high order infeasible interior point methods for solving sufficient linear complementarity problems in this paper we develop systematically infeasible interior point methods of arbitrarily high order for solving horizontal linear complementarity problems that are sufficient in the sense ofcottle pang and venkateswaran 1989 the results apply to degenerate problems and problem shaving no strictly complementary solution variants of these methods are described that eventually avoid recentering steps and for which all components of the approximate solutions converge superlinearly at a high order and other variants which even terminate with a solution of the complementarity problem after finitely many steps

a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

on the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm in this paper we focus on the problem of identifying the index sets p x i xi 0 n x i fi x 0 and c x i xi fi x 0 for a solution x of the monotone nonlinear complementarity problem ncp f the correct identification of these sets is important from both theoretical and practical points of view such an identification enables us to remove complementarity conditions from the ncp and locally reduce the ncp to a system which can be dealt with more easily we present a new technique that utilizes a sequence generated by the proximal point algorithm ppa using the superlinear convergence property of ppa we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

a matched peak inversion approach for ocean acoustic travel time tomography a new approach for the inversion of travel time data is proposed based on the matching between model arrivals and observed peaks using the linearized model relations between sound speed and arrival time perturbations about a set of background states arrival times and associated errors are calculated on a fine grid of model states discretizing the sound speed parameter space each model state can explain identify a number of observed peaks in a particular reception lying within the uncertainty intervals of the corresponding predicted arrival times the model states that explain the maximum number of observed peaks are considered as the more likely parametric descriptions of the reception these model states can be described in terms of mean values and variances providing a statistical answer matched peak solution to the inversion problem a basic feature of the matched peak inversion approach is that each reception can be treated independently i e no constraints are posed from previous reception identification or inversion results accordingly there is no need for initialization of the inversion procedure and furthermore discontinuous travel time data can be treated the matched peak inversion method is demonstrated by application to 9 month long travel time data from the thetis 2 tomography experiment in the western mediterranean sea

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

new active set identification for general constrained optimization and minimax problems abstract the purpose of this paper is to discuss the problem of identifying the active constraints for general constrained nonlinear programming and constrained minimax problems at an isolated local solution facchinei fischer and kanzow 4 proposed an effective technique which can identify the active set in a neighborhood of an isolated local solution for nonlinear programming and han jian and li 5 improved and extended this to minimax problems in this work a new active constraint identification set is constructed not only is it tighter than the previous two identification sets but also it can be used effectively in penalty algorithms without strict complementarity and linear independence it is shown that the new identification technique can accurately identify the active constraints of nonlinear programming and constrained minimax problems some numerical examples illustrate the identification technique

local behavior of an iterative framework for generalized equations with nonisolated solutions an iterative framework for solving generalized equations with nonisolated solutions is presented for generalized equations with the structure where is a multifunction and f is single valued the framework covers methods that at each step solve subproblems of the type the multifunction approximates f around s besides a condition on the quality of this approximation two other basic assumptions are employed to show q superlinear or q quadratic convergence of the iterates to a solution a key assumption is the upper lipschitz continuity of the solution set map of the perturbed generalized equation moreover the solvability of the subproblems is required conditions that ensure these assumptions are discussed in general and by means of several applications they include monotone mixed complementarity problems karush kuhn tucker systems arising from nonlinear programs and nonlinear equations particular results deal with error bounds and upper lipschitz continuity properties for these problems

on the identification of zero variables in an interior point framework we consider column sufficient linear complementarity problems and study the problem of identifying those variables that are zero at a solution to this end we propose a new computationally inexpensive technique that is based on growth functions we analyze in detail the theoretical properties of the identification technique and test it numerically the identification technique is particularly suited to interior point methods but can be applied to a wider class of methods

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

rank two stable sheaves with odd determinant on fano threefolds of genus nine according to mukai and iliev a smooth prime fano threefold x of genus 9 is associated with a surface mathbb p mathcal v ruled over a smooth plane quartic vargamma and the derived category of vargamma embeds into that of x by a theorem of kuznetsov we use this setup to study the moduli spaces of rank 2 stable sheaves on x with odd determinant for each c 2 ge 7 we prove that a component of their moduli space mathsf m x 2 1 c 2 is birational to a brill noether locus of vector bundles with fixed rank and degree on vargamma having enough sections when twisted by mathcal v for c 2 7 we prove that mathsf m x 2 1 7 is isomorphic to the blow up of the picard variety text pic 2 vargamma along the curve parametrizing lines contained in x

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

on the accurate identification of active set for constrained minimax problems abstract in this paper the problem of identifying the active constraints for constrained nonlinear programming and minimax problems at an isolated local solution is discussed the correct identification of active constraints can improve the local convergence behavior of algorithms and considerably simplify algorithms for inequality constrained problems so it is a useful adjunct to nonlinear optimization algorithms facchinei et al f facchinei a fischer c kanzow on the accurate identification of active constraints siam j optim 9 1998 14 32 introduced an effective technique which can identify the active set in a neighborhood of a solution for nonlinear programming in this paper we first improve this conclusion to be more suitable for infeasible algorithms such as the strongly sub feasible direction method and the penalty function method then we present the identification technique of active constraints for constrained minimax problems without strict complementarity and linear independence some numerical results illustrating the identification technique are reported

ocean acoustic tomography based on peak arrivals the recently introduced notion of peak arrivals athanassoulis and skarsoulis j acoust soc am 97 3575 3588 1995 defined as the significant local maxima of the arrival pattern is studied here as a modeling basis for performing ocean tomography peak arrivals constitute direct theoretical counterparts of experimentally observed peaks and offer a complete modeling of experimental observables even in cases where ray or modal arrivals cannot be resolved the coefficients of the resulting peak inversion system relating travel time with sound speed perturbations are explicitly calculated in the case of range independent environments using normal mode theory to apply the peak inversion scheme to tomography the peak identification and tracking problem is examined from a statistical viewpoint maximum likelihood and least square solutions are derived and discussed the particular approach adopted treats the identification and tracking problem in close relation to the inversion procedure all possibi

second order perturbations of peak arrival times due to sound speed variations the concept of peak arrivals is used to study the second order arrival time behavior of broadband tomographic signals due to sound speed variations peak arrivals defined as the local maxima of the pressure amplitude at the receiver in the time domain are the exact theoretical counterparts of the experimental observables in ocean acoustic tomography and they offer a uniform description of all the peaks in the arrival pattern independent of whether they can be resolved as ray modal arrivals or not closed form expressions for the first and second sound speed derivatives of peak arrival times are derived in terms of sound speed derivatives of the green s function using normal mode theory the sound speed derivatives of the green s function are analytically expressed for the case of a range independent medium in terms of eigenvalues and eigenfunctions at the reference state numerical examples are given for range independent environments motivated from the thetis 2 experiment showing different kinds of

on the cohomology of brill noether loci over quot schemes abstract let c be a smooth projective irreducible curve over an algebraic closed field k of characteristic 0 we consider brill noether loci over the moduli space of morphisms from c to a grassmannian g m n of m planes in k n and the corresponding quot schemes of quotients of a trivial vector bundle on c compactifying the spaces of morphisms we study in detail the case in which m 2 n 4 we prove results on the irreducibility and dimension of these brill noether loci and we address explicit formulas for their cohomology classes we study the existence problem of these spaces which is closely related with the problem of classification of vector bundles over curves

linked symplectic forms and limit linear series in rank 2 with special determinant abstract we generalize the prior linked symplectic grassmannian construction applying it to prove smoothing results for rank 2 limit linear series with fixed special determinant on chains of curves we apply this general machinery to prove new results on nonemptiness and dimension of rank 2 brill noether loci in a range of degrees

linear series on metrized complexes of algebraic curves a metrized complex of algebraic curves over an algebraically closed field kappa is roughly speaking a finite metric graph gamma together with a collection of marked complete nonsingular algebraic curves c v over kappa one for each vertex v of gamma the marked points on c v are in bijection with the edges of gamma incident to v we define linear equivalence of divisors and establish a riemann roch theorem for metrized complexes of curves which combines the classical riemann roch theorem over kappa with its graph theoretic and tropical analogues from amini and caporaso adv math 240 1 23 2013 baker and norine adv math 215 2 766 788 2007 gathmann and kerber math z 259 1 217 230 2008 and mikhalkin and zharkov tropical curves their jacobians and theta functions contemporary mathematics 203 231 2007 providing a common generalization of all of these results for a complete nonsingular curve x defined over a non archimedean field mathbb k together with a strongly semistable model mathfrak x for x over the valuation ring r of mathbb k we define a corresponding metrized complex mathfrak c mathfrak x of curves over the residue field kappa of mathbb k and a canonical specialization map tau mathfrak c mathfrak x from divisors on x to divisors on mathfrak c mathfrak x which preserves degrees and linear equivalence we then establish generalizations of the specialization lemma from baker algebra number theory 2 6 613 653 2008 and its weighted graph analogue from amini and caporaso adv math 240 1 23 2013 showing that the rank of a divisor cannot go down under specialization from x to mathfrak c mathfrak x as an application we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to eisenbud and harris invent math 85 337 371 1986 using this link we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove among other things that any degeneration of a mathfrak g r d in a regular family of semistable curves is a limit mathfrak g r d on the special fiber

new error bounds for the linear complementarity problem recently mangasarian and solodov 1993 showed that every nonlinear complementarity problem ncp is equivalent to the unconstrained minimization of a certain implicit lagrangian in particular it was shown that this implicit lagrangian is nonnegative everywhere and its set of zeros coincides with the solution set of the original ncp in this paper we consider the linear complementarity problem lcp and show that the distance to the solution set of the lcp from any point sufficiently close to the set can be bounded above by the square root of the implicit lagrangian for the lcp in other words the square root of the implicit lagrangian is a local error bound for the lcp our proof is based on showing that the square root of the implicit lagrangian is equivalent to the residual function used in a known local error bound robinson 1981 luo and tseng 1992 when the matrix associated with the lcp is nondegenerate the new error bound is in fact global this extends the error bound result of mathias and

perturbations in travel time and ray geometry due to mesoscale disturbances a comparison of exact and approximate calculations exact calculations of the changes in eigenray travel time and geometry due to range dependent mesoscale disturbances are compared to three approximations in the first approximation the sound speed perturbation is integrated along the unperturbed path in the second approximation the range average of the sound speed perturbation is integrated along the unperturbed path the third approximation uses the range average raytrace to estimate changes of travel time and geometry calculations made both for a measured section through a mesoscale eddy and for a simplified model of an eddy with variable strength position and boundary width suggest that i while none of the approximations tested is uniformly the best the first approximation is to be preferred ii the eddy position relative to the ray path is quite significant and iii the exact travel time perturbations are not sensitive to whether the eddy boundaries are assumed sharp or gradual for an eddy perturbation of 9 m s 1 9 c or rossby wave displacement of about 200 m at 700 m depth the travel time approximations are typically in error by 10 20 the associated change in turning depth from the unperturbed state to either the perturbed or the range average state is typically 100 200 m these errors are significant but not large and suggest that it will be beneficial to iterate linear inverse schemes used to estimate sound speed perturbations from travel time perturbations

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

the picard group of the universal moduli space of vector bundles on stable curves abstract we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its picard group as a consequence we obtain an explicit description of the picard groups of the universal moduli stack of vector bundles on smooth curves and of the schmitt s compactification over the stack of stable curves we prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group in particular we give necessary and sufficient conditions for the existence of poincare bundles over the universal curve of an open substack of the rigidification generalizing a result of mestrano ramanan

the proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem in this paper we consider a proximal point algorithm ppa for solving monotone nonlinear complementarity problems ncp ppa generates a sequence by solving subproblems that are regularizations of the original problem it is known that ppa has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems however it is not always easy to solve subproblems or to check those criteria in this paper we adopt the generalized newton method proposed by de luca facchinei and kanzow to solve subproblems and adopt some ncp functions to check the criteria then we show that the ppa converges globally provided that the solution set of the problem is nonempty moreover without assuming the local uniqueness of the solution we show that the rate of convergence is superlinear in a genuine sense provided that the limit point satisfies the strict complementarity condition

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

the severi bound on sections of rank two semistable bundles on a riemann surface let c be a compact riemann surface of genus g 1 for a line bundle l on c of degree d with 0 d 2g 2 the classical theorem of clifford states that the dimension h 0 l of the space of global sections of l satisfies h 0 l d 2 1 it is also quite easy to list the cases which give equality apart from the trivial bundle and the canonical bundle these exist only on hyperelliptic curves this theorem can be extended to a semistable bundle e on c of rank n and degree d with 0 d n 2g 2 when it states that h 0 e d 2 n now let e be a semistable bundle of rank 2 in this case the authors obtain a more refined result depending on the segre invariant s e which is defined as the minimum value of dege l degl for l a line subbundle of e in particular the semistability condition becomes s e 0 the main theorem is that if dege d and s d 4g 4 s then h 0 e d s 2 where 1 or 2 note that d s is always even in a previous paper e arrondo and the second author conjectured this inequality with 1 with a small number of exceptions similar to those for rank 1 however this is not correct and 2 is sometimes necessary the authors call the bundles which attain this bound severi bundles the number is obtained as follows we define polynomials k r n n the krawtchouk polynomials by means of the generating function r k r n n z r 1 z n 1 z n n let r d s 2 1 then we can take 1 if k r g 2g s 0 and 2 otherwise the connection with krawtchouk polynomials is intriguing since these are used in coding theory and recent work of t johnsen has established links between coding theory and semistable bundles of rank 2 on riemann surfaces these links make essential use of the segre invariant s the key to the proof of the inequality is the following lemma let e be a bundle of rank 2 and degree d on a curve of genus g having at least r 1 g independent sections if k r g 2g 2r 2 d 0 then there is a nonzero section of e vanishing at r points this is proved using a computation of chern classes formal properties of k r n n and the porteous formula the authors give examples to show that for any allowable values of g s d the bound 1 can be attained in fact on a hyperelliptic curve a good deal is known about the vanishing of k r n n thus providing a restricted list of candidates for the invariants of severi bundles but these values do not all arise from actual bundles the severi bundles with d s 0 are all known and the authors compute those with d s 2 and d s 4 in particular the bundles obtained by i grzegorczyk ulam quart 3 1996 no 2 41 ff approx 3 pp electronic are severi bundles with g 5 s 4 d 8 the authors also obtain sharp bounds for the numbers of independent sections of non semistable bundles

stability of bundles with a given filtration on a smooth curve let x be a smooth projective curve of genus g 2 here we construct stable vector bundles on x equipped with a filtration with suitable numerical properties and with stable graded subquotients

universal families of extensions if x denotes a projective variety over a field k and jr and y coherent sheaves on x the vector space exti sr y parametrizes the extensions of f by v on x and there is a universal extension of p t by p y on xx exti y 5 such that for every k rational point u of exti ir y its restriction to x x u is just the extension represented by u moreover p p extk st v parametrizes the classes of nonsplitting extensions of 9 by y on x module k and there is a universal family of extensions of pt f by pf5 p 1 on x x p over p such that for every k rational point p of p its restriction to x x p represents just the class of extensions given by p it is the purpose of this paper not only to prove these two statements but to generalize them to the relative case of a flat projective morphism f x y of noetherian schemes they turn out to be a consequence of grothendieck s universal properties of vector bundles and projective bundles cp 2 section i 91 special cases of these results appeared several times in the literature to the best of my knowledge first in 5 lemma 3 11 and second in 6 lemma 2 41 the technical tool is the base change theory for relative ext sheaves which is outlined in section 1 in section 2 some relations between relative ext sheaves and families of extensions are given section 3 contains the construction of the universal family of extensions and section 4 the construction of the universal family of classes of nonsplitting extensions concerning the notation f x y means a flat projective morphism of noetherian schemes and r and ce coherent ox modules flat over y the proofs of sections 2 to 4 are more generally correct if only there is a base change theory for relative ext sheaves for example if one uses the base change results of 1 valid for arbitrary proper flat morphisms of complex spaces one gets the corresponding results in this case too 101 oozl 8693183 3 00