a newton method for convex regression data smoothing and quadratic programming with bounded constraints this paper formulates systems of piecewise linear equations derived from the karush kuhn tucker conditions for constrained convex optimization problems as unconstrained minimization problems in which the objective function is a multivariate quadratic spline such formulations provide new ways of developing efficient algorithms for many optimization problems such as the convex regression problem the least distance problem the symmetric monotone linear complementarily problem and the convex quadratic programming problem with bounded constraints theoretical results a description of an algorithm and its implementation and numerical results are presented along with a stability analysis

robust inversion of seismic data using the huber norm the huber function or huber norm is one of several robust error measures which interpolates between smooth l2 treatment of small residuals and robust l1 treatment of large residuals since the huber function is differentiable it may be minimized reliably with a standard gradient based optimizer we propose to minimize the huber function with a quasi newton method that has the potential of being faster and more robust than conjugate gradient methods when solving nonlinear problems tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the huber function gives far more robust model estimates than does a least squares fit with or without damping

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

the sharp lipschitz constants for feasible and optimal solutions of a perturbed linear program abstract the purpose of this paper is to derive the sharp lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right hand side perturbations the lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

on reduced convex qp formulations of monotone lcps techniques for transforming convex quadratic programs qps into monotone linear complementarity problems lcps and vice versa are well known we describe a class of lcps for which a reduced qp formulation one that has fewer constraints than the standard qp formulation is available we mention several instances of this class including the known case in which the coefficient matrix in the lcp is symmetric

hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion we study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms ut f u x e xb xu x2 xu 2l 1 any l 1 the flux function f u being mth order growth at infinity it is shown that if e e tend to 0 then the sequence ue of the smooth solutions converges to the unique entropy solution u l 0 t lq r to the conservation law ut f u x 0 in lk 0 t lp r k p q the proof relies on the methods of compensated compactness young measures and entropy measure valued solutions some new a priori estimates are carried out in particular our result includes the convergence result by schonbek when b l 1 and lefloch and natalini when l 1

a conjugate gradient method for the unconstrained minimization of strictly convex quadratic splines in this paper we show that an analogue of the classical conjugate gradient method converges linearly when applied to solving the problem of unconstrained minimization of a strictly convex quadratic spline since a strictly convex quadratic program with simple bound constraints can be reformulated as unconstrained minimization of a strictly convex quadratic spline the conjugate gradient method is used to solve the unconstrained reformulation and find the solution of the original quadratic program in particular if the solution of the original quadratic program is nondegenerate then the conjugate gradient method finds the solution in a finite number of iterations

a cooperative recurrent neural network for solving l1 estimation problems with general linear constraints the constrained l1 estimation is an attractive alternative to both the unconstrained l1 estimation and the least square estimation in this letter we propose a cooperative recurrent neural network crnn for solving l1 estimation problems with general linear constraints the proposed crnn model combines four individual neural network models automatically and is suitable for parallel implementation as a special case the proposed crnn includes two existing neural networks for solving unconstrained and constrained l1 estimation problems respectively unlike existing neural networks with penalty parameters for solving the constrained l1 estimation problem the proposed crnn is guaranteed to converge globally to the exact optimal solution without any additional condition compared with conventional numerical algorithms the proposed crnn has a low computational complexity and can deal with the l1 estimation problem with degeneracy several applied examples show that the proposed crnn can obtain more accurate estimates than several existing algorithms

the inverse scattering problem at fixed energy for the three dimensional schrodinger equation with an exponentially decreasing potential the relations between the faddeev functions and the functions of classical scattering theory are found in the complex domain at fixed energy for the three dimensional case without assumption of smallness of the potential it is proved that the exponentially decreasing potential is uniquely determined by its scattering amplitude at fixed energy

discrete time neural network for fast solving large linear l_ 1 estimation problems and its application to image restoration there is growing interest in solving linear l1 estimation problems for sparsity of the solution and robustness against non gaussian noise this paper proposes a discrete time neural network which can calculate large linear l1 estimation problems fast the proposed neural network has a fixed computational step length and is proved to be globally convergent to an optimal solution then the proposed neural network is efficiently applied to image restoration numerical results show that the proposed neural network is not only efficient in solving degenerate problems resulting from the nonunique solutions of the linear l1 estimation problems but also needs much less computational time than the related algorithms in solving both linear l1 estimation and image restoration problems

a generalized least absolute deviation method for parameter estimation of autoregressive signals this paper proposes a generalized least absolute deviation glad method for parameter estimation of autoregressive ar signals under non gaussian noise environments the proposed glad method can improve the accuracy of the estimation of the conventional least absolute deviation lad method by minimizing a new cost function with parameter variables and noise error variables compared with second and high order statistical methods the proposed glad method can obtain robustly an optimal ar parameter estimation without requiring the measurement noise to be gaussian moreover the proposed glad method can be implemented by a cooperative neural network nn which is shown to converge globally to the optimal ar parameter estimation within a finite time simulation results show that the proposed glad method can obtain more accurate estimates than several well known estimation methods in the presence of different noise distributions

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

an inverse scattering problem with part of the fixed energy phase shifts assume that q r is a real valued compactly supported potential q r 0 for let be an arbitrary fixed subset of non negative integers such that and be fixed energy phase shifts corresponding to q r the main result is

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

l sub 1 and l sub infinity minimization via a variant of karmarkar s algorithm simple iterative algorithms are presented for l sub 1 and l sub infinity minimization regression based on a variant of karmarkar s linear programming algorithm although these algorithms are based on entirely different theoretical principles to the popular irls iteratively reweighted least squares algorithm they have almost identical matrix operations also presented are the results of a monte carlo study comparing the numerical convergence properties of the karmarkar algorithm for l sub 1 minimization to those of an irls and a simplex algorithm the test problem involves l sub 1 estimation of ar autoregressive model parameters the karmarkar algorithm outperformed irls by achieving higher numerical accuracy in fewer iterations techniques for reducing the computational cost per iteration of the karmarkar l sub 1 algorithm are discussed

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

interior point methods for massive support vector machines we investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low rank update to a positive semidefinite matrix several formulations of the support vector machine fit into this category an interesting feature of these particular problems is the volume of data which can lead to quadratic programs with between 10 and 100 million variables and if written explicitly a dense q matrix our code is based on ooqp an object oriented interior point code with the linear algebra specialized for the support vector machine application for the targeted massive problems all of the data is stored out of core and we overlap computation and input output to reduce overhead results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable

extended least correlation estimates for errors in variables non linear models this paper introduces a method of parameter estimation working on errors in variables polynomial non linear models in which all measurements are corrupted by noise the first step is to develop the linear regression models which are equivalent to polynomial non linear systems a main idea is to extend the parameter vector by even order components of noise and to augment the regression vector by appropriate constants or measurements applying the method of least correlation which has a capability to cope with errors in variables linear models to the equivalent model with extended parameters and augmented regressors yields an extended least correlation estimator analysis shows that for non linear systems with third or lower order polynomials the parameters estimated by the proposed method asymptotically converge to the true values numerical examples also support analytical results applications of the approach to volterra models hammerstein models and weiner non linear systems are included

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

robust estimation when more than one variable per equation of condition has error summary it is well known that in the errors in variables estimation problem the results will have unnecessary asymptotic bias unless the algorithm is properly formulated similar difficulties can be expected with robust estimation techniques that are based on extending least squares to a noneuclidean metric this paper presents an algorithm for robust estimation for the nonlinear model errors in variables case the properties of the estimates produced by the algorithm are investigated and a numerical example using data on galaxies is given

parameter identification of quasibrittle materials as a mathematical program with equilibrium constraints the identification of the cohesive fracture parameters of quasibrittle materials forms the focus of the present work this important and challenging task is formulated as an inverse problem requiring the solution of a special type of constrained optimization problem known as a mathematical program with equilibrium constraints mpec a key feature of which is the presence of complementarity conditions involving the orthogonality of two nonnegative vectors we propose a simple scheme based on transforming the mpec into a standard nonlinear programming problem through appropriate smoothing of the complementarity constraints for solving the parameter identification problem some computational results using actual experimentally obtained data are presented to assess the effectiveness of the approach

robust line estimation with errors in both variables abstract the estimator holding the central place in the theory of the multivariate errors in the variables ev model results from performing orthogonal regression on variables rescaled according to the covariance matrix of the errors our simulations on the univariate model essentially relegate this estimator to use only in large samples on data with no trace of outlier contamination a modification requiring a robust preliminary slope is proposed that essentially sets out the generalization to ev of the robust w estimator in regression it is demonstrated that the modification is robust to outlier contamination even in small samples given a sufficiently good preliminary estimator candidates for a preliminary slope estimator based on the data are discussed and the performance of one under simulation is examined

formula for the radius of the support of the potential in terms of scattering data let q r r x be a real valued square integrable compactly supported function and be the smallest interval containing the support of q r let be the corresponding scattering amplitude at a fixed positive energy let be the phase shifts at k 1 it is proved that provided that q r does not change sign in some arbitrary small neighbourhood of a

inverse scattering with fixed energy and an inverse eigenvalue problem on the half line recently a g ramm 1999 has shown that a subset of phase shifts 1 l 0 1 determines the potential if the indices of the known shifts satisfy the miintz condition l 0 l l 1 l we prove the necessity of this condition in some classes of potentials the problem is reduced to an inverse eigenvalue problem for the half line schrodinger operators

a finite smoothing algorithm for linear l_1 estimation in this paper a new method for solving the linear l 1 problem is described analysed and tested the method is based on smoothing the nondifferentiable l 1 function the smoothing can be done in a well onditioned manner since the method has finite convergence extensive numerical tests demonstrate significant superiority to existing simplex type codes furthermore the tests show that the new algorithm is very well suited for vector processing

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

measurement error model collinearities collinearities can occur among the error free true predictors in measurement error models just as they occur among predictors in traditional regression models the coefficient estimators for measurement error models suffer the same ill effects of collinearities as do least squares estimators for linear measurement error models collinearities are a property of the secondorder moment matrix of the unobservable error free predictors for nonlinear models they are a property of the second order moment matrix of derivatives of the nonlinear function in this article collinearities among the predictors in measurement error models are defined and diagnostics for their detection are discussed the collinearity diagnostics introduced are similar to those used in traditional regression models but they are applied to estimated second order moment matrices of the error free predictors rather than to the second order moment matrix of the observed predictors examples are discussed one of which demonstrates th

partial identification of the potential from phase shifts we consider the three dimensional inverse scattering with fixed energy in the spherically symmetrical case we give a characterization of the sequences of phase shifts for two potentials which can be different only in a ball of radius a in other words we study how the large distance interaction influences the asymptotical behavior of the phase shifts we also characterize the tail of the potential by the growth order of the scattering amplitude f t for large t

robust estimation in the errors in variables model summary orthogonal regression analogues of m estimates of regression called here orthogonal regression m estimates are defined these estimates are shown to be consistent at elliptical errors in variables models and robust if the corresponding loss function is bounded the orthogonal regression analogues of regression scale estimates called here orthogonal regression s estimates are considered as well in particular they provide a robust estimate for the scale of the orthogonal residuals a crucial quantity in the computation of orthogonal regression m estimates finally we present an algorithm for computing orthogonal regression s and m estimates and the results of a small monte carlo experiment

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

fitting data with errors in all variables using the huber m estimator this article is concerned with the problem of data fitting where the model is nonlinear in the free parameters using the huber m estimator under the assumption that there are significant errors in all the variables an efficient algorithm is developed some numerical examples are given

noise suppression by removing singularities a method is proposed for suppressing gaussian noise by extracting local singularities the method is based on truncating riemann series decomposition whose components naturally characterize different orders of holder regularity the approach yields a single step filtering technique whose performance is comparable to three step wavelet decomposition and thresholding techniques

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm