization will appear as a separate article elsewhere the author wishes to express his sincere appreciation to professor frank raymond for his guidance in this work

facets of the clique partitioning polytope a subseta of the edge set of a graphg v e is called a clique partitioning ofg is there is a partition of the node setv into disjoint setsw 1 w k such that eachw i induces a clique i e a complete but not necessarily maximal subgraph ofg and such thata 1 uv u v w i u v given weightsw e for alle e the clique partitioning problem is to find a clique partitioninga ofg such that e a w e is as small as possible this problem known to be hard see wakabayashi 1986 comes up for instance in data analysis and here the underlying graphg is typically a complete graph in this paper we study the clique partitioning polytope of the complete graphk n i e is the convex hull of the incidence vectors of the clique partitionings ofk n we show that triangles 2 chorded odd cycles 2 chorded even wheels and other subgraphs ofk n induce facets of the theoretical results described here have been used to design an empirically efficient cutting plane algorithm with which large real world instances of the clique partitioning problem could be solved these computational results can be found in grotschel and wakabayashi 1989

the real cohomology of virtually nilpotent groups abstract in this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group the main ingredient in our setup consists of a polynomial crystallographic action of this group as any finitely generated virtually nilpotent group admits such an action which can be constructed quite easily the approach we present applies to all these groups our main result is an algorithmic way of computing these cohomology spaces as a first application we prove a kind of poincare duality also in the nontorsion free case and we derive explicit formulas in the virtually abelian case

convexity and generalized second order derivatives for locally lipschitz functions abstract we define the generalized second order directional derivatives by means of the clarke generalized gradient for locally lipschitz functions then we give characterization of convexity and state a new sufficient optimality condition

on generic second order gateaux differentiability let f x be a function on a banach space x we say that f is strictly gateaux differentiable if it is gateaux differentiable at every point of x and the mapping x h x h is continuous recall that a convex gateaux differentiable function is strictly gateaux differentiable in the case of a locally lipschitz function our definition coincides with more standard ones it requires that f be norm to weak star continuous

second order necessary conditions for optimality in nonsmooth nonlinear programming the notion of a second order directional derivative introduced by ben tal and zowe and a generalization of this notion by penot are used to state new second order necessary optimality conditions for a nonsmooth nonlinear programming problem with a finite number of inequality and equality constraints

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

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

superlinearly convergent approximate newton methods for lc 1 optimization problems in the literature the proof of superlinear convergence of approximate newton or sqp methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions sometimes the second order derivatives of those functions are required to be lipschitzian in this paper we present approximate newton or sqp methods for solving nonlinear programming problems whose objective and constraint functions have locally lipschitzian derivatives and establishq superlinear convergence of these methods under the assumption that these derivatives are semismooth this assumption is weaker than the second order differentiability the extended linear quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives

some applications of variational inequalities in nonsmooth analysis abstract in this paper we apply some results on variational inequalities to the clarke generalized gradient and the generalized second order directional derivative f x u v and obtain some variational inequalities of them

on second order conditions in unconstrained optimization the main purpose of this paper is to establish the second order nonsmooth sufficient unconstrained optimality condition for so called l stable at some point functions and in this way to generalize some previous results in this direction we provide the comparisons with other results by examples

the clique partitioning problem abstract we study the problem of clique partitioning a graph we prove a new general upper bound result on the number of clique partitions this upper bound is the best possible given information of just the vertices and the number of edges next we show that there exists an optimal partition in which one of the cliques is a maximal clique finally we present two new efficient methods to clique partition a graph since the clique partitioning of a graph is equivalent to the coloring of the complement of the graph any coloring algorithm can also be used to clique partition a graph we present detailed statistics comparing the performance of our algorithms against two of the best known coloring algorithms and against a recently published clique partitioning algorithm both functional and timing comparisons are given and we show that our algorithms compare very favorably on both counts

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

polynomial structures on polycyclic groups we know by recent work of benoist and of burde grunewald that there exist polycyclic by finite groups g of rank h the examples given were in fact nilpotent admitting no properly discontinuous affine action on rh on the other hand for such g it is always possible to construct a properly discontinuous smooth action of g on rh our main result is that any polycyclic by finite group g of rank h contains a subgroup of finite index acting properly discontinuously and by polynomial diffeomorphisms of bounded degree on rh moreover these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group g

on the uniqueness of some affinely flat infra nilmanifolds i p irn irk u p irn irk since pi irn irk pi irn ir k each element of i 1 ir irk is completely determined by k polynomials of total degree at most i in n variables let g be a group and suppose we have a morphism 4 g g l k ir and a homomorphism p g aff ir we can provide a g module structure for p ir f k in the following way for g e g and pepi ir irk we define

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

approximating a second order directional derivative for nonsmooth convex functions for a lower semicontinuous convex function f the approximate second order directional derivative d delta mapsto f varepsilon x 0 d delta is defined through the varepsilon directional derivative f varepsilon x d the function v d x mapsto v d x f varepsilon x d is for all varepsilon 0 locally lipschitz on inf domf and at those points where it is not differentiable v d admits a directional derivative v d x 0 delta for all delta which we precisely denote by f varepsilon x 0 d delta the objective of the present work is two fold to classify all the possible differentiability properties of v d according to the behavior of the function lambda mapsto f x 0 lambda d on r and to study the existence or nonexistence of the limit of f varepsilon x 0 d delta when varepsilon to 0

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

the auslander conjecture for nil affine crystallographic groups let n be a simply connected connected real nilpotent lie group of finite dimension n we study subgroups in aff n n aut n acting properly discontinuously and cocompactly on n this situation is a natural generalization of the so called affine crystallographic groups we prove that for all dimensions 1 n 5 the generalized auslander conjecture holds i e that such subgroups are virtually polycyclic

polynomial alternatives for the group of affine motions it is known that every polycyclic by finite group even if it admits no affine structure allows a polynomial structure of bounded degree a major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of mathbb r n in contrast to the group of affine motions is no longer a finite dimensional lie group in this paper we construct a family of finite dimensional lie groups even linear algebraic groups of polynomial diffeomorphisms which we call weighted groups of polynomial diffeomorphisms it turns out that every polycyclic by finite group admits a polynomial structure via these weighted groups in the nilpotent and other case s we can sharpen by specifying a nice set of weights the existence results obtained in earlier work we introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding mal cev completion this and other properties provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e g in the work of fried goldman and hirsch

clique web facets for multicut polytopes let g v e be a graph an edge set uv e u si v sj i j where s1 sk is a partition of v is called a multicut with k shores we investigate the polytopes mck n and mck n that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores respectively of the complete graph kn we introduce a large class of inequalities called clique web inequalities valid for these polytopes and provide a quite complete characterization of those clique web inequalities that define facets of mck n and mck n using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes we also exhibit a class of clique web facets for which the separation problem can be solved in polynomial time

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