stochastic perturbation of reduced gradient grg methods for nonconvex programming problems in this paper we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints a reduced gradient and grg generalized reduced gradient descent methods involving stochastic perturbation are proposed and we give a mathematical result establishing the convergence to a global minimizer numerical examples are given in order to show that the method is effective to calculate namely we consider classical tests such as the statistical problem the octagon problem the mixture problem and an application to the linear optimal control servomotor problem

the numerical range of products of normal matrices abstract in an earlier paper the author developed a formula for the trace class multiplier norm of a matrix of rank at most 2 in this article applications of this formula are given in the main result we suppose that f 1 f n and g 1 g n are given sets of complex numbers a description is given of the union of the numerical ranges of the product fg as f and g run over all n n normal matrices with the given sets as eigenvalues

simulating the dynamics of flexible bodies and vortex sheets we present a numerical method for the dynamics of a flexible body in an inviscid flow with a free vortex sheet the formulation is implicit with respect to body variables and explicit with respect to the free vortex sheet we apply the method to a flexible foil driven periodically in a steady stream we give numerical evidence that the method is stable and accurate for a relatively small computational cost a continuous form of the vortex sheet regularization permits continuity of the flow across the body s trailing edge nonlinear behavior arises gradually with respect to driving amplitude and is attributed to the rolling up of the vortex sheet flow quantities move across the body in traveling waves and show large gradients at the body edges we find that in the small amplitude regime the phase difference between heaving and pitching which maximizes trailing edge deflection also maximizes power output the phase difference which minimizes trailing edge deflection maximizes efficiency

a new accelerating method for globally solving a class of nonconvex programming problems abstract in this paper we combine the new global optimization method proposed by jiao h jiao a branch and bound algorithm for globally solving a class of nonconvex programming problems nonlinear anal 70 2 2008 1113 1123 with a suitable deleting technique to propose a new accelerating global optimization algorithm for solving a class of nonconvex programming problems np this technique offers a possibility to cut away a large part of the currently investigated region in which the global optimal solution of np does not exist and can be seen as an accelerating device for the global optimization algorithm of the nonconvex programming problems compared with the method in the above cited reference numerical results show that the computational efficiency is obviously improved by using this new technique in the number of iterations the required list length and the overall execution time of the algorithm

a branch and bound algorithm for globally solving a class of nonconvex programming problems abstract a branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems np for minimizing the problem linear lower bounding functions llbfs of objective function and constraint functions are constructed then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value the proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems and finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm

an extremal problem in fourier analysis with applications to operator theory a certain minimal extrapolation problem for fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators in this paper we estimate the value of the constant in the fourier transform problem by an analytic reformulation

a practicable contraction approach for the sum of the generalized polynomial ratios problem abstract in this paper a practicable contraction approach is proposed for solving the sum of the generalized polynomial ratios problem p with generalized polynomial constraints due to the intrinsic difficulty of problem p less work has been devoted to solving this problem the proposed approach is based on reducing the original nonconvex problem p as a standard geometric programming gp problem by utilizing simple transformation and contraction strategies the resulting optimization problem can be solved effectively by utilizing the solutions of a series of gp problems the tractability and effectiveness of the proposed successive contraction approach are demonstrated by several numerical examples and the performance comparison of the proposed approach and other methods published is also presented in terms of solution quality

reducing transformation and global optimization abstract in this paper we give new results on the alienor method of dimension reduction this technique is used to solve multidimensional global optimization problems of type min x x f x where f is a non convex lipschitz function and x a compact set of r n n 2 defined by lipschitz constraints the idea is to construct an dense curve h in the feasible set x the global minimum of f on x is then approximated by the global minimum of f on the curve h that is our problem has become a one dimensional problem which can be solved by the piyavskii shubert method examples of these curves and numerical implementations on several test functions are given

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

regularizing a vortex sheet near a separation point current methods for computing vortex sheet separation use a regularization parameter which is discontinuous from the body to the vortex sheet we propose two methods for reducing the errors associated with the discontinuity and improving convergence with respect to the regularization parameter the velocity smoothing method is the simpler of the two and removes the discontinuity in regularization from one of the two equations where it occurs the tapered smoothing method removes the discontinuity from both equations in a model problem both methods are found to converge much more rapidly with exponents 3 2 and 2 versus 1 2 for the standard method as the regularization parameter tends to zero unsteady algorithms are proposed for evolving the free sheet using the two methods and are tested in a benchmark problem accuracy is significantly improved for similar computational expense

global optimization for sum of linear ratios problem with coefficients the global optimization of the sum of linear ratios problem has attracted the interest of researchers and practitioners for a number of years since these types of optimization problems are nonconvex various specialized algorithms have been proposed for globally solving these problems however these algorithms are only for the case that sum of linear ratios problem without coefficients and may be difficult to implement in this paper a branch and bound algorithm is proposed for globally solving the sum of linear ratios problem with coefficients by utilizing an equivalent problem and linearization technique the initial nonconvex programming problem is reduced to a sequence of linear programming problems the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are given to show the feasibility and effectiveness of the proposed algorithm

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

a fast chebyshev method for simulating flexible wing propulsion abstract we develop a highly efficient numerical method to simulate small amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid we allow the wing s elastic modulus and mass density to vary arbitrarily with an eye towards optimizing these distributions for propulsive performance the method to determine the wing kinematics is based on chebyshev collocation of the 1d beam equation as coupled to the surrounding 2d fluid flow through small amplitude analysis of the euler equations with trailing edge vortex shedding the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1d wing kinematics a class of semi analytical solutions permits fast evaluation of this operator with o n log n operations where n is the number of collocation points on the wing this is in contrast to the minimum o n 2 cost of a direct 2d fluid solver the coupled wing fluid problem is thus recast as a pde with nonlocal operator which we solve using a preconditioned iterative method these techniques yield a solver of near optimal complexity o n log n allowing one to rapidly search the infinite dimensional parameter space of all possible material distributions and even perform optimization over this space

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

spectral variation of normal matrices abstract it has been a durable conjecture that the distance appropriately defined between the spectra of two normal matrices is bounded by the operator norm of their difference we report on some numerical studies that show that this conjecture is in general false in fact even in the 3 x 3 case the spectral distance may exceed the norm distance by between 1 and 2 percent we also describe the background of the conjecture and offer an analysis of extremal counterexamples

similarity preserving linear maps on matrices abstract we determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n hermitian matrices sub majorization preserving linear maps on hermitian matrices are also determined

a branch and bound algorithm to globally solve the sum of several linear ratios in this paper we propose a branch and bound algorithm to globally solve the sum of several linear fractional functions over a polytope for minimizing problem a linear lower bounding function llbf of the objective function is constructed then a linear programming is obtained which is solved by a simplex algorithm and provides the lower bounding of the optimal value the proposed branch and bound algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems the numerical experiment is reported to show the effectiveness and feasibility of the proposed algorithm also this method is extended to solve the maximizing problems

numerical solution of integral equations 1 a survey of boundary integral equation methods for the numerical solution of laplace s equation in three dimensions 2 superconvergence 3 perturbed projection methods for various classes of operator and integral equations 4 numerical solution of parallel processors of two point boundary value problems of astrodynamics 5 introduction to the numerical solution of cauchy singular integral equations 6 convergence theorems for singular integral equations 7 planing surfaces 8 abel integral equations

perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations abstract for selfadjoint matrices in an indefinite inner product possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation genericity is understood in the sense of algebraic geometry special attention is paid to the perturbation behavior of the sign characteristic typically under such a perturbation for every given eigenvalue the largest jordan block of the eigenvalue is destroyed and in case the eigenvalue is real all other jordan blocks keep their sign characteristic the new eigenvalues i e those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix are typically simple and in some cases information is provided about their sign characteristic if the new eigenvalue is real the main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product a version of the brunovsky canonical form and on general results concerning rank one perturbations obtained

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

optimal measures for toeplitz matrices abstract we extend and refine a result of r mceachin concerning 2 2 toeplitz matrices even in the complex case the schur norm of such a matrix is equal to the total variation of an appropriately chosen representing measure the measure may be supported on two or three points from this phenomenon we derive formulas for the schur norms of the matrices we point out a connection with the extreme points of the unit ball in a certain function space

trace class multipliers and spectral variation of normal matrices abstract in this article we show how to estimate the trace multiplier norm of a rank 2 matrix as an application an alternative proof of a theorem of holbrook et al maximal spectral distance linear algebra appl 249 1996 197 205 on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given

the spectral variation for two matrices with spectra on two intersecting lines abstract it is shown that for a pair a b of n n matrices where the eigenvalues of a lie on a straight line l 1 and the eigenvalues of b lie on a ray with initial point on l 1 the spectral distance between a and b is bounded from above by the spectral norm of their difference the more general case where the eigenvalues of b lie on a straight line intersecting l 1 is also discussed

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

an outcome space finite algorithm for solving linear multiplicative programming this paper presents an outcome space finite algorithm for solving linear multiplicative programming in each iteration of which a convex quadratic programming is only solved in the paper we give a global optimization condition on a class of multiplicative programming problems and prove that the proposed algorithm is finite terminative and gain a global optimal solution of the former problem when it stops it can be shown by the numerical results that the proposed algorithm is effective and the computational results can be gained in short time

on permutations convex hulls and normal operators abstract a spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite dimensional space this is used to prove the following if a and b are normal operators on an n dimensional complex hilbert space h with eigenvalues given by 1 n and 1 n respectively and if a b is also normal then a b max s n diag k k for any unitarily invariant norm on l h

norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils abstract we start by proving a lower bound for the lp operator norm of a submatrix with sufficiently large dimensions and some relations between the lp norm and the lp operator norm we then study the lp norm and the lp operator norm of certain matrices arising in estimating the eigenvalue generalized eigenvalue singular value and generalized singular value variations of matrices matrix pencils and matrix pairs finally as applications of our bounds upon the lp norm and lp operator norm of certain matrices we give several new bounds for the variations of the spectra of matrices and matrix pencils these new bounds are generalizations of the weyl lidskii theorem for hermitian matrices and the bhatia davis theorem for unitary matrices and of some results obtained by the author recently for diagonalizable pencils with real spectra

global optimization of generalized linear fractional programming with nonlinear constraints this paper presents a branch and bound algorithm for globally solving a wide class of generalized linear fractional programming problems glfp this class includes such problems as minimizing a sum or error for product of a finite number of ratios of linear functions linear multiplicative programming polynomial programming etc over nonconvex feasible region first a problem q is derived which is equivalent to problem glfp in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and constraint functions of problem q over the feasible region convergent property of the presented algorithm is proved and numerical results are given to show the feasibility of the proposed algorithm

vortex design problem in this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity the problem is formulated as a pde constrained optimization in which the solutions are found using a gradient based descent method recognizing such euler flows as free boundary problems the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients in explicit tracking of interfaces vortex boundaries this method offers an alternative to grid based techniques such as the level set methods and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique we develop and validate this approach using the design of 2d equilibrium euler flows with finite area vortices as a model problem it is also discussed how the proposed methodology can be applied to euler flows featuring other vorticity distributions such as vortex sheets and to time dependent phenomena

hypothesis testing on invariant subspaces of the symmetric group part i quantum sanov s theorem and arbitrarily varying sources we report a proof of the quantum sanov theorem by the elementary application of basic facts about representations of the symmetric group together with a complete characterization of the optimal error exponent in a situation where the null hypothesis is given by an arbitrarily varying quantum source instead our approach differs from previous ones in two ways first it supports a reasoning inspired by the method of types second the measurement scheme we propose to distinguish the two alternatives not only does that job asymptotically perfectly but also yields additional information about the null hypothesis an example of that is given the measurement is composed of projections onto permutation invariant subspaces thus providing a direct link between one of the most basic tasks in quantum information on the one hand and fundamental objects in representation theory on the other we additionally connect to the representation theory by proving a relation between kostka numbers and quantum states and to state estimation via a generalization of a well known spectral estimation theorem to non i i d sequences

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

global optimization of signomial geometric programming using linear relaxation various local optimization approaches have been developed for solving signomial geometric programming sgp problems but up to now less work has been devoted to solving global optimization of sgp due to the inherent difficulty this paper considers the global minimum of sgp that arise in various practice problems by utilizing an exponential variable transformation and tangential hypersurfaces and convex envelop approximations a linear relaxation of sgp is then obtained thus initial nonconvex nonlinear problem sgp is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region of the objective function the proposed algorithm is convergent to the global minimum of sgp by means of the subsequent solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve global minimum of sgp on a microcomputer

geometry and the norms of hadamard multipliers abstract the norm of a matrix b as a hadamard multiplier is the norm of the map x x b where is the hadamard or entrywise product of matrices we give a geometric interpretation to a criterion of haagerup and use this interpretation to find an explicit formula for the hadamard multiplier norms of real 2 2 matrices for hermitian matrices with one positive eigenvalue we give necessary and sufficient conditions for the norm to be the magnitude of the largest entry

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2