penalty barrier multiplier methods for convex programming problems we study a class of methods for solving convex programs which are based on nonquadratic augmented lagrangians for which the penalty parameters are functions of the multipliers this gives rise to lagrangians which are nonlinear in the multipliers each augmented lagrangian is specified by a choice of a penalty function varphi and a penalty updating function pi the requirements on varphi are mild and allow for the inclusion of most of the previously suggested augmented lagrangians more importantly a new type of penalty barrier function having a logarithmic branch glued to a quadratic branch is introduced and used to construct an efficient algorithm convergence of the algorithms is proved for the case of pi being a sublinear function of the dual multipliers the algorithms are tested on large scale quadratically constrained problems arising in structural optimization

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

normal form of a metric admitting a parallel field of planes local coordinate normal forms for two classes of metrics are obtained the first class is comprised of those metrics that have a field of degenerate but not necessarily null planes thereby extending a result of a g walker the second class of metrics are artinian metrics that possess a pair of parallel complementary null distributions this latter class is shown to be equivalent to a certain class of symplectic forms

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

the holonomy lie algebras of neutral metrics in dimension four the lie algebra isomorphism between su 1 1 su 1 1 and o 2 2 is used to obtain a list of subalgebras of the latter the resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the lie algebra of the holonomy group of a neutral metric in four dimensions the conclusions taken in conjunction with previously known results furnish a classification of such lie subalgebras of o 2 2 with only one case remaining unresolved

espaces abstraits de morphismes et mutations i try to extend here some results of the paper moduli spaces of decomposable morphisms of sheaves and quotients by non reductive groups let e f be decomposable sheaves on a smooth projective variety i find more linearizations of the natural action of the group mathop rm aut nolimits e times mathop rm aut nolimits f on mathop rm hom nolimits e f such that a good quotient of the open set of semi stable morphisms exists to obtain this one associates to mathop rm hom nolimits e f another space of morphisms mathop rm hom nolimits e f in such a way that there is a natural bijection between the set of orbits of an open subset of mathop rm hom nolimits e f and the set of orbits of an open subset of mathop rm hom nolimits e f we can in this case deduce the existence of good quotients of some open subsets of mathop rm hom nolimits e f from the existence of good quotients of the corresponding open subsets of mathop rm hom nolimits e f

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

the theorem of grauert mulich spindler the boundedness of semi stable sheaves on p is a simple corollary to the above see 13 if one looks into the proof of this theorem by spindler and the techniques by van de ven 14 and barth 1 he can generalize immediately the theorem in the following way let e be a torsion free coherent sheaf on a non singular projective variety x over an algebraically closed field k and ox 1 a very ample invertible sheaf on x if g l y n dimx are sufficiently general member of a very ample linear subsystem l of iox 1 l then y yl i1 is a non singular subvariety of codimension n in x and elf is torsion free

stable rank 2 reflexive sheaves on with small and applications we investigate the moduli spaces of stable rank two reflexive sheaves on p3 with small chern classes as an application to curves of low degree in p3 we prove the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational introduction in the past few years the subject of vector bundles on projective spaces and in particular the case of rank 2 on p3 has received much attention several basic theorems have been proved e g the existence of the moduli space of stable vector bundles m though so far very few of these moduli spaces have been studied in detail the interest of vector bundles partly stems from their connection with curves however the class of curves obtained in this way is rather restricted recently hartshorne srs has focused attention on reflexive sheaves of rank 2 on p3 on the one hand most results proved for vector bundles also turn out to be true for reflexive sheaves on the other hand reflexive sheaves have two significant advantages first they correspond to quite general curves second and most importantly one has the reduction step introduced by hartshorne which is an effective tool in studying vector bundles by relating them to simpler reflexive sheaves in this paper we investigate the moduli spaces of stable rank 2 reflexive sheaves on p3 with c2 3 for c2 2 we prove the moduli spaces are irreducible nonsingular and rational we also classify the related unstable planes for c2 3 in most cases we show that the moduli space is irreducible and unirational as one of the applications to curves of low degree in p3 cf srs 4 1 we prove that the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational hence as a corollary we conclude that the variety of moduli 1g of curves of genus g is unirational for g 5 6 7 8 in 1 we give facts on multiple lines and deduce a criterion for unstable planes 2 contains our classification of semistable sheaves with c2 2 3 is about sheaves with c2 3 4 gives the applications to curves acknowledgment i would like to thank my thesis advisor robin hartshorne he has been very generous in his advice and time in addition he attracted to received by the editors september 1 1982 and in revised form march 18 1983 1980 mathematics subject classification primary 14f05 secondary 14d22 i1984 american mathematical society 0002 9947 84 1 00 25 per page

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities in this paper we analyze and discuss the well posedness of two new variants of the so called sweeping process introduced by moreau in the early 70s moreau in sem anal convexe montpellier 1971 with motivation in plasticity theory the first new variant is concerned with the perturbation of the normal cone to the moving convex subset c t c t supposed to have a bounded variation by a lipschitz mapping under some assumptions on the data we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation the second variant for which a large analysis is made concerns a first order sweeping process with velocity in the moving set c t c t this class of problems subsumes as a particular case the evolution variational inequalities widely used in applied mathematics and unilateral mechanics duvaut and lions in inequalities in mechanics and physics springer berlin 1976 assuming that the moving subset c t c t has a continuous variation for every t in 0 t t 0 t with c 0 c 0 bounded we show that the problem has at least a lipschitz continuous solution the well posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator we also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model krej i in eur j appl math 2 281 292 1991 to the planning procedure in mathematical economy henry in j math anal appl 41 179 186 1973 and cornet in j math anal appl 96 130 147 1983 and to nonregular electrical circuits containing nonsmooth electronic devices like diodes acary et al nonsmooth modeling and simulation for switched circuits lecture notes in electrical engineering springer new york 2011 the theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof our methodology is based only on tools from convex analysis like other papers in this collection we show in this presentation how elegant modern convex analysis was influenced by moreau s seminal work

existence stability and optimality for optimal control problems governed by maximal monotone operators abstract we study optimal control problems governed by maximal monotone differential inclusions with mixed control state constraints in infinite dimensional spaces we obtain some existence results for this kind of dynamics and construct the discrete approximations that allows us to strongly approximate optimal solutions of the continuous type optimal control problems by their discrete counterparts our approach allows us to apply our results for a wide class of mappings that are applicable in mechanics and material sciences

a spinor approach to walker geometry a four dimensional walker geometry is a four dimensional manifold m with a neutral metric g and a parallel distribution of totally null two planes this distribution has a natural characterization as a projective spinor field subject to a certain constraint spinors therefore provide a natural tool for studying walker geometry which we exploit to draw together several themes in recent explicit studies of walker geometry and in other work of dunajski 11 and plebanski 30 in which walker geometry is implicit in addition to studying local walker geometry we address a global question raised by the use of spinors

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

penalization in non classical convex programming via variational convergence a penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method proofs of convergence rely on variational convergence theory

walker s theorem without coordinates we provide a coordinate free version of the local classification due to walker q j math 1 69 1950 of null parallel distributions on pseudo riemannian manifolds the underlying manifold is realized locally as the total space of a fiber bundle each fiber of which is an affine principal bundle over a pseudo riemannian manifold all structures just named are naturally determined by the distribution and the metric in contrast with the noncanonical choice of coordinates in the usual formulation of walker s theorem

yosida moreau regularization of sweeping processes with unbounded variation lett c t be a hausdorff continuous multifunction with closed convex values in a hilbert spacehsuch thatc t has nonempty interior for allt we show that the yosida moreau regularizations of the sweeping process with moving setc t i e the solutions of formula are strongly pointwisely convergent as 0 to the solution of the corresponding sweeping process formally written as formula

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

the space of stability conditions on abelian threefolds and on some calabi yau threefolds we describe a connected component of the space of stability conditions on abelian threefolds and on calabi yau threefolds obtained as the crepant resolution of a finite quotient of an abelian threefold our proof includes the following essential steps 1 we simultaneously strengthen a conjecture by the first two authors and toda and prove that it follows from a more natural and seemingly weaker statement this conjecture is a bogomolov gieseker type inequality involving the third chern character of tilt stable two term complexes on smooth projective threefolds we extend it from complexes of tilt slope zero to arbitrary tilt slope 2 we show that this stronger conjecture implies the so called support property of bridgeland stability conditions and the existence of an explicit open subset of the space of stability conditions 3 we prove our conjecture for abelian threefolds thereby reproving and generalizing a result by maciocia and piyaratne important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall crossing closely related to the support property

a horrocks theorem for reflexive sheaves abstract in this paper we define m tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology we prove that the rank of any m tail reflexive sheaf e on p n is greater or equal to n m m we completely describe m tail reflexive sheaves on p n of minimal rank and we construct huge families of m tail reflexive sheaves of higher rank

asymptotic analysis for penalty and barrier methods in convex and linear programming we consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature we provide a systematic way to generate penalty and barrier functions in this class and we analyze the existence of primal and dual optimal paths generated by these penalty methods as well as their convergence to the primal and dual optimal sets for linear programming we prove that these optimal paths converge to single points

coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm we study the asymptotic behavior of the continuous flow generated by coupling the steepest descent method with a general class of penalty schemes for convex programming we establish convergence of the trajectories towards primal optimal solutions as well as convergence of some naturally associated dual paths the results are extended to the sequences generated by an implicit discretization scheme which corresponds to the coupling of an inexact proximal point iteration with the penalty schemes

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required

representing stable complexes on projective spaces abstract we give an explicit proof of a bogomolov type inequality for c 3 of reflexive sheaves on p 3 then using resolutions of rank two reflexive sheaves on p 3 we prove that the closed points of some strata of the moduli of rank two complexes that are both pt stable and dual pt stable can be given by the structure of quotient stacks using monads we apply the same techniques to p 2 and obtain similar results for some strata of the moduli of bridgeland stable complexes

primal and dual convergence of a proximal point exponential penalty method for linear programming we consider the diagonal inexact proximal point iteration frac u k u k 1 lambda k in partial varepsilon k f u k r k nu k where f x r ctx r exp a i x b i r is the exponential penalty approximation of the linear program min ctx ax b we prove that under an appropriate choice of the sequences k e k and with some control on the residual k for every r k 0 the sequence u k converges towards an optimal point u of the linear program we also study the convergence of the associated dual sequence i k exp a i u k b i r k towards a dual optimal solution