on the generalized lanczos trust region method the so called trust region subproblem gets its name in the trust region method in optimization and also plays a vital role in various other applications several numerical algorithms have been proposed in the literature for solving small to medium size dense problems as well as for large scale sparse problems the generalized lanczos trust region gltr method proposed by n i m gould s lucidi m roma and p l toint siam j optim 9 1999 pp 561 580 is a natural extension of the classical lanczos method for the linear system to the trust region subproblem in this paper we first analyze the convergence of gltr to reveal its convergence behavior in theory and then propose new stopping criteria that can be integrated into gltr for better numerical performance specifically we develop a priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution and argue that these bounds can be efficiently estimated numerically and serve as stopping criteria f

an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

energy spectra of the schrodinger equation and the differential quadrature method improvement of the solution using particle swarm optimization in our earlier work we have developed the differential quadratic method dq to find the physical structures of the schrodinger equation in which the interpolating points of tchebychev have been used in this paper the particle swarm optimization pso for short has been suggested as a means to improve qualitatively the solutions this approach is carefully handled and tested with some numerical examples

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

limited memory bfgs systems with diagonal updates abstract in this paper we investigate a formula to solve systems of the form b i x y where b is a limited memory bfgs quasi newton matrix and is a positive constant these types of systems arise naturally in large scale optimization such as trust region methods as well as doubly augmented lagrangian methods we show that provided a simple condition holds on b 0 and the system b i x y can be solved via a recursion formula that requires only vector inner products this formula has complexity m 2 n where m is the number of l bfgs updates and n m is the dimension of x

oracle based robust optimization via online learning robust optimization is a common optimization framework under uncertainty when problem parameters are unknown but it is known that they belong to some given uncertainty set in the robust optimization framework a min max problem is solved wherein a solution is evaluated according to its performance on the worst possible realization of the parameters in many cases a straightforward solution to a robust optimization problem of a certain type requires solving an optimization problem of a more complicated type which might be np hard in some cases for example solving a robust conic quadratic program such as those arising in a robust support vector machine svm with an ellipsoidal uncertainty set leads in general to a semidefinite program in this paper we develop a method for approximately solving a robust optimization problem using tools from online convex optimization where at every stage a standard nonrobust optimization program is solved our algorithms find an approximate robust solution usin

newton type methods for non convex optimization under inexact hessian information we consider variants of trust region and adaptive cubic regularization methods for non convex optimization in which the hessian matrix is approximated under certain condition on the inexact hessian and using approximate solution of the corresponding sub problems we provide iteration complexity to achieve varepsilon approximate second order optimality which have been shown to be tight our hessian approximation condition offers a range of advantages as compared with the prior works and allows for direct construction of the approximate hessian with a priori guarantees through various techniques including randomized sampling methods in this light we consider the canonical problem of finite sum minimization provide appropriate uniform and non uniform sub sampling strategies to construct such hessian approximations and obtain optimal iteration complexity for the corresponding sub sampled trust region and adaptive cubic regularization methods

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

a subspace minimization method for the trust region step we consider methods for large scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two norm trust region constraint the steihaug toint method uses the conjugate gradient method to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary however if the conjugate gradient method is used with a preconditioner the steihaug toint method requires that the trust region norm be defined in terms of the preconditioning matrix if a different preconditioner is used for each subproblem the shape of the trust region can change substantially from one subproblem to the next which invalidates many of the assumptions on which standard methods for adjusting the trust region radius are based in this paper we propose a method that allows the trust region norm to be defined independently of the preconditioner the method solves the inequality constrained trust region subproblem over a sequence of evolving low dimensional subspaces each subspace includes an accelerator direction defined by a regularized newton method for satisfying the optimality conditions of a primal dual interior method a crucial property of this direction is that it can be computed by applying the preconditioned conjugate gradient method to a positive definite system in both the primal and dual variables of the trust region subproblem numerical experiments on problems from the cuter test collection indicate that the method can require significantly fewer function evaluations than other methods in addition experiments with general purpose preconditioners show that it is possible to significantly reduce the number of matrix vector products relative to those required without preconditioning

the exact solution of two new types of schrodinger equation a compact solution of a special biconfluent heun equation is obtained by means of a laplace transform the eigenvalues and eigenfunctions of two new types of schrodinger equation are deduced by appropriate variable transformations the associated potential function in each case can be arranged to have a double well

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

a note on computing the smallest conic singular value abstract the goal of this note is to study the smallest conic singular value of a matrix from a lagrangian duality viewpoint and provide an efficient method for its computation

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

analysis of a quadratic programming decomposition algorithm we analyze a decomposition algorithm for minimizing a quadratic objective function separable in mathbf x 1 and mathbf x 2 subject to the constraint that mathbf x 1 and mathbf x 2 are orthogonal vectors on the unit sphere our algorithm consists of a local step where we minimize the objective function in either variable separately while enforcing the constraints followed by a global step where we minimize over a subspace generated by solutions to the local subproblems we establish a local convergence result when the global minimizers are nondegenerate our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint the analysis is connected with a new domain decomposition algorithm for electronic structure calculations

quasi newton trust region algorithm for non smooth least squares problems1this work was supported by cnpq of brazil and the national natural science foundation of china 1 in this paper we present a quasi newton trust region algorithm for non smooth least squares problems which possess the smooth plus non smooth decomposition this method uses a smooth subproblem with second order information to approximate the locally lipschitzian function which is important to increase the convergence rate of the algorithm and deal with the large residual non smooth least squares problems the global and superlinear convergence of this algorithm are established

a normal form theorem around symplectic leaves we prove the poisson geometric version of the local reeb stability from foliation theory and of the slice theorem from equivariant geometry which is also a generalization of conn s linearization theorem

separating multipaths by global optimization of a multidimensional matched filter a transmitted signal can arrive at a receiver via several refracted fermat paths if the paths are independent in the fresnel sense then the received signal can be modeled as the sum of amplitude scaled and time shifted copies of a predetermined replica plus white noise we present an algorithm that uses the replica to determine the time shifts and amplitudes for each path it is referred to as an n dimensional matched filter algorithm by analogy with the well known matched filter algorithm the cross correlation between the received signal and the replica oscillates near the center frequency of the transmitted signal this causes the n dimensional matched filter output to have many local maxima that are not globally optimal the time shifts and amplitude scalings for the fermat paths are determined by maximizing the output of the n dimensional matched filter the algorithm is more robust and efficient than others currently available simulated realizations of received signals were generated with multipath and noise characteristics similar to an ocean acoustic transmission case these realizations were then separated into arrival times and corresponding amplitudes by the algorithm the results of these tests and the general limitations of the algorithm are discussed

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

nonholonomic mechanics and connections over a bundle map a general notion of connections over a vector bundle map is considered and applied to the study of mechanical systems with linear nonholonomic constraints and a lagrangian of kinetic energy type in particular it is shown that the description of the dynamics of such a system in terms of the geodesics of an appropriate connection can be easily recovered within the framework of connections over a vector bundle map also the reduction theory of these systems in the presence of symmetry is discussed from this perspective

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

an efficient chebyshev lanczos method for obtaining eigensolutions of the schrodinger equation on a grid a grid method for obtaining eigensolutions of bound systems is presented in this the block lanczos method is applied to a cheby shev approximation of exp h where is the range of eigenvalues we are interested in with this choice a preferential convergence of the eigenvectors corresponding to low lying eigenvalues ofhis achieved the method is used to solve a variety of one two and three dimensional problems to apply the kinetic energy operator we use the fast sine transform instead of the fast fourier transform thus fullfilling a priori the box boundary conditions we further extend the chebyshev approximation to treat general functions of matrices thus allowing its application to cases for which no analytical expressions of the expansion coefficients are available

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

the generalized trust region subproblem solution complexity and convex hull results we consider the generalized trust region subproblem gtrs of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint a lifting of this problem recasts the gtrs as minimizing a linear objective subject to two nonconvex quadratic constraints our first main contribution is structural we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil this result may be of interest in building relaxations for nonconvex quadratic programs moreover this result allows us to reformulate the gtrs as the minimization of two convex quadratic functions in the original space our next set of contributions is algorithmic we present an algorithm for solving the gtrs up to an epsilon additive error based on this reformulation we carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms notably our algorithms run in linear in the size of the input time furthermore our algorithm for computing an epsilon optimal solution has a slightly improved running time dependence on epsilon over the state of the art algorithm our analysis shows that the dominant cost in solving the gtrs lies in solving a generalized eigenvalue problem establishing a natural connection between these problems finally generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality interval and hollow constrained variants of the gtrs this gives the first linear time algorithm in the literature for these variants of the gtrs

on the complexity of steepest descent newton s and regularized newton s methods for nonconvex unconstrained optimization problems it is shown that the steepest descent and newton s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to o epsilon 2 to drive the norm of the gradient below epsilon this shows that the upper bound of o epsilon 2 evaluations known for the steepest descent is tight and that newton s method may be as slow as the steepest descent method in the worst case the improved evaluation complexity bound of o epsilon 3 2 evaluations known for cubically regularized newton s methods is also shown to be tight

connections in poisson geometry i holonomy and invariants we discuss contravariant connections on poisson manifolds for vector bun dles the corresponding operational notion of a contravariant derivative had been introduced by i vaisman we show that these connections play an important role in the study of global properties of poisson manifolds and we use them to define poisson holonomy and new invariants of poisson manifolds

numerically complemented analytic method for solving the time independent one dimensional schrodinger equation a general method of solving the one dimensional schrodinger equation is developed the first step is to construct an exactly solvable reference potential of several smoothly joined morse type components which should be a good approximation to a given potential the exact solutions for that reference hamiltonian are then combined with a nonperturbative approach r g gordon j chem phys 51 14 1969 which enables us to numerically solve the energy eigenvalue problem for the original potential to any desired accuracy a full description of the analytical procedures is given and examples of both exact and numerical solutions are presented

variational analysis of an extended eigenvalue problem abstract for a symmetric matrix b rn n and a vector a rn the maximal extended eigenvalue a max x r n s t b i x a x t x 1 is known to arise in optimality conditions for the mathematical programming problemp a given by max x t b x 2 a t x x t x 1 as well as in extended rayleigh ritz type results pertaining to the one parameter family of nonsymmetric border perturbations ofb given by a t a b a t a t nonsmooth analysis is employed in order to describe the function with special emphasis on its sensitivity near the origin further connections with p a are drawn

line integral alternating minimization algorithm for dual energy x ray ct image reconstruction we propose a new algorithm called line integral alternating minimization liam for dual energy x ray ct image reconstruction instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates liam allows for a tunable discrepancy between the basis material projections and the basis sinograms a parameter is introduced that controls the size of this discrepancy and with this parameter the new algorithm can continuously go from a two step approach to the joint estimation approach liam alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm an edge preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by liam while varying the regularization parameters to identify good choices the results from the dual energy alternating minimization algorithm applied to the same data were used for comparison using a small fraction of the computation time of dual energy alternating minimization liam achieves better accuracy of the component images in the presence of poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

minimizing a quadratic over a sphere a new method the sequential subspace method ssm is developed for the problem of minimizing a quadratic over a sphere in our scheme the quadratic is minimized over a subspace which is adjusted in successive iterations to ensure convergence to an optimum when a sequential quadratic programming iterate is included in the subspace convergence is locally quadratic numerical comparisons with other recent methods are given

complete dictionary recovery over the sphere ii recovery by riemannian trust region method we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y boldsymbol a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint and hence is naturally phrased in the language of manifold optimization in a companion paper we have showed that with high probability our nonconvex formulation has no spurious local minimizers and around any saddle point the objective function has a negative directional curvature in this paper we take advantage of the particular geometric structure and describe a riemannian trust region algorithm that provably converges to a local minimizer with from arbitrary initializations such minimizers give excellent approximations to the rows of x 0 the rows are then recovered by a linear programming rounding and deflation

iterative methods for nearly singular linear systems iterative methods are developed and studied for near singular linear systems cx b our approach called the transformed minimal residual algorithm tmres is derived from any convergent iterative scheme sxk 1 txk b associated with a splitting c s t in each step of tmres the transformed residual s 1 b cx is minimized over a krylov space generated by s 1 t the original iterative scheme typically converges slowly when c is nearly singular while a krylov space generated by s 1 t often contains a much better approximation to a solution tmres is algebraically equivalent to the generalized minimal residual algorithm gmres preconditioned by s 1 although there are numerical differences since a different matrix s 1 c is used to generate the krylov space in preconditioned gmres special attention is given to sparsity and convergence issues related to linear systems of the form bf aa tr sigma bf i bf x bf b where sigma ge 0

a trust region approach to the regularization of large scale discrete forms of ill posed problems we consider large scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill posed problem and the right hand side contains noise special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution we pose the regularization problem as a quadratically constrained least squares problem this formulation is equivalent to tikhonov regularization and we note that it is also a special case of the trust region subproblem from optimization we analyze the trust region subproblem in the regularization case and we consider the nontrivial extensions of a recently developed method for general large scale subproblems that will allow us to handle this case the method relies on matrix vector products only has low and fixed storage requirements and can handle the singularities arising in ill posed problems we present numerical results on test problems on an inverse interpolation problem with field data and on a model seismic inversion problem with field data

the berwald type linearization of generalized connections we study the existence of a natural linearization process for generalized connections on an affine bundle it is shown that this leads to an affine generalized connection over a prolonged bundle which is the analogue of what is called a connection of berwald type in the standard theory of connections various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalized connections on such bundles

a derivative free algorithm for least squares minimization we develop a framework for a class of derivative free algorithms for the least squares minimization problem these algorithms are designed to take advantage of the problem structure by building polynomial interpolation models for each function in the least squares minimization under suitable conditions global convergence of the algorithm is established within a trust region framework promising numerical results indicate the algorithm is both efficient and robust numerical comparisons are made with standard derivative free software packages that do not exploit the special structure of the least squares problem or that use finite differences to approximate the gradients

the momentum map in poisson geometry every action on a poisson manifold by poisson diffeomorphisms lifts to a hamiltonian action on its symplectic groupoid which has a canonically defined momentum map we study various properties of this momentum map as well as its use in reduction

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

on the spherical quasi convexity of quadratic functions abstract in this paper the spherical quasi convexity of quadratic functions on spherically convex sets is studied several conditions characterizing the spherical quasi convexity of quadratic functions are presented in particular conditions implying spherical quasi convexity of quadratic functions on the spherical positive orthant are given some examples are provided as applications of the obtained results

note on generalized connections and affine bundles we develop an alternative view on the concept of connections over a vector bundle map which consists of a horizontal lift procedure to a prolonged bundle we further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalized connection

solving the trust region subproblem using the lanczos method the approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods when the number of variables is large the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust region boundary in this paper we investigate ways of continuing the process once the boundary has been encountered the key is to observe that the trust region problem within the currently generated krylov subspace has a very special structure which enables it to be solved very efficiently we compare the new strategy with existing methods the resulting software package is available as hsl vf05 within the harwell subroutine library

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

hidden convexity in some nonconvex quadratically constrained quadratic programming we consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints we then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints which is also shown to be equivalent to a minimax convex problem in both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts special cases and applications are also discussed we outline interior point polynomial time algorithms for the solution of the equivalent convex programs

global convergence of ssm for minimizing a quadratic over a sphere in an earlier paper minimizing a quadratic over a sphere siam j optim 12 2001 188 208 we presented the sequential subspace method ssm for minimizing a quadratic over a sphere this method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed we showed in this earlier paper that when the subspace contains a vector obtained by applying one step of newton s method to the first order optimality system ssm is locally quadratically convergent even when the original problem is degenerate with multiple solutions and with a singular jacobian in the optimality system in this paper we prove nonlocal convergence of ssm to a global minimizer whenever each ssm subspace contains the following three vectors i the current iterate ii the gradient of the cost function evaluated at the current iterate and iii an eigenvector associated with the smallest eigenvalue of the cost function hessian for nondegenerate problems the convergence rate is at least linear when vectors i iii are included in the ssm subspace

an affine scaling reduced preconditional conjugate gradient path method for linear constrained optimization abstract this paper presents an affine scaling reduced preconditional conjugate gradient path approach in association with nonmonotonic interior backtracking line search technique for the linear constrained optimization employing the affine scaling preconditional conjugate gradient to form the curvilinear path and using interior backtracking line search technique each iterate switches to trial step of strict interior feasibility the nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature theoretical analysis are given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions

computing correlated equilibria in multi player games we develop polynomial time algorithms for finding correlated equilibria a well studied notion of rationality that generalizes the nash equilibrium in a broad class of succinctly representable multiplayer games encompassing graphical games anonymous games polymatrix games congestion games scheduling games local effect games as well as several generalizations our algorithm is based on a variant of the existence proof due to hart and schmeidler and employs linear programming duality the ellipsoid algorithm markov chain steady state computations as well as application specific methods for computing multivariate expectations over product distributions for anonymous games and graphical games of bounded tree width we provide a different polynomial time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game in contrast to our sweeping positive results for computing an arbitrary correlated equilibrium we prove that optimizing over correlated equilibria is np hard in all of the other classes of games that we consider

integrability of poisson brackets we discuss the integration of poisson brackets motivated by our recent solution to the integrability problem for general lie brackets we give the precise obstructions to integrating poisson manifolds describing the integration as a symplectic quotient in the spirit of the poisson sigma model of cattaneo and felder for regular poisson manifolds we express the obstructions in terms of variations of symplectic areas improving on results of alcalde cuesta and hector we apply our results and our point of view to decide about the existence of complete symplectic realizations to the integrability of submanifolds of poisson manifolds and to the study of dual pairs morita equivalence and reduction

global convergence of trust region interior point algorithms for infinite dimensional nonconvex minimization subject to pointwise bounds a class of interior point trust region algorithms for infinite dimensional nonlinear optimization subject to pointwise bounds in l p banach spaces 2 le p le infty is formulated and analyzed the problem formulation is motivated by optimal control problems with l p controls and pointwise control constraints the interior point trust region algorithms are generalizations of those recently introduced by coleman and li siam j optim 6 1996 pp 418 445 for finite dimensional problems many of the generalizations derived in this paper are also important in the finite dimensional context all first and second order global convergence results known for trust region methods in the finite dimensional setting are extended to the infinite dimensional framework of this paper