a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

metric subregularity and the proximal point method we examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators we begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator this result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

analysis of two active set type methods to solve unilateral contact problems in this work two active set type methods are considered in order to solve a mathematical problem which describes the frictionless contact between a deformable body and a perfectly rigid obstacle the so called signorini problem these methods are the primal dual active set method and the projection iterative method our aim here is to analyze these two active set type methods and to carry out a comparison with the well known augmented lagrangian method by considering two representative contact problems in the case of large and small deformation after presenting the mechanical formulation in the hyperelasticity framework we establish weak formulations of the problem and the existence result of the weak solution is recalled then we give the finite element approximation of the problem and a description of the numerical methods is presented the main result of this work is to provide a convergence result for the projection iterative method finally we present numerical simulations which illustrate the behavior of the solution and allow the comparison of the numerical methods

the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

null surgery on knots in l spaces let k be a knot in an l space y with a dehn surgery to a surface bundle over s 1 we prove that k is rationally fibered that is the knot complement admits a fibration over s 1 as part of the proof we show that if k c y has a dehn surgery to s 1 x s 2 then k is rationally fibered in the case that k admits some s 1 x s 2 surgery k is floer simple that is the rank of hfk y k is equal to the order of h 1 y by combining the latter two facts we deduce that the induced contact structure on the ambient manifold y is tight in a different direction we show that if k is a knot in an l space y then any thurston norm minimizing rational seifert surface for k extends to a thurston norm minimizing surface in the manifold obtained by the null surgery on k i e the unique surgery on k with b 1 0

characterization of the robust isolated calmness for a class of conic programming problems this paper is devoted to studying the robust isolated calmness of the karush kuhn tucker kkt solution mapping for a large class of interesting conic programming problems including most commonly known ones arising from applications at a locally optimal solution under the robinson constraint qualification we show that the kkt solution mapping is robustly isolated calm if and only if both the strict robinson constraint qualification and the second order sufficient condition hold this implies among others that at a locally optimal solution the second order sufficient condition is needed for the kkt solution mapping to have the aubin property

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

on the generic properties of convex optimization problems in conic form we prove that strict complementarity primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties in this paper we say generic to mean that the set of data possessing the desired property or properties has strictly larger hausdorff dimension than the set of data that does not our proof is elementary and it employs an important result due to larman 7 on the boundary structure of convex bodies

a practical relative error criterion for augmented lagrangians this paper develops a new error criterion for the approximate minimization of augmented lagrangian subproblems this criterion is practical since it is readily testable given only a gradient or subgradient of the augmented lagrangian it is also relative in the sense of relative error criteria for proximal point algorithms in particular it uses a single relative tolerance parameter rather than a summable parameter sequence our analysis first describes an abstract version of the criterion within rockafellar s general parametric convex duality framework and proves a global convergence result for the resulting algorithm specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented lagrangian method with a novel inexact solution condition for the subproblems finally we present computational results drawn from the cute test set including many nonconvex problems indicating that the approach works well in practice

a relaxed constant positive linear dependence constraint qualification and applications in this work we introduce a relaxed version of the constant positive linear dependence constraint qualification cpld that we call rcpld this development is inspired by a recent generalization of the constant rank constraint qualification by minchenko and stakhovski that was called rcrcq we show that rcpld is enough to ensure the convergence of an augmented lagrangian algorithm and that it asserts the validity of an error bound we also provide proofs and counter examples that show the relations of rcrcq and rcpld with other known constraint qualifications in particular rcpld is strictly weaker than cpld and rcrcq while still stronger than abadie s constraint qualification we also verify that the second order necessary optimality condition holds under rcrcq

formulation and analysis of two energy consistent methods for nonlinear elastodynamic frictional contact problems energy conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability furthermore some physical phenomena such as friction can generate dissipation then in this work we present and analyse two energy consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon the first approach permits one to enforce respectively the kuhn tucker and persistency conditions during each time step by combining an adapted continuation of the newton method and a lagrangean formulation in addition the second method which is based on the work in p hauret p le tallec energy controlling time integration methods for nonlinear elastodynamics and low velocity impact comput methods appl mech eng 195 2006 4890 4916 represents a specific penalisation of the unilateral contact conditions some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods

ozsvath szabo invariants and tight contact three manifolds ii let s 3 k be the oriented 3 manifold obtained by rational r surgery on a knot k s 3 using the contact ozsv ath szab o invariants we prove for a class of knots k containing all the algebraic knots that s 3 k carries positive tight contact structures for every r 6 g s k 1 where gs k is the slice genus of k this implies in particular that the brieskorn spheres 2 3 4 and 2 3 3 carry tight positive contact structures as an application of our main result we show that for each m2n there exists a seifert bered rational homology 3 sphere mm carrying at least m pairwise non isomorphic tight nonllable contact structures

a parallelizable augmented lagrangian method applied to large scale non convex constrained optimization problems we contribute improvements to a lagrangian dual solution approach applied to large scale optimization problems whose objective functions are convex continuously differentiable and possibly nonlinear while the non relaxed constraint set is compact but not necessarily convex such problems arise for example in the split variable deterministic reformulation of stochastic mixed integer optimization problems we adapt the augmented lagrangian method framework to address the presence of nonconvexity in the non relaxed constraint set and to enable efficient parallelization the development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions however deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized pivotal in our modification to the augmented lagrangian method is an integration of the simplicial decomposition method and the nonlinear block gauss seidel method an adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted under mild conditions optimal dual convergence is proven and we report computational results on test instances from the stochastic optimization literature we demonstrate improvement in parallel speedup over a baseline parallel approach

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

the rate of convergence of the augmented lagrangian method for nonlinear semidefinite programming we analyze the rate of local convergence of the augmented lagrangian method in nonlinear semidefinite optimization the presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices an implicit function theorem for semismooth functions and variational analysis on the projection operator in the symmetric matrix space without requiring strict complementarity we prove that under the constraint nondegeneracy condition and the strong second order sufficient condition the rate of convergence is linear and the ratio constant is proportional to 1 c where c is the penalty parameter that exceeds a threshold overline c 0

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

two new weak constraint qualifications and applications we present two new constraint qualifications cqs that are weaker than the recently introduced relaxed constant positive linear dependence rcpld cq rcpld is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally a major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a cq this is done in the first new cq which we call the constant rank of the subspace component crsc cq this new cq also preserves many of the good properties of rcpld such as local stability and the validity of an error bound we also introduce an even weaker cq called the constant positive generator cpg which can replace rcpld in the analysis of the global convergence of algorithms we close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods sequential quadratic programming augmented lagrangians

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

handlebody construction of stein surfaces the topology of stein surfaces and contact 3 manifolds is studied by means of handle decompositions a simple characterization of homeomorphism types of stein surfaces is obtained they correspond to open handlebodies with all handles of index 2 an uncountable collection of exotic l4 s is shown to admit stein structures new invariants of contact 3 manifolds are produced including a complete and computable set of invariants for determining the homotopy class of a 2 plane field on a 3 manifold these invariants are applicable to seiberg witten theory several families of oriented 3 manifolds are examined namely the seifert fibered spaces and all surgeries on various links in s3 and in each case it is seen that most members of the family are the

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

on the nonergodic convergence rate of an inexact augmented lagrangian framework for composite convex programming in this paper we consider the linearly constrained composite convex optimization problem whose objective is a sum of a smooth function and a possibly nonsmooth function we propose an inexact aug

sufficient optimality conditions hold for almost all nonlinear semidefinite programs we derive a new genericity result for nonlinear semidefinite programming nlsdp namely almost all linear perturbations of a given nlsdp are shown to be nondegenerate here nondegeneracy for nlsdp refers to the transversality constraint qualification strict complementarity and second order sufficient condition due to the presence of the second order sufficient condition our result is a nontrivial extension of the corresponding results for linear semidefinite programs sdp from alizadeh et al math program 77 2 ser b 111 128 1997 the proof of the genericity result makes use of forsgren s derivation of optimality conditions for nlsdp in forsgren math program 88 1 ser a 105 128 2000 due to the latter approach the positive semidefiniteness of a symmetric matrix g x depending continuously on x is locally equivalent to the fact that a certain schur complement s x of g x is positive semidefinite this yields a reduced nlsdp by considering the new semidefinite constraint s x succeq 0 s x 0 instead of g x succeq 0 g x 0 while deriving optimality conditions for the reduced nlsdp the well known and often mentioned h term in the second order sufficient condition vanishes this allows us to access the proof of the genericity result for nlsdp

a new active set algorithm for box constrained optimization an active set algorithm asa for box constrained optimization is developed the algorithm consists of a nonmonotone gradient projection step an unconstrained optimization step and a set of rules for branching between the two steps global convergence to a stationary point is established for a nondegenerate stationary point the algorithm eventually reduces to unconstrained optimization without restarts similarly for a degenerate stationary point where the strong second order sufficient optimality condition holds the algorithm eventually reduces to unconstrained optimization without restarts a specific implementation of the asa is given which exploits the recently developed cyclic barzilai borwein cbb algorithm for the gradient projection step and the recently developed conjugate gradient algorithm cg descent for unconstrained optimization numerical experiments are presented using box constrained problems in the cuter and minpack 2 test problem libraries

on the classification of tight contact structures ii we present complete classification results for tight contact structures on two classes of 3 manifolds i torus bundles which fiber over the circle and ii circle bundles which fiber over closed surfaces

error estimates for the three field formulation with bubble stabilization in this paper we prove convergence and error estimates for the so called 3 field formulation using piecewise linear finite elements stabilized with boundary bubbles optimal error bounds are proved in l 2 and in the broken h 1 norm for the internal variable u and in suitable weighted l 2 norms for the other variables a and

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

a practical general approximation criterion for methods of multipliers based on bregman distances this paper demonstrates that for generalized methods of multipliers for convex programming based on bregman distance kernels including the classical quadratic method of multipliers the minimization of the augmented lagrangian can be truncated using a simple generally implementable stopping criterion based only on the norms of the primal iterate and the gradient or a subgradient of the augmented lagrangian at that iterate previous results in this and related areas have required conditions that are much harder to verify such as e optimality with respect to the augmented lagrangian or strong conditions on the convex program to be solved here only existence of a kkt pair is required and the convergence properties of the exact form of the method are preserved the key new element in the analysis is the use of a full conjugate duality framework as opposed to mainly examining the action of the method on the standard dual function of the convex program an existence result for the iterates stronger than those possible for the exact form of the algorithm is also included

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

local convergence of exact and inexact augmented lagrangian methods under the second order sufficient optimality condition we establish local convergence and rate of convergence of the classical augmented lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second order sufficient optimality condition in particular no constraint qualifications of any kind are needed previous literature on the subject required in addition the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second order sufficient condition that said the classical results allow the initial multiplier estimate to be far from the optimal one at the expense of proportionally increasing the threshold value for the penalty parameters although our primary goal is to avoid constraint qualifications if the stronger assumptions are introduced then starting points far from the optimal multiplier are allowed within our analysis as well using only the second order sufficient optimality condition for penalty parameters large

convergence properties of an augmented lagrangian algorithm for optimization with a combination of general equality and linear constraints we consider the global and local convergence properties of a class of augmented lagrangian methods for solving nonlinear programming problems in these methods linear and more general constraints are handled in different ways the general constraints are combined with the objective function in an augmented lagrangian the iteration consists of solving a sequence of subproblems in each subproblem the augmented lagrangian is approximately minimized in the region defined by the linear constraints a subproblem is terminated as soon as a stopping condition is satisfied the stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints in this paper we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as show that potentially troublesome penalty parameters remain bounded away from zero

hybrid finite element methods for the signorini problem we study three mixed linear finite element methods for the numerical simulation of the two dimensional signorini problem applying falk s lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them two of these finite elements provide optimal results under reasonable regularity assumptions on the signorini solution and the numerical investigation shows that the third method also provides optimal accuracy

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

a note on upper lipschitz stability error bounds and critical multipliers for lipschitz continuous kkt systems we prove a new local upper lipschitz stability result and the associated local error bound for solutions of parametric karush kuhn tucker systems corresponding to variational problems with lipschitzian base mappings and constraints possessing lipschitzian derivatives and without any constraint qualifications this property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the lagrange multiplier associated to the primal solution which is weaker than second order sufficiency all this extends several results previously known only for optimization problems with twice differentiable data or assuming some constraint qualifications in addition our results are obtained in the more general variational setting

relative error approximate versions of douglas rachford splitting and special cases of the admm we derive a new approximate version of the alternating direction method of multipliers admm which uses a relative error criterion the new version is somewhat restrictive and allows only one of the two subproblems to be minimized approximately but nevertheless covers commonly encountered special cases the derivation exploits the long established relationship between the admm and both the proximal point algorithm ppa and douglas rachford dr splitting for maximal monotone operators along with a relative error of the ppa due to solodov and svaiter in the course of analysis we also derive a version of dr splitting in which one operator may be evaluated approximately using a relative error criterion we computationally evaluate our method on several classes of test problems and find that it significantly outperforms several alternatives on one problem class

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

tight contact structures on some small seifert fibered 3 manifolds we classify tight contact structures on the small seifert fibered 3 manifolds m 1 r1 r2 r3 with ri 0 1 q and r1 r2 the result is obtained by combining convex surface theory with computations of contact ozsvath szabo invariants we also show that some of the tight contact structures on the manifolds considered are nonfillable justifying the use of heegaard floer theory

linear rate convergence of the alternating direction method of multipliers for convex composite programming in this paper we aim to prove the linear rate convergence of the alternating direction method of multipliers admm for solving linearly constrained convex composite optimization problems under a mild calmness condition which holds automatically for convex composite piecewise linear quadratic programming we establish the global q linear rate of convergence for a general semi proximal admm with the dual step length being taken in 0 1 51 2 2 this semi proximal admm which covers the classic one has the advantage to resolve the potentially nonsolvability issue of the subproblems in the classic admm and possesses the abilities of handling the multi block cases efficiently we demonstrate the usefulness of the obtained results when applied to two and multi block convex quadratic semidefinite programming

coupling finite element and spectral methods first results a poisson equation on a rectangular domain is solved by coupling two methods the domain is divided in two squares a finite element approximation is used on the first square and a spectral discretization is used on the second one two kinds of matching conditions on the interface are presented and compared in both cases error estimates are proved

a unified approach to error bounds for structured convex optimization problems error bounds which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems in this paper we present a new framework for establishing error bounds for a class of structured convex optimization problems in which the objective function is the sum of a smooth convex function and a general closed proper convex function such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning signal processing and statistics using our framework we show that a number of existing error bound results can be recovered in a unified and transparent manner to further demonstrate the power of our framework we apply it to a class of nuclear norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity type regularity condition we then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition we believe that our approach will find further applications in the study of error bounds for structured convex optimization problems

the augmented lagrangian method for equality and inequality constraints in hilbert spaces in this paper we consider an augmented lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a hilbert space an inequality constraint whose image space is finite dimensional and an affine inequality constraint whose image space is in an infinite dimensional hilbert space we obtain local convergence of this method without imposing strict complementarity conditions when the equality as well as the inequality constraint with finite dimensional image space are augmented to the author s knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional

an efficient hessian based algorithm for solving large scale sparse group lasso problems the sparse group lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level in this paper we develop an efficient augmented lagrangian method for large scale non overlapping sparse group lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth newton method theoretically we prove that if the penalty parameter is chosen sufficiently large the augmented lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence the dual infeasibility and the duality gap of the primal and dual objective functions computationally we derive explicitly the generalized jacobian of the proximal mapping associated with the sparse group lasso regularizer and exploit fully the underlying second order sparsity through the semismooth newton method the efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets

mixed finite element methods for unilateral problems convergence analysis and numerical studies in this paper we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics the discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant lagrange multipliers in the saddle point formulation a priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved

holomorphic disks and knot invariants abstract we define a floer homology invariant for knots in an oriented three manifold closely related to the heegaard floer homologies for three manifolds defined in an earlier paper we set up basic properties of these invariants including an euler characteristic calculation and a description of the behavior under connected sums then we establish a relationship with hf for surgeries along the knot applications include calculation of hf of three manifolds obtained by surgeries on some special knots in s 3 and also calculation of hf for certain simple three manifolds which fiber over the circle

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

book review implicit functions and solution mappings a view from variational analysis second edition by a l dontchev and r t rockafellar springer new york 2014

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

rationally convex domains and singular lagrangian surfaces in mathbb c 2 we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in mathbb c 2 we recall some classical obstructions and prove some deeper ones related to symplectic and contact topology we explain the close connection to lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces our proof also gives a complete characterization of lagrangian surfaces with open whitney umbrellas answering a question first posed by givental in 1986

holomorphic triangles and invariants for smooth four manifolds abstract in earlier articles the authors introduced invariants for closed oriented three manifolds defined using a variant of lagrangian floer homology in the symmetric products of riemann surfaces the aim of this article is to introduce invariants of oriented smooth four manifolds built using these floer homology groups this four dimensional theory also endows the corresponding three dimensional theories with additional structure an absolute grading of certain of its floer homology groups

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method

some properties of the augmented lagrangian in cone constrained optimization a large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions in this paper we discuss augmented lagrangian duality for such optimization problems we formulate the augmented lagrangian dual problems and study conditions ensuring existence of the corresponding augmented lagrange multipliers we also discuss sensitivity of optimal solutions to small perturbations of augmented lagrange multipliers

khovanov homology open books and tight contact structures abstract we define the reduced khovanov homology of an open book s and identify a distinguished contact element in this group which may be used to establish the tightness or non fillability of contact structures compatible with s our construction generalizes the relationship between the reduced khovanov homology of a link and the heegaard floer homology of its branched double cover as an application we give combinatorial proofs of tightness for several contact structures which are not stein fillable lastly we investigate a comultiplication structure on the reduced khovanov homology of an open book which parallels the comultiplication on heegaard floer homology defined in baldwin 2008 4

a note on fejer monotone sequences in product spaces and its applications to the dual convergence of augmented lagrangian methods in a recent math program paper eckstein and silva proposed a new error criterion for the approximate solutions of augmented lagrangian subproblems based on a saddle point formulation of the primal and dual problems they proved that dual sequences generated by augmented lagrangians under this error criterion are bounded and that their limit points are dual solutions in this note we prove a new result about the convergence of fejer monotone sequences in product spaces which seems to be interesting by itself and as a consequence we obtain the full convergence of the dual sequence generated by augmented lagrangians under eckstein and silva s criterion

ozsvath szabo invariants and fillability of contact structures recently ozsvath and szabo defined an invariant of contact structures with values in the heegaard floer homology groups they also proved that a version of the invariant with twisted coefficients is non trivial for weakly symplectically fillable contact structures in this article we show that their non vanishing result does not hold in general for the contact invariant with untwisted coefficients as a consequence of this fact heegaard floer theory can distinguish between weakly and strongly symplectically fillable contact structures

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac