quadratic optimization with orthogonality constraint explicit łojasiewicz exponent and linear convergence of retraction based line search and stochastic variance reduced gradient methods the problem of optimizing a quadratic form over an orthogonality constraint qp oc for short is one of the most fundamental matrix optimization problems and arises in many applications in this paper we characterize the growth behavior of the objective function around the critical points of the qp oc problem and demonstrate how such characterization can be used to obtain strong convergence rate results for iterative methods that exploit the manifold structure of the orthogonality constraint i e the stiefel manifold to find a critical point of the problem specifically our primary contribution is to show that the ojasiewicz exponent at any critical point of the qp oc problem is 1 2 such a result is significant as it expands the currently very limited repertoire of optimization problems for which the ojasiewicz exponent is explicitly known moreover it allows us to show in a unified manner and for the first time that a large family of retraction based line search methods will converge linearly to a critical point of the qp oc problem then as our secondary contribution we propose a stochastic variance reduced gradient svrg method called stiefel svrg for solving the qp oc problem and present a novel ojasiewicz inequality based linear convergence analysis of the method an important feature of stiefel svrg is that it allows for general retractions and does not require the computation of any vector transport on the stiefel manifold as such it is computationally more advantageous than other recently proposed svrg type algorithms for manifold optimization

gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

new algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds there are many applications related to singly linearly constrained quadratic programs subjected to upper and lower bounds in this paper a new algorithm based on secant approximation is provided for the case in which the hessian matrix is diagonal and positive definite to deal with the general case where the hessian is not diagonal a new efficient projected gradient algorithm is proposed the basic features of the projected gradient algorithm are 1 a new formula is used for the stepsize 2 a recently established adaptive non monotone line search is incorporated and 3 the optimal stepsize is determined by quadratic interpolation if the non monotone line search criterion fails to be satisfied numerical experiments on large scale random test problems and some medium scale quadratic programs arising in the training of support vector machines demonstrate the usefulness of these algorithms

extrema of sums of heterogeneous quadratic forms abstract we analyze the following problem arising in various situations in multivariate statistical analysis we are given k symmetric positive definite n n matrices a1 a2 ak k n and we would like to maximize the function ki 1 xtiaixi under the constraint that x1 x2 xk rn form an orthonormal system some theoretical results as well as an algorithm are presented

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

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

dynamic programming and maximum principle for discrete goursat systems abstract we obtain the dynamic programming equations for discrete goursat systems we prove that they are necessary and sufficient for optimality and we prove formulas for the exact or approximate solution of these equations using dynamic programming arguments we obtain semi dynamic programming equations and then we prove a maximum principle of pontryagin s type

minimal subspace rotation on the stiefel manifold for stabilization and enhancement of projection based reduced order models for the compressible navier stokes equations for a projection based reduced order model rom of a fluid flow to be stable and accurate the dynamics of the truncated subspace must be taken into account this paper proposes an approach for stabilizing and enhancing projection based fluid roms in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace attention is focused on the full non linear compressible navier stokes equations in specific volume form as a step toward a more general formulation for problems with generic non linearities unlike traditional approaches no empirical turbulence modeling terms are required and consistency between the rom and the navier stokes equation from which the rom is derived is maintained mathematically the approach is formulated as a trace minimization problem on the stiefel manifold the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems including a problem involving laminar flow over an airfoil with a high angle of attack and a channel driven cavity flow problem

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

inequalities for m matrices and inverse m matrices in this paper we establish some determinantal inequalities concerning m matrices and inverse m matrices the main results are as follows 1 if a aij is either an n n m matrix or inverse m matrix then for any permutation i1 i2 in of 1 2 n a deta i 1naii s 2n1 ai1i2 ais 1isaisi1 ai1i1ai2i2 aisis b deta i 1naii if and only if a is essentially triangular 2 if a aij is an n n m matrix b bij is an n n inverse m matrix a b denotes the hadamard product of a and b then a b is an m matrix and for any permutation i1 i2 in of 1 2 n det a b det ab s 2naisisdeta i1 i2 is 1 deta i1 is 1 is bisisdetb i1 i2 is 1 detb i1 is 1 is 1

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

solutions to two functional equations arising in dynamic programming this paper deals with the existence uniqueness and iterative approximation of solutions for two functional equations arising in dynamic programming of multistage decision processes the results presented in this paper extend improve and unify the results due to bellman bhakta and choudhury bhakta and mitra liu liu and ume and others

existence and uniqueness of solutions of functional equations arising in dynamic programming abstract in this paper we study solvability of two functional equations arising in dynamic programming of multistage decision processes by using boyd and wong fixed point theorem some existence and uniqueness theorems of solutions and iterative approximation for solving these class of functional equations are established the results presented here extend improve and unify the corresponding results due to bellman bhakta and mitra bhakta and choudhary liu and kang liu et al jiang et al and others we also discuss some illustrative examples to highlight the realized improvements

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

some determinantal inequalities for hadamard product of matrices abstract the main result of this paper is the following if both a aij and b bij are m matrices or positive definite real symmetric matrices of order n the hadamard product of a and b is denoted by a b and ak and bk k 1 2 n are the k k leading principal submatrices of a and b respectively then det a b det ab k 2 n a kk det a k 1 det a k b kk det b k 1 det b k 1

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

a gradient algorithm for stereo matching without correspondence in this paper we study the problem of feature point matching via a technique well known in geometry leading to a new gradient algorithm in machine vision the procedure does not require one to solve explicit correspondences between feature points but relies on a specific form of spatial averaging in this sense the estimation procedure is robust in this paper we describe the algorithm and report simulation results

scaled toda like flows abstract this paper discusses the class of isospectral flows x x a x where denotes the hadamard product and is the lie bracket the presence of a allows arbitrary and independent scaling for each element in the matrix x the time 1 mapping of the scaled toda like flow still enjoys a qr like iteration the scaled structure includes the classical toda flow brockett s double bracket flow and other interesting flows as special cases convergence proof is thus unified and simplified the effect of scaling on a variety of applications is demonstrated by examples

algebraic aspects of brockett type equations abstract it is well known that soliton equations can be written in lax form where the bracket of two linear differential n kdv or pseudo differential kp operators appears in this work we introduce a new hierarchy each equation of which is defined by a double bracket of two pseudo differential operators these double bracket equations arose originally in the study by brockett of the steepest descent equations corresponding to certain least squares matching and sorting problems we deal with some algebraic properties of these equations in particular we show that as in the classical case they are related to the presence of an infinite sequence of first integrals

super brockett equations a graded gradient integrable system rather recently equations of lax type defined by a double commutator the so called brockett equations have received considerable attention in this paper we prove that a supersymmetric version of a brockett hierarchy is an infinite dimensional integrable gradient system as far as we know this is the only graded system of this type existing in the literature

existence theorems of solutions for certain classes of functional equations arising in dynamic programming in this paper we study the existence uniqueness and iterative approximation of solutions for certain classes of functional equations arising in dynamic programming of multistage decision processes the results presented here generalize improve and unify the corresponding results of other authors

model indexing and object recognition using 3d viewpoint invariance object recognition and location of 3d objects from range data are described based on invariant representations of characteristic views of single objects depth data is processed to remove outliers then smoothed and segmented into surface patches fitted by polynomial equations mathematical invariants are derived from groups of these surface patches and used for indexing a model database stored in the form of a kd tree the pose of the known object may then be determined from the euclidean centre and orientation of the higher order surfaces formed from the feature groups the approach is complete scalable and extendable the robustness of the method is evaluated in terms of the stability and discriminatory power of the representation and the accuracy of the determined pose

an oppenheim type inequality for a block hadamard product abstract we prove an oppenheim type determinantal inequality for a block hadamard product of two block commuting positive semidefinite matrices this solves a conjecture of gunther and klotz 2012 in 2

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

total linear least squares and the algebraic riccati equation abstract it is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation which turns into an algebraic riccati equation when the matrix of unknowns is square if there is an additional symmetry constraint on the solution the optimal solution is given by the anti stabilizing solution of this riccati equation

grain rotations and distortions in the asymptotic variance of vacancy of the boolean model abstract we consider the asymptotic variance of vacancy avv in the high intensity small grain boolean model subjecting the grains to rotations or more generally linear distortions gives rise to a function which maps distortion distributions to the avv of the corresponding boolean model we mainly study continuity properties of this function where we use the l 1 wasserstein metric on distortion distributions an important role in the formulation and derivation of our results is played by notions of symmetry commonly used in multivariate analysis and stochastic simulation such as conjugation invariance and group models

robust algorithms for object localization model based localization the task of estimating an object s pose from sensed and corresponding model features is a fundamental task in machine vision exact constant time localization algorithms have been developed for the case where the sensed features and the model features are the same type still it is not uncommon for the sensed features and the model features to be of different types i e sensed data points may correspond to model faces or edges previous localization approaches have handled different model and sensed features of different types via sampling and synthesizing virtual features to reduce the problem of matching features of dissimilar types to the problem of matching features of similar types unfortunately these approaches may be suboptimal because they introduce artificial errors other localization approaches have reformulated object localization as a nonlinear least squares problem where the error is between the sensed data and model features in image coordinates the euclidean image error metric unfortunately all of the previous approaches which minimized the euclidean image error metric relied on gradient descent methods to find the global minima and gradient descent methods may suffer from problems of local minima in this paper we describe an exact efficient solution to the nonlinear least squares minimization problem based upon resultants linear algebra and numerical techniques on a sparc 20 our localization algorithm runs in a few microseconds for rectilinear polygonal models a few milliseconds for generic polygonal models and one second for generalized polygonal models models composed of linear edges and circular arcs

on minimization on stiefel manifolds abstract the minimization of a smooth function f r kn r under the constraint that vectors x 1 x 2 x k r n k n form an orthonormal system seems to be a new and interesting global optimization problem with important theoretical and practical applications the set of feasible points determines a differentiable manifold introduced by stiefel in 1935 based on the nice geometric structure the optimality conditions are obtained by the global lagrange multiplier rule and global optimality conditions based on local information which make the advantages of using the riemannian geometry in difficult smooth optimization problems clear

bayesian segmentation via asymptotic partition functions asymptotic approximations to the partition function of gaussian random fields are derived textures are characterized via gaussian random fields induced by stochastic difference equations determined by finitely supported stationary linear difference operators adjusted to be nonstationary at the boundaries it is shown that as the scale of the underlying shape increases the log normalizer converges to the integral of the log spectrum of the operator inducing the random field fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator induced random field model matrix analysis techniques are proposed for handling textures with variable orientation examples of texture parameters estimated from training data via asymptotic maximum likelihood are shown isotropic models involving powers of the laplacian and directional models involving partial derivative mixtures are explored parameters are estimated for mitochondria and actin myocin complexes in electron micrographs and clutter in forward looking infrared images deformable template models are used to infer the shape of mitochondria in electron micrographs with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

a dynamical systems approach to weighted graph matching graph matching is a fundamental problem that arises frequently in the areas of distributed control computer vision and facility allocation in this paper we consider the optimal graph matching problem for weighted graphs which is computationally challenging due the combinatorial nature of the set of permutations contrary to optimization based relaxations to this problem in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices in particular since permutation matrices are orthogonal matrices with nonnegative elements we define two gradient flows in the space of orthogonal matrices the first minimizes the cost of weighted graph matching over orthogonal matrices whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations the combination of the two dynamical systems converges to a permutation matrix which provides a suboptimal solution to the weighted graph matching problem finally our approach is shown to be promising by illustrating it on nontrivial problems

a control theoretic approach to the design of zero finding numerical methods in this paper it is shown how standard iterative methods for solving linear and nonlinear equations can be designed from the point of view of control appropriate choices of control lyapunov functions clfs lead to both continuous and discrete time versions of the newton raphson and conjugate gradient algorithms as well as new variants

orthogonalization of vectors with minimal adjustment two transformations are proposed that give orthogonal components with a one to one correspondence between the original vectors and the components the aim is that each component should be close to the vector with which it is paired orthogonality imposing a constraint the transformations lead to a variety of new statistical methods including a unified approach to the identification and diagnosis of collinearities a method of setting prior weights for bayesian model averaging and a means of calculating an upper bound for a multivariate chebychev inequality one transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors and is determined using a new algorithm that also provides the decomposition of a positive definite matrix in terms of a diagonal matrix and a correlation matrix the algorithm is shown to converge to a global optimum

nonmonotone algorithm for minimization on closed sets with applications to minimization on stiefel manifolds a nonmonotone levenberg marquardt based algorithm is proposed for minimization problems on closed domains by preserving the feasible set s geometry throughout the process the method generates a feasible sequence converging to a stationary point independently of the initial guess as an application a specific algorithm is derived for minimization on stiefel manifolds and numerical results involving a weighted orthogonal procrustes problem are reported

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

adaptive multilevel inexact sqp methods for pde constrained optimization we present a class of inexact adaptive multilevel trust region sqp methods for the efficient solution of optimization problems governed by nonlinear pdes the algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1 implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2 implementable accuracy requirements for iterative solvers of the linearized pde and adjoint pde on the current grid we prove global convergence to a stationary point of the infinite dimensional problem moreover we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations numerical results are presented

a procrustes problem on the stiefel manifold an orthogonal procrustes problem on the stiefel manifold is studied where a matrix q with orthonormal columns is to be found that minimizes aq b rm f for an l times m matrix a and an l times n matrix b with l geq m and m n based on the normal and secular equations and the properties of the stiefel manifold necessary conditions for a global minimum as well as necessary and sufficient conditions for a local minimum are derived

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter