a network penalty method we consider the minimum cost network flow problem and describe how the non linear penalty function methods of conn and bartels can be specialized to a combinatorial algorithm for this problem we report on preliminary computational results which show that this method can require fewer pivots than the simplex method while the amount of work required for each pivot is comparable the algorithm can be proven finite using a modification of cunningham s strongly feasible basis pivoting rule

real algebraic hypersurfaces in complex projective varieties let m be a compact affine nonsingular real algebraic variety of dimension m for the background material on real algebraic geometry the reader may refer to the book 2 by an affine real algebraic variety we mean a locally ringed space isomorphic to an algebraic subset of ir for some n equipped with the sheaf of n valued regular functions cf 2 sect 3 2 denote by halgi m z 2 the subgroup of the homology group h i m z 2 which consists of all homology classes represented by zariski closed real algebraic hypersurfaces of m cf 2 sect 11 3 let h2ig m z 2 be the image of h g l m z 2 under the poincar6 duality isomorphism hm i m 7z 2 hi m z 2 the group h2ig m 7 2 is an important invariant with numerous interesting applications it has been recently extensively studied in 4 cf also the cited paper for a survey of earlier results and applications in this paper we investigate the group hlalg m z 2 where m is the underlying real algebraic variety of a complex projective nonsingular variety more precisely every complex projective variety v that is up to isomorphism an algebraic subset of cp n for some n endowed with the sheaf of c valued regular functions can be in the obvious way considered as a real algebraic variety and as such will be denoted by v of course dim v 2 dim v moreover v is an affine real algebraic variety cf 2 proposition 3 4 8 the very interesting and natural class of real varieties of type v has never been the subject of any serious investigation with a recent exception 3 7 7 if v is nonsingular then v is also nonsingular and we can consider the group hlalg v z 2 and in particular define d v dimz 2 h21g v e 2

real algebraic cycles on complex projective varieties if we identify cn with r2n then to any quasi affine complex algebraic variety v of dimension d corresponds in a natural way a real algebraic variety of dimension 2d in the sense of bcr which we will denote by vr since each complex morphism v v can be considered as a real morphism vr v r and all complex algebraic varieties are obtained by glueing affine pieces together we can extend this construction in a natural way to arbitrary complex algebraic varieties x in order to obtain xr the underlying real algebraic structure of x if f x x is a morphism of complex algebraic varieties then it is also in a natural way a regular mapping of real structures xr x r which by abuse of notation will be denoted by f as well if y is a real algebraic variety of dimension n and hk y z 2z is the singular homology group in dimension k with respect to the strong topology then for all k n there is a subgroup denoted by h alg k y z 2z consisting of the fundamental classes of k dimensional real algebraic subsets of y see bcr for a definition of the fundamental class of a real algebraic set and bk1 for more information on h alg k we will study the group h alg 2d 1 y z 2z for y xr where x is a nonsingular projective complex algebraic variety of dimension d in particular y is a nonsingular real algebraic variety of dimension n 2d that is compact for the strong topology it will be more convenient to consider h 1 alg y z 2z which is by definition the image of h alg n 1 y z 2z under the poincare duality isomorphism hn 1 y z 2z h 1 y z 2z then the homomorphism

vector bundles and regulous maps let x be a compact nonsingular affine real algebraic variety we prove that every pre algebraic vector bundle on x becomes algebraic after finitely many blowing ups using this theorem we then prove that the stiefel whitney classes of any pre algebraic mathbb r vector bundle on x are algebraic we also derive that the chern classes of any pre algebraic mathbb c vector bundles and the pontryagin classes of any pre algebraic mathbb r vector bundle are blow mathbb c algebraic we also provide several results on line bundles on x

optimal coalition formation and surplus distribution two sides of one coin abstract a fuzzy coalitional game represents a situation in which players can vary the intensity at which they participate in the coalitions accessible to them as opposed to the treatment as a binary choice in the non fuzzy crisp game building on the property not made use of so far in the literature of fuzzy games that a fuzzy game can be represented as a convex program this paper shows that the optimum of such a program determines the optimal coalitions as well as the optimal rewards for the players two sides of one coin furthermore this program is seen to provide a unifying framework for representing the core the least core and the fuzzy nucleolus among others next we derive conditions for uniqueness of core rewards and to deal with non uniqueness we introduce a family of parametric perturbations of the convex program that encompasses a large number of well known concepts for selection from the core including the dutta ray solution dutta and ray 1989 the equal sacrifice solution yu 1973 the equal division solution selten 1972 and the tau value tijs 1981 we also generalize the concept of the grand coalition of contracting players by allowing for multiple technologies and we specify the conditions for this allocation to be unique and egalitarian finally we show that our formulation offers a natural extension to existing models of production economies with threats and division rules for common surplus

the topology of real algebraic sets with isolated singularities in this paper we give a topological classification of real algebraic sets with isolated singularities showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points the question of which topological spaces are homeomorphic to real algebraic sets solutions of polynomial equations in euclidean space has been long studied in 1936 seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set 121 and in 1952 nash extended this result to all smooth compact manifolds 11 in 1973 tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set 13 so at least compact nonsingular algebraic sets are classified little has been done with singular algebraic sets however since the transversality arguments used by seifert nash and tognoli no longer apply except in some special cases one could use stability of singularities such as kuiper 71 and akbulut 1 used to show certain nonsmoothable pl manifolds are algebraic sets or one could use the projective version of seifert nash tognoli as king did 6 but one could still not hope these techniques would allow even a characterization of isolated singularities to get around this problem we take a cue from hironaka s resolution of singularities 4 the idea is to take a topological resolution of a space if it exists we can apply transversality techniques seifertnash tognoli to the resolved space and then blow down algebraically and end up with the original space as an algebraic set it seems likely that this technique allows one to classify all algebraic sets but in any case we show that it classifies all algebraic sets with isolated singularities in future papers we will use this technique to show for instance that all compact pl manifolds are homeomorphic to real algebraic sets 17 and that 2 dimensional real algebraic sets are topologically characterized as polyhedra satisfying sullivan s even local euler characteristic condition 16

real algebraic morphisms represent few homotopy classes we study the problem of representing homotopy classes of maps between real algebraic varieties of regular maps

on real algebraic morphisms into even dimensional spheres given affine real algebraic varieties x and y let us denote by r x y the set of regular mappings real algebraic morphisms from x into y for definitions and notions of real algebraic geometry see 2 where the theory of real algebraic varieties is treated systematically our aim in this paper is to study the set m x s of regular mappings from affine nonsingular real algebraic varieties x into s the unit sphere ezn 1x2 1 earlier we obtained several results in this direction 3 4 cf also 2 chap 13 in particular it was shown in 4 that given a compact connected nonsingular orientable real algebraic subset x of rp of odd dimension k either each smooth mapping from x to sk is homotopic to a regular mapping or precisely those mappings of even topological degree have this property now we shall study the case of regular mappings into even dimensional spheres strangely enough the situation then is radically different from that mentioned above for odd dimensional spheres for a very large class of compact smooth manifolds of even dimension 2k most algebraic models x of these manifolds have the property that every regular mapping from x into s2k is null homotopic the meaning of most will be made precise later cf remark 1 6 theorem 2 1 example 2 3 this class of manifolds contains all compact co hypersurfaces of r2k 1

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

a second derivative sqp method global convergence sequential quadratic programming sqp methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems although second derivative information may often be calculated there is little practical theory that justifies exact hessian sqp methods in particular the resulting quadratic programming qp subproblems are often nonconvex and thus finding their global solutions may be computationally nonviable this paper presents a second derivative sqp method based on quadratic subproblems that are either convex and thus may be solved efficiently or need not be solved globally additionally an explicit descent constraint is imposed on certain qp subproblems which guides the iterates through areas in which nonconvexity is a concern global convergence of the resulting algorithm is established

torsion cohomology classes and algebraic cycles on complex projective manifolds abstract atiyah and hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold which are not poincare dual to an algebraic cycle we notice that the order of these classes must be small compared to the dimension of the manifold however building upon a construction of kollar one can provide such examples with arbitrary high prime order the dimension being fixed this method also provides examples of torsion algebraic cycles which are non trivial in the griffiths group and lie in a arbitrary high level of the h saito filtration on chow groups

on approximating submanifolds by algebraic sets and a solution to the nash conjecture in n nash showed that any compact smooth submanifold m of ir is e isotopic to a union of connected components of a real algebraic variety if n 2 dim m e isotopic just means that there are arbitrarily small smooth isotopies to components of real algebraic varieties for smaller n however he was not able to get such a strong result but conjectured it was still true in w a proof of this conjecture was given which unfortunately had a gap later i and t2 gave correct proofs in the case where n is bigger than roughly 3 dim m 2 among other results in this paper we give a proof of this conjecture of nash in particular we will show

a second derivative sqp method local convergence and practical issues gould and robinson siam j optim 20 2010 pp 2023 2048 proved global convergence of a second derivative sqp method for minimizing the exact ell 1 merit function for a fixed value of the penalty parameter this result required the properties of a so called cauchy step which was itself computed from a so called predictor step in addition they allowed for the additional computation of a variety of optional accelerator steps that were intended to improve the efficiency of the algorithm the main purpose of this paper is to prove that a nonmonotone variant of the algorithm is quadratically convergent for two specific realizations of the accelerator step this is verified with preliminary numerical results on the hock and schittkowski test set once fast local convergence is established we consider two specific aspects of the algorithm that are important for an efficient implementation first we discuss a strategy for defining the positive definite matrix b k used in computing the predictor step that is based on a limited memory bfgs update second we provide a simple strategy for updating the penalty parameter based on approximately minimizing the ell 1 penalty function over a sequence of increasing values of the penalty parameter

stably irrational hypersurfaces of small slopes we show that over any uncountable field of characteristic different from two a very general hypersurface of dimension n 2 and degree at least log 2 n 2 is not stably rational this significantly improves earlier results of kollar and totaro as a byproduct of our proof we obtain new counterexamples to the integral hodge conjecture answering a question of voisin and colliot thelene voisin

semialgebraic topology over a real closed field ii basic theory of semialgebraic spaces our first goal in this paper is to develop a basic language for the theory of 4 4 semialgebraic spaces over an arbitrary real closed field r 6 9 this language seems to be very convenient for topological considerations in the space x r of rational points of an algebraic variety x over r we then study paths and path components in semialgebraic spaces 10 13 expanding and completing the results of part i of the paper 6 we use the notations and terminology from part i without further explanation we refer the reader to the introduction 1 in part i to get an idea about the program pursued here we hope to continue our work in the near future by research on the homology cohomology and homotopy of semialgebraic spaces a n essential purpose of the present paper is to lay firm ground for all that thus we feel it necessary to proceed carefully and even pedantically in all foundational matters especially in 6 and 7

a line search exact penalty method with bi object strategy for nonlinear constrained optimization the exact penalty methods are very popular because of their ability to handle degenerate problems and inconsistent constraint linearizations this paper presents a line search exact penalty method with bi object strategy lsbo for nonlinear constrained optimization in the algorithm lsbo the penalty parameter is selected at every iteration such that the sufficient progress toward feasibility and optimality is guaranteed along the search direction in contrast with classical exact penalization approaches lsbo method has two goals to determine whether the current iteration is successful or not one is improving the feasibility and the other is reducing the value of the objective function moreover the penalty parameter is only related to the information at the current iterate point the sequence of the penalty parameter is non monotone which does not affect the global convergence in theory and is seen to be advantageous in practice it is shown that the algorithm enjoys favorable global convergence properties under the weaker assumptions numerical experiments illustrate the behavior of the algorithm on various difficult situations penalty factor is only related to the information at the current iterate point the sequence of the penalty parameter is non monotone the search direction is related to the penalty factor the acceptable criterion is not related to the penalty factor method can handle degenerate problems and inconsistent constraint linearizations

embedding semi algebraic spaces let r be any real closed field which we take to be totally ordered and let a denote n dimensional affine r space an affine semi algebraic set is a subset of some a which has a finite description in terms of polynomial equalities and inequalitities a semi algebraic space is an object obtained by pasting finitely many semi algebraic sets together along open semi algebraic subsets and may be thought of as the semi algebraic version of an integral separated r scheme of finite type semi algebraic spaces were introduced in 2 by delfs and knebusch and continue to be the objects of intensive and fruitful investigation by these authors the question which we address is can every semi algebraic space be embedded in some affine space this question was raised in 2 and the answer is provided by

automatic decrease of the penalty parameter in exact penalty function methods this paper presents an analysis of the involvement of the penalty parameter in exact penalty function methods that yields modifications to the standard outer loop which decreases the penalty parameter typically dividing it by a constant the procedure presented is based on the simple idea of making explicit the dependence of the penalty function upon the penalty parameter and is illustrated on a linear programming problem with the l1 exact penalty function and an active set approach the procedure decreases the penalty parameter when needed to the maximal value allowing the inner minimization algorithm to leave the current iterate it moreover avoids unnecessary calculations in the iteration following the step in which the penalty parameter is decreased we report on preliminary computational results which show that this method can require fewer iterations than the standard way to update the penalty parameter this approach permits a better understanding of the performance of exact penalty methods

improved bounds for some nonstandard problems in generalized heat conduction abstract we consider the maxwell cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time we obtain l 2 bounds for the temperature and heat flux by means of lagrange identities these bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations

rigidity and moduli space in real algebraic geometry let r be a real closed field and let x be an affine algebraic variety over r we say that x is universally map rigid umr for short if for each irreducible affine algebraic variety z over r the set of nonconstant rational maps from z to x is finite a bijective map open image in new window from an affine algebraic variety over r to x is called weak change of the algebraic structure of x if it is regular and 1 is a nash map which preserves nonsingular points we prove the following rigidity theorem every affine algebraic variety over r is umr up to a weak change of its algebraic structure let us introduce another notion let y be an affine algebraic variety over r we say that x and y are algebraically unfriendly if all the rational maps from x to y and from y to x are trivial i e zariski locally constant from the preceding theorem we infer that if dim x 1 then there exists a set open image in new window of weak changes of the algebraic structure of x such that for each t s r with t s open image in new window and open image in new window are algebraically unfriendly this result implies the following expected fact for each nonsingular affine algebraic variety x over r of positive dimension the natural nash structure of x does not determine the algebraic structure of x in fact the moduli space of birationally nonisomorphic nonsingular affine algebraic varieties over r which are nash isomorphic to x has the same cardinality of r this result was already known under the special assumption that r is the field of real numbers and x is compact and nonsingular

the six grothendieck operations on o minimal sheaves in this paper we develop the formalism of the grothendieck six operations on o minimal sheaves the grothendieck formalism allows us to obtain o minimal versions of i derived projection formula ii universal coefficient formula iii derived base change formula iv kunneth formula v local and global verdier duality

primal methods are better than dual methods for solving overdetermined linear systems in the l sense it is generally believed that methods based upon solving the dual of the linear programming formulation of the linear l infty problem are superior to methods based upon the primal formulation a primal approach is presented along with evidence that for random problems primal methods are to be preferred it is then shown that the method generally considered to be the best algorithm for discrete linear approximation in the l infty norm actually owes much of its efficiency to effective choice of a starting point this phenomenon is explained with the aid of results from classical minimax theory which are then used to determine a suitable initial point for a primal method with this enhancement it would appear that the primal approach is also preferable for problems arising from function approximations

exact penalty functions in constrained optimization in this paper formal definitions of exactness for penalty functions are introduced and sufficient conditions for a penalty function to be exact according to these definitions are stated thus providing a unified framework for the study of both nondifferentiable and continuously differentiable penalty functions in this framework the best known classes of exact penalty functions are analyzed and new results are established concerning the correspondence between the solutions of the constrained problem and the unconstrained minimizers of the penalty functions

algebraic realization of manifolds with group actions abstract let g be a compact lie group and m a closed smooth g manifold with some restrictions on g or on the isotropy groups of the action we show that m is algebraically realized i e there exists a non singular real algebraic g variety which is equivariantly diffeomorphic to m

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

piecewise regular maps let v w be real algebraic varieties that is up to isomorphism real algebraic sets and x subseteq v some subset a map from x into w is said to be regular if it can be extended to a regular map defined on some zariski locally closed subvariety of v that contains x furthermore a continuous map f x rightarrow w is said to be piecewise regular if there exists a stratification mathscr s of v such that for every stratum s in mathscr s the restriction of f to each connected component of x cap s is a regular map by a stratification of v we mean a finite collection of pairwise disjoint zariski locally closed subvarieties whose union is equal to v assuming that the subset x of v is compact we prove that every continuous map from x into a grassmann variety or a unit sphere can be approximated by piecewise regular maps as an application we obtain a variant of the algebraization theorem for topological vector bundles if the variety v is compact and nonsingular we prove that each continuous map from v into a unit sphere is homotopic to a piecewise regular map of class mathscr c k where k is an arbitrary nonnegative integer

semialgebraic topology over a real closed field i paths and components in the set of rational points of an algebraic variety we fix once and for all a real closed base field r by a variety x over r we mean a separated algebraic scheme x over r for all problems attacked here we could equally well assume that x is also reduced since we are only interested in the set x r of rational points of x notice that for every closed point x of x the residue class field k x ojmx either coincides with r i e x is rational or is isomorphic to r t we call the points x of x with k x r the real points of x and the other closed points the complex points of x r is a topological field a basis of open sets being given by the open intervals

on real algebraic models of smooth manifolds an affine nonsingular real algebraic variety x diffeomorphic to a smooth manifold m is said to be an algebraic model of m the remarkable theorem of nash tognoli asserts that each compact smooth manifold m has an algebraic model 16 or 6 theorem 14 1 10 in fact there exists an infinite family x n of irreducible algebraic models of m such that xt and xj are birationally nonisomorphic for i j 10 cf also 7 for a proof in a special case in view of these results it seems natural and interesting to investigate algebro geometric properties of various algebraic models of a given smooth manifold this paper addresses a few questions of this type for notions and results of real algebraic geometry we refer the reader to the book 6 given a compact affine nonsingular real algebraic variety x denote by h x z 2 the subgroup of hk x z 2 of the homology classes represented by zariski closed fc dimensional algebraic subvarieties of x 6 chapter 11 or 11 let h x z 2 be the image of h k xfz 2 9 d dimx under the poincare duality isomorphism hd k x z 2 h x z 2 although the groups hlg x z 2 are one of the most important invariants of x a sample of applications can be found in 1 2 3 6 8 9 our knowledge of their behavior is still rather limited here we consider the following problem let m be a compact smooth manifold and let g be a subgroup of h m z 2 when are there an algebraic model x of m and a diffeomorphism c x m such that the induced isomorphism p h m z 2 h x z 2

on the integral hodge conjecture for real varieties i we formulate the real integral hodge conjecture a version of the integral hodge conjecture for real varieties and raise the question of its validity for cycles of dimension 1 on uniruled and calabi yau threefolds and on rationally connected varieties we relate it to the problem of determining the image of the borel haefliger cycle class map for 1 cycles with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the chow group of 1 cycles of real threefolds new results about these problems are obtained along the way

qualitative aspects in dual phase lag thermoelasticity we consider the system of dual phase lag thermoelasticity proposed by chandrasekharaiah and tzou first we prove that the solutions of the problem are generated by a semigroup of quasi contractions thus the problem of the third order in time is well posed then the exponential stability is investigated finally the spatial behavior of solutions is analyzed in a semi infinite cylinder and a result on the domain of influence is obtained

infeasibility detection and sqp methods for nonlinear optimization this paper addresses the need for nonlinear programming algorithms that provide fast local convergence guarantees regardless of whether a problem is feasible or infeasible we present a sequential quadratic programming method derived from an exact penalty approach that adjusts the penalty parameter automatically when appropriate to emphasize feasibility over optimality the superlinear convergence of such an algorithm to an optimal solution is well known when a problem is feasible the main contribution of this paper however is a set of conditions under which the superlinear convergence of the same type of algorithm to an infeasible stationary point can be guaranteed when a problem is infeasible numerical experiments illustrate the practical behavior of the method on feasible and infeasible problems

bigraded equivariant cohomology of real quadrics we give a complete description of the bigraded bredon co homology ring of smooth projective real quadrics with coefficients in the constant mackey functor z these invariants are closely related to the integral motivic cohomology ring which is not known for these va rieties some of the results and techniques introduced can be applied to other geometrically cellular real varieties

on two classes of nonstandard parabolic problems abstract in this paper the authors continue an investigation of nonstandard parabolic problems this investigation was motivated by one of the suggested methods for stabilizing ill posed problems for evolution equations two different methods are employed methods which in a sense complement one another

algebraic cycles and approximation theorems in real algebraic geometry let af be a compact c manifold a theorem of nash tognoli asserts that m has an algebraic model that is m is diffeomorphic to a nonsingular real algebraic set x let fv afx z 2 denote the subgroup of hk x z 2 of the cohomology classes determined by algebraic cycles of codi mension k on x assuming that m is connected orientable and dim m 5 we prove in this paper that a subgroup g of h2 m z 2 is isomorphic to h x z 2 for some algebraic model x of m if and only if w2 tm is in g and each element of g is of the form w2k for some real vector bundle over m where w2 stands for the second stiefel whitney class a result of this type was previously known for subgroups g of hx m z 2

a resolution theorem for homology cycles of real algebraic varieties the theorem is the best possible in the sense that the components of nonsingular algebraic subsets in the conclusion cannot be replaced by nonsingular algebraic subsets see remark 14 it is well known that the topology of the singularities of 7 2 homology cycles is closely related to real algebraic structures on v ak 2 ak3 therefore it is natural to study to what extent general homology cycles differ from the cycles induced by submanifolds the main theorem of this paper implies that all the singularities of homology cycles can be smoothed by a blowing up process this result should be viewed as a homology version of the resolution theorem h the reason this result is not an easy consequence of h is that not all homology cycles of v may be represented by algebraic subsets there are real obstructions e g ak2 even if they were the map resolving them may introduce new bad homology cycles upstairs the result turns out to be a useful device in topologically classifying real algebraic sets ak6 it also gives interesting corollaries in algebraic topology ak2 another result of this paper theorem 4 says that any smooth map f w v between nonsingular algebraic sets can be approximated by a rational map after changing w by a rational diffeomorphism if the homotopy class of f contains a rational representative this strengthens proposition 2 3 of aks for given algebraic sets vcf and w e n m a function f v w is called an entire rational function if f x p x q x where p ir and q ir ir

algebraic models of smooth manifolds in fact we conjecture that such a family can be chosen uncountable in view of this theorem it seems natural and interesting to investigate algebrogeometric properties of various algebraic models of a given smooth manifold let us now describe one of these problems which will be the main object of study in this paper for notions definitions and results of real algebraic geometry we refer the reader to the book 9 given a compact nonsingular real affine algebraic variety x denote by h ig x 7z 2 the subgroup of nk x z 2 of homology classes represented by zariski closed k dimensional algebraic subvarieties of x cf 9 chap 1 1 let h lg x z 2 be the image of h a k x 71 2 d dimx under the poincar6 duality isomorphism hd k x 71 2 o hk x 71 2 these groups h ig x 71 2 are undoubtedly one of the most important invariants of an algebraic variety x still our knowledge of the behavior of h g x 71 2 is rather limited the main open problem is the following

discontinuous piecewise linear optimization a theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known a penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions it is straightforwardly extendable using standard nonlinear programming techniques tononlinear discontinuous piecewise differentiable functions the descent algorithm which is elaborated uses active set and projected gradient approaches it is a generalization of the ideas used by conn to deal with nonsmoothness in thel1 exact penalty function and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part the constrained case is reduced to the unconstrained minimization of a piecewise linear l1 exact penalty function we also discuss how the algorithm is modified when it encounters degenerate points preliminary numerical results are presented the algorithm is applied to discontinuous optimization problems from models in industrial engineering 1998 the mathematical programming society inc published by elsevier science b v

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6