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

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

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

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

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

two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

sequence folding lattice tiling and multidimensional coding folding a sequence b into a multidimensional box is a well known method which is used as a multidimensional coding technique the operation of folding is generalized in a way that the sequence b can be folded into various shapes and not just a box the novel definition of folding is based on a lattice tiling for the given shape s and a direction in the d dimensional integer grid necessary and sufficient conditions that a lattice tiling for s combined with a direction define a folding of a sequence into s are derived the immediate and most impressive applications are some new lower bounds on the number of dots in two dimensional synchronization patterns asymptotically optimal such patterns were known only for rectangular shapes we show asymptotically optimal such patterns for a large family of hexagons this is also generalized for multidimensional synchronization patterns the best known patterns in terms of dots for circles and other polygons are also given the technique and its application for two dimensional synchronization patterns raises some interesting problems in discrete geometry we will also discuss these problems it is also shown how folding can be used to construct multidimensional error correcting codes finally by using the new definition of folding new types of multidimensional pseudo random arrays with various shapes are generated

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

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

on graphs with distance laplacian spectral radius of multiplicity n 3 abstract let 1 l 2 l n l be the distance laplacian eigenvalues of a connected graph g and m i l the multiplicity of i l it is well known that the graphs with m 1 l n 1 are complete graphs recently the graphs with m 1 l n 2 have been characterised by celso et al in this paper we completely determine the graphs with m 1 l n 3

simple and robust binary self location patterns a simple method to generate a 2 d binary grid pattern which allows for absolute and accurate self location in a finite planar region is proposed the pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly

proof for four conjectures about the distance laplacian and distance signless laplacian eigenvalues of a graph abstract the distance laplacian matrix l g of a graph g is defined to be l g diag tr d g where d g denotes the distance matrix of g and diag tr denotes the diagonal matrix of the vertex transmissions in g similarly the distance signless laplacian matrix of g is defined as q g diag tr d g the eigenvalues of l g and q g are called the distance laplacian and distance signless laplacian eigenvalues respectively in this paper four conjectures proposed by m aouchche and p hansen about the largest and the second largest distance laplacian eigenvalues and the second largest distance signless laplacian eigenvalue of a graph are proved

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

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

constructions for perfect maps and pseudorandom arrays a construction of perfect maps i e periodic r v binary arrays in which each n m binary matrix appears exactly once is given a similar construction leads to arrays in which only the zero n m matrix does not appear and to a construction in which only a few n m binary matrices do not appear a generalization to the nonbinary case is given the constructions involve an interesting problem in shift register theory the solution is given for almost all the case of this problem

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

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

on the truncated conjugate gradient method in this paper we consider the truncated conjugate gradient method for minimizing a convex quadratic function subject to a ball trust region constraint it is shown that the reduction in the objective function by the solution obtained by the truncated cg method is at least half of the reduction by the global minimizer in the trust region

algorithms for the generation of full length shift register sequences two algorithms are presented for the generation of full length shift register cycles also referred to as de bruijn sequences the first algorithm generates 2 k cdot g n k full cycles of length 2 n using 3n k cdot g n k bits of storage where k is a free parameter in the range 1 leq k leq 2 n 4 2 and g n k is of the order of n 2 log k the second algorithm generates about 2 n 2 4 full cycles of length 2 n using about n 2 2 bits of storage in both algorithms the time required to produce the next bit from the last n bits is close to n a possible application to the construction of stream ciphers is indicated

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

error correction of multidimensional bursts we present several methods and constructions to generate binary codes for correction of a multidimensional cluster error whose shape can be a box error a lee sphere error or an error with an arbitrary shape our codes have very low redundancy close to optimal and a large range of parameters of arrays and clusters our main results are summarized as follows 1 a construction of two dimensional codes capable to correct a rectangular error with considerably more flexible parameters from previously known constructions this construction is easily generalized for d dimensions 2 a novel method based on d colorings of the d dimensional space for constructing d dimensional codes correcting a d dimensional cluster error of various shapes 3 a transformation of the d dimensional space into another d dimensional space in a way that a d dimensional lee sphere is transformed into a shape located in a d dimensional box of a relatively small size 4 applying the coloring method to correct more efficiently a two dimensional error whose shape is a lee sphere 5 a construction of d dimensional codes capable to correct a d dimensional cluster error of size b in which the number of erroneous positions is relatively small compared to b 6 we present a code which corrects a d dimensional arbitrary cluster error with relatively small redundancy

on the distance laplacian spectra of graphs abstract the distance laplacian matrix of a connected graph g is defined in 2 3 and it is proved that for a graph g on n vertices if the complement of g is connected then the second smallest distance laplacian eigenvalue is strictly greater than n in this article we consider the graphs whose complement is a tree or a unicyclic graph and characterize the graphs among them having n 1 as the second smallest distance laplacian eigenvalue we prove that the largest distance laplacian eigenvalue of a path is simple and the corresponding eigenvector has the similar property like that of a fiedler vector

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

quickest sequence phase detection a phase detection sequence is a length n cyclic sequence such that the location of any length k contiguous subsequence can be determined from a noisy observation of that subsequence in this paper we derive bounds on the minimal possible k in the limit of n to infty and describe some sequence constructions we further consider multiple phase detection sequences where the location of any length k contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences we study the optimal trade offs between the lengths of the sequences and describe some sequence constructions we compare these phase detection problems to their natural channel coding counterparts and show a strict separation between the fundamental limits in the multiple sequence case both adversarial and probabilistic noise models are addressed

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

vandermonde matrices with chebyshev nodes abstract for an n n vandermonde matrix v n j i 1 1 ij n with translated chebyshev zero nodes it is discovered that v n t admits an explicit qr decomposition with the r factor consisting of the coefficients of the translated chebyshev polynomials this decomposition then leads to an exact expression for the frobenius condition number of its submatrix v k n j i 1 1 i k 1 j n so called a rectangular vandermonde matrix bounds on individual singular value and more it is explained how these results can be used to establish asymptotically optimal lower bounds on condition numbers of real rectangular vandermonde matrices and nearly optimally conditioned real rectangular vandermonde matrices on a given interval extensions are also made for v n with nodes being zeros of any translated orthogonal polynomials other than chebyshev ones similar results hold for v n with translated chebyshev extreme nodes too owing to that v n t admits an explicit qr like decomposition close formulas of or tight bounds on the residuals are also presented for the conjugate gradient method the minimal residual method and the generalized minimal residual method on certain linear systems ax b with a having eigenvalues the same as the nodes mentioned above as a by product they yield positive definite linear systems for which the residuals by the conjugate gradient method are always comparable to the existing error bounds for all iteration steps

on meinardus examples for the conjugate gradient method the conjugate gradient cg method is widely used to solve a positive definite linear system ax b of order n it is well known that the relative residual of the kth approximate solution by cg with the initial approximation x 0 0 is bounded above by 2 k k k k 1 with k k 1 k 1 where k k a a 2 a 1 2 is a s spectral condition number in 1963 meinardus numer math 5 1963 pp 14 23 gave an example to achieve this bound for k n 1 but without saying anything about all other 1 k n 1 this very example can be used to show that the bound is sharp for any given k by constructing examples to attain the bound but such examples depend on k and for them the k 1 th residual is exactly zero therefore it would be interesting to know if there is any example on which the cg relative residuals are comparable to the bound for all 1 k n 1 there are two contributions in this paper 1 a closed formula for the cg residuals for all 1 k n 1 on meinardus example is obtained and in particular it implies that the bound is always within a factor of 2 of the actual residuals 2 a complete characterization of extreme positive linear systems for which the kth cg residual achieves the bound is also presented

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

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

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

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

an interior affine scaling projective algorithm for nonlinear equality and linear inequality constrained optimization in this paper we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints the general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints the horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint by introducing the fletcher s penalty function as the merit function trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems the global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions a nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases

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

a method for constructing decodable de bruijn sequences we present two related methods of construction for de bruijn 1946 sequences both based on interleaving smaller de bruijn sequences sequences obtained using these construction methods have the advantage that they can be decoded very efficiently i e the position within the sequence of any particular window can be found very simply sequences with simple decoding algorithms are of considerable practical importance in position location applications

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

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

convergence to second order stationary points in inequality constrained optimization we propose a new algorithm for the nonlinear inequality constrained minimization problem and prove that it generates a sequence converging to points satisfying the kkt second order necessary conditions for optimality the algorithm is a line search algorithm using directions of negative curvature and it can be viewed as a nontrivial extension of corresponding known techniques from unconstrained to constrained problems the main tools employed in the definition and in the analysis of the algorithm are a differentiable exact penalty function and results from the theory of lc1 functions

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

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

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

on piecewise quadratic newton and trust region problems some recent algorithms for nonsmooth optimization require solutions to certain piecewise quadratic programming subproblems two types of subproblems are considered in this paper the first type seeks the minimization of a continuously differentiable and strictly convex piecewise quadratic function subject to linear equality constraints we prove that a nonsmooth version of newton s method is globally and finitely convergent in this case the second type involves the minimization of a possibly nonconvex and nondifferentiable piecewise quadratic function over a euclidean ball characterizations of the global minimizer are studied under various conditions the results extend a classical result on the trust region problem

graphs with some distance laplacian eigenvalue of multiplicity n 3 abstract the eigenvalues of the distance laplacian matrix of a connected graph g are called the distance laplacian eigenvalues of g denote by g n n 3 the set of all connected graphs on n vertices with some distance laplacian eigenvalue of multiplicity n 3 in this paper all graphs belonging to g n n 3 are characterized completely which solves the remaining problems in fernandes et al 2018 11

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

an interior point method for solving box constrained underdetermined nonlinear systems a method introduced recently by bellavia macconi and morini for solving square nonlinear systems with bounds is modified and extended to cope with the underdetermined case the algorithm produces a sequence of interior iterates and is based on globalization techniques due to coleman and li global and local convergence results are proved and numerical experiments are presented

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

relation between the matching number and the second largest distance laplacian eigenvalue of a graph abstract let g be a connected simple graph with matching number m g the second largest distance laplacian eigenvalue of g is denoted by 2 g in this paper we investigate the relation between the matching number and the second largest distance laplacian eigenvalue of g establishing the lower bounds of 2 g in terms of m g moreover all the extremal graphs attaining the lower bounds are completely characterized