stability in the presence of degeneracy and error estimation given an approximation to a local minimizer to a nonlinear optimization problem and to associated multipliers we obtain a tight error estimate in terms of the violation of the first order conditions our results apply to degenerate optimization problems where independence of the active constraint gradients and strict complementarity can be violated

a bundle filter method for nonsmooth convex constrained optimization for solving nonsmooth convex constrained optimization problems we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points the resulting algorithm inherits some attractive features from both approaches on the one hand it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods on the other hand the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods some encouraging preliminary computational results are also reported

a framework of constraint preserving update schemes for optimization on stiefel manifold this paper considers optimization problems on the stiefel manifold x mathsf t x i p xtx ip where x in mathbb r n times p x rn p is the variable and i p ip is the p p by p p identity matrix a framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of x x and the null space of x mathsf t xt while this general framework can unify many existing schemes a new update scheme with low complexity cost is also discovered then we study a feasible barzilai borwein like method under the new update scheme the global convergence of the method is established with an adaptive nonmonotone line search the numerical tests on the nearest low rank correlation matrix problem the kohn sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method in particular the new method performs remarkably well for the nearest low rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used scf iteration for the kohn sham total energy minimization

line search filter methods for nonlinear programming local convergence a line search method is proposed for nonlinear programming using fletcher and leyffer s filter method which replaces the traditional merit function a simple modification of the method proposed in a companion paper siam j optim 16 2005 pp 1 31 introducing second order correction steps is presented it is shown that the proposed method does not suffer from the maratos effect so that fast local convergence to second order sufficient local solutions is achieved

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

a filter active set algorithm for ball sphere constrained optimization problem in this paper we propose a filter active set algorithm for the minimization problem over a product of multiple ball sphere constraints by making effective use of the special structure of the ball sphere constraints a new limited memory bfgs l bfgs scheme is presented the new l bfgs implementation takes advantage of the sparse structure of the jacobian of the constraints and generates curvature information of the minimization problem at each iteration only two or three reduced linear systems are required to solve for the search direction the filter technique combined with the backtracking line search strategy ensures the global convergence and the local superlinear convergence can also be established under mild conditions the algorithm is applied to two specific applications the nearest correlation matrix with factor structure and the maximal correlation problem our numerical experiments indicate that the proposed algorithm is competitive with some recently custom designed methods for each ind

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

sequential systems of linear equations method for general constrained optimization without strict complementarity in this paper based on an l1 l hybrid exact penalty function we proposed an infeasible ssle method for general constrained optimization an automatic adjustment rule is incorporated in the algorithm for the choice of the penalty parameter which ensures that the penalty parameter be updated only finitely many times we also extend the facchinei fischer kanzow active set identification technique to general constrained optimization and a corresponding identification function is given at each iteration only two or three reduced linear equations with the same coefficients are solved to obtain the search direction under the linear independence condition the sequence generated by the new algorithm globally converges to a kkt point in particular the convergence rate is proved to be one step superlinear without assuming the strict complementarity and under a condition weaker than the strong second order sufficiency condition some preliminary numerical results indicate that the new algorithm is quite promising

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

on the regularized determinant for non invertible elliptic operators we propose a technique for regularizing the determinant of a non invertible elliptic operator restricted to the complement of its nilpotent elements we apply this approach to the study of chiral changes in the fermionic path integral variables

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

on the breuil mezard conjecture we give a new local proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of qp when p 5 and the representation has scalar endomorphisms

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

a global convergence analysis of an algorithm for large scale nonlinear optimization problems in this paper we give a global convergence analysis of a basic version of an sqp algorithm described in p t boggs a j kearsley and j w tolle siam j optim 9 1999 pp 755 778 for the solution of large scale nonlinear inequality constrained optimization problems several procedures and options have been added to the basic algorithm to improve the practical performance some of these are also analyzed the important features of the algorithm include the use of a constrained merit function to assess the progress of the iterates and a sequence of approximate merit functions that are less expensive to evaluate it also employs an interior point quadratic programming solver that can be terminated early to produce a truncated step

on the global convergence of a filter sqp algorithm a mechanism for proving global convergence in sqp filter methods for nonlinear programming nlp is described such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the main point of interest is to demonstrate how convergence for nlp can be induced without forcing sufficient descent in a penalty type merit function the proof relates to a prototypical algorithm within which is allowed a range of specific algorithm choices associated with the hessian matrix representation updating the trust region radius and feasibility restoration

a feasible sequential linear equation method for inequality constrained optimization in this paper by means of the concept of the working set which is an estimate of the active set we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems at each iteration of the proposed algorithm we first solve one system of linear equations with a coefficient matrix of size m m where m is the number of constraints to compute the working set we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix unlike existing qp free algorithms the subproblem is concerned with only the constraints corresponding to the working set the constraints not in the working set are neglected consequently the dimension of each subproblem is not of full dimension without assuming the isolatedness of the stationary points we prove that every accumulation point of the sequence generated by the proposed algorithm is a kkt point of the problem moreover after finitely many iterations the working set becomes independent of the iterates and is essentially the same as the active set of the kkt point in other words after finitely many steps only those constraints which are active at the solution will be involved in the subproblem under some additional conditions we show that the convergence rate is two step superlinear or even q superlinear we also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems

global convergence of filter methods for nonlinear programming we present a general filter algorithm that allows a great deal of freedom in the step computation each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter from the current point we prove its global convergence assuming that the step must be efficient in the sense that near a feasible nonstationary point the reduction of the objective function is large we show that this condition is reasonable by presenting two classical ways of performing the step which satisfy it in the first one the step is obtained by the inexact restoration method of martinez and pilotta in the second the step is computed by sequential quadratic programming

lattices in the cohomology of u 3 arithmetic manifolds under hypotheses required for the taylor wiles method we prove for forms of u 3 which are compact at infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given by the lambda isotypic part of completed cohomology is a local invariant of the galois representation attached to lambda when this galois representation is residually irreducible locally at places dividing p as a crucial input we establish corresponding mod p multiplicity one results our main innovation is the combination of integral hecke theory and the taylor wiles method

a sequential quadratic optimization algorithm with rapid infeasibility detection we present a sequential quadratic optimization sqo algorithm for nonlinear constrained optimization the method attains all of the strong global and fast local convergence guarantees of classical sqo methods but has the important additional feature that fast local convergence is guaranteed when the algorithm is employed to solve infeasible instances a two phase strategy carefully constructed parameter updates and a line search are employed to promote such convergence the first phase subproblem determines the reduction that can be obtained in a local model of an infeasibility measure when the objective function is ignored the second phase subproblem then seeks to minimize a local model of the objective while ensuring that the resulting search direction attains a reduction in the local model of the infeasibility measure that is proportional to that attained in the first phase the subproblem formulations and parameter updates ensure that near an optimal solution the algorithm reduces to a classica

constrained best euclidean distance embedding on a sphere a matrix optimization approach the problem of data representation on a sphere of unknown radius arises from various disciplines such as statistics spatial data representation psychology constrained multidimensional scaling and computer science machine learning and pattern recognition the best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some euclidean distance constraints it is those spherical and euclidean distance constraints that present an enormous challenge to the existing algorithms in this paper we reformulate the problem as an euclidean distance matrix optimization problem with a low rank constraint we then propose an iterative algorithm that uses a quadratically convergent newton cg method at each step we study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized jacobian that ensure the quadratic convergence of the newton method we use some classic examples from the spherical multidimensional scaling to demonstrat

construction of some families of 2 dimensional crystalline representations we construct explicitly some analytic families of etale modules which give rise to analytic families of 2 dimensional crystalline representations as an application of our constructions we verify some conjectures of breuil on the reduction modulo p of those representations and extend some results of deligne edixhoven fontaine and serre on the representations arising from modular forms

reductions of galois representations of slope 1 abstract we compute the reductions of irreducible crystalline two dimensional representations of g q p of slope 1 for primes p 5 and all weights we describe the semisimplification of the reductions completely in particular we show that the reduction is often reducible we also investigate whether the extension obtained is peu or tres ramifiee in the relevant reducible non semisimple cases the proof uses the compatibility between the p adic and mod p local langlands correspondences and involves a detailed study of the reductions of both the standard and non standard lattices in certain p adic banach spaces

functional determinants for ordinary differential operators the authors study the analytic properties of regularized functional determinants for ordinary differential operators to this end a new regularization scheme is introduced relations between different approaches are analysed and the connection between the presence of divergent traces and regularization differences is discussed

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

a superlinearly convergent strongly sub feasible ssle type algorithm with working set for nonlinearly constrained optimization in this paper by means of a new efficient identification technique of active constraints and the method of strongly sub feasible direction we propose a new sequential system of linear equations ssle algorithm for solving inequality constrained optimization problems in which the initial point is arbitrary at each iteration we first yield the working set by a pivoting operation and a generalized projection then three or four reduced linear equations with a same coefficient are solved to obtain the search direction after a finite number of iterations the algorithm can produced a feasible iteration point and it becomes the method of feasible directions moreover after finitely many iterations the working set becomes independent of the iterates and is essentially the same as the active set of the kkt point under some mild conditions the proposed algorithm is proved to be globally strongly and superlinearly convergent finally some preliminary numerical experiments are reported to show that the algorithm is practicable and effective

spectral density of the correlation matrix of factor models a random matrix theory approach we studied the eigenvalue spectral density of the correlation matrix of factor models of multivariate time series by making use of the random matrix theory we analytically quantified the effect of statistical uncertainty on the spectral density due to the finiteness of the sample we considered a broad range of models ranging from one factor models to hierarchical multifactor models

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

generalization of sturm liouville theory to a system of ordinary differential equations with dirac type spectrum the sturm liouville theory is generalized to dirac equation like systems of ordinary differential equations it is shown how the comparison theorem and conversion to integral equations can be generalized

functional determinants for general sturm liouville problems simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest we extend the range of situations for which these hold to cover systems of self adjoint operators of the sturm liouville type with arbitrary linear boundary conditions the results hold whether or not the operators have negative eigenvalues the physically important case of functional determinants of operators with a zero mode but where that mode has been extracted is studied in detail for the same range of situations as when no zero mode exists the method of proof uses the properties of generalized zeta functions the general form of the final results is the same for the entire range of problems considered

black box galois representations abstract we develop methods to study 2 dimensional 2 adic galois representations of the absolute galois group of a number field k unramified outside a known finite set of primes s of k which are presented as black box representations where we only have access to the characteristic polynomials of frobenius automorphisms at a finite set of primes using suitable finite test sets of primes depending only on k and s we show how to determine the determinant det whether or not is residually reducible and further information about the size of the isogeny graph of whose vertices are homothety classes of stable lattices the methods are illustrated with examples for k q and for k imaginary quadratic being the representation attached to a bianchi modular form these results form part of the first author s thesis 2

on a multivariate eigenvalue problem part i algebraic theory and a power method multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated by using the method of lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem for over 30 years an iterative method proposed by horst psychometrika 26 1961 pp 129 149j has been used for solving the multivariate eigenvalue problem yet the theory of convergence has never been complete the number of solutions to the multivariate eigenvalue problem also remains unknown this paper contains two new results by using the degree theory a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved a convergence property of horst s method by forming it as a generalization of the so called power method is then proved the discussion leads to new formulations of nume

a qp free globally convergent locally superlinearly convergent algorithm for inequality constrained optimization an algorithm is proposed for the minimization of a smooth function subject to smooth inequality constraints unlike sequential quadratic programming type methods this algorithm does not involve the solution of quadratic programs but merely that of linear systems of equations locally the iteration can be viewed as a perturbation of a quasi newton iteration on both the primal and dual variables for the solution of the equalities in the kuhn tucker first order conditions of optimality it is observed that provided the current iterate is feasible and the current multiplier estimates are strictly positive the primal component of the quasi newton direction is a direction of descent for the objective function this fact is used to induce global convergence without the need of a surrogate merit function a careful bending of the search direction prevents any maratos like effect and local superlinear convergence is proven while the algorithm requires that an initial feasible point be available the succe

a feasible active set qp free method for nonlinear programming we propose a monotone descent active set qp free method for inequality constrained optimization that ensures the feasibility of all iterates and allows for iterates on the boundary of the feasible set the study is motivated by the facchinei fischer kanzow active set identification technique for nonlinear programming and variational inequalities f facchinei a fischer and c kanzow siam j optim 9 1999 pp 14 32 distinguishing features of the proposed method compared with existing qp free methods include lower subproblem costs and a fast convergence rate under milder assumptions specifically four reduced linear systems with a common coefficient matrix involving only constraints in a working set are solved at each iteration to determine the working set the method makes use of multipliers from the last iteration eliminating the need to compute a new estimate and no additional linear systems are solved to select linearly independent constraint gradients a new technique is presented to avoid possible ill conditioned newton systems caused by dual degeneracy it is shown that the method converges globally to kkt points under the linear independence constraint qualification licq and the asymptotic rate of convergence is q superlinear under an additional strong second order sufficient condition ssosc without strict complementarity