linear programming for the 0 1 quadratic knapsack problem abstract in this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we propose a new method for computing an upper bound this method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0 1 variables but nonredundant in continuous variables the obtained upper bound is better than the bounds given by other known methods we also propose an algorithm for computing a good feasible solution this algorithm is an elaboration of the heuristic methods proposed by chaillou hansen and mahieu and by gallo hammer and simeone the relative error between this feasible solution and the optimum solution is generally less than 1 we show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum finally we propose a branch and bound algorithm for solving the quadratic knapsack problem and report extensive computational tests

type ii hidden symmetries through weak symmetries for nonlinear partial differential equations abstract the type ii hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a lie point symmetry in b abraham shrauner k s govinder provenance of type ii hidden symmetries from nonlinear partial differential equations j nonlinear math phys 13 2006 612 622 abraham shrauner and govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries the lie point symmetries of a model equation and the two dimensional burgers equation and their descendants were used to identify the hidden symmetries in this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries we have considered the same models presented in b abraham shrauner k s govinder provenance of type ii hidden symmetries from nonlinear partial differential equations j nonlinear math phys 13 2006 612 622 as well as the wdvv equations of associativity in two dimensional topological field theory which reduces in the case of three fields to a single third order equation of monge ampere type we have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using lie symmetries however when is reduced by using nonclassical symmetries the reduced partial differential equation gains lie symmetries

lie subalgebras reduction of order and group invariant solutions one of the great utilities of lie symmetries of differential equations is in their use to reduce the order of ordinary differential equations and partial differential equations to ordinary differential equations this process is guided by the lie algebra of the symmetries admitted by the equation under study it is important in these reductions to use appropriate symmetries this will ensure that a sufficient number remain after the initial reduction to allow subsequent reductions by considering two and three dimensional subalgebras of the lie algebra of the symmetries of differential equations we indicate the fate of these symmetries under various reductions in doing so it is clear as to which symmetries need be prioritized in reduction

hidden and contact symmetries of ordinary differential equations the lie algebra for the maximal contact symmetries of third order ordinary differential equations odes is examined for type i and ii hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries ones invariant under the group associated with the ten dimensional maximal lie algebra may produce type i hidden symmetries for two parameter subgroups and type ii hidden symmetries for certain solvable non abelian three parameter subgroups in the third order odes when they are reduced in order a new class of type ii hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths two examples of odes invariant under subgroups of the ten parameter group under which y 0 is invariant demonstrate the new class of type ii hidden symmetries

parametric convex quadratic relaxation of the quadratic knapsack problem abstract we consider a parametric convex quadratic programming cqp relaxation for the quadratic knapsack problem qkp this relaxation maintains partial quadratic information from the original qkp by perturbing the objective function to obtain a concave quadratic term the nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to matrix space we present a primal dual interior point method to optimize the perturbation of the quadratic function in a search for the tightest upper bound for the qkp we prove that the same perturbation approach when applied in the context of semidefinite programming sdp relaxations of the qkp cannot improve the upper bound given by the corresponding linear sdp relaxation the result also applies to more general integer quadratic problems finally we propose new valid inequalities on the lifted matrix variable derived from cover and knapsack inequalities for the qkp and present separation problems to generate cuts for the current solution of the cqp relaxation our best bounds are obtained alternating between optimizing the parametric quadratic relaxation over the perturbation and applying cutting planes generated by the valid inequalities proposed

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

lagrangean methods for the 0 1 quadratic knapsack problem abstract it is well known that the lagrangean decomposition provides better bounds than the lagrangean relaxation does nevertheless the lagrangean decomposition bound is harder to compute than the lagrangean relaxation bound thus one might wonder what is the best lagrangean method to use in a branch and bound algorithm in this paper we give an answer to such a question for the 0 1 quadratic knapsack problem we first study the lagrangean decomposition for this problem and give new necessary optimality conditions for the dual problem which allow us to elaborate a heuristic method for solving the lagrangean decomposition dual problem we then introduce this method in chaillou hansen mahieu s branch and bound algorithm where upper bounds were computed by lagrangean relaxation

variational restoration of nonflat image features models and algorithms we develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features nonflat image features are those that live on riemannian manifolds instead of on the euclidean spaces familiar examples include the orientation feature from optical flows or gradient flows that lives on the unit circle s1 the alignment feature from fingerprint waves or certain texture images that lives on the real projective line mathbb rp 1 and the chromaticity feature from color images that lives on the unit sphere s2 in this paper we apply the variational method to denoise and restore general nonflat image features mathematical models for both continuous image domains and discrete domains or graphs are constructed riemannian objects such as metric distance and levi civita connection play important roles in the models computational algorithms are also developed for the resulting nonlinear equations the mathematical framework can be applied to res

master partial differential equations for a type ii hidden symmetry an approach for determining a class of master partial differential equations from which type ii hidden point symmetries are inherited is presented as an example a model nonlinear partial differential equation pde reduced to a target pde by a lie symmetry gains a lie point symmetry that is not inherited hidden from the original pde on the other hand this type ii hidden symmetry is inherited from one or more of the class of master pdes the class of master pdes is determined by the hidden symmetry reverse method the reverse method is extended to determine symmetries of the master pdes that are not inherited we indicate why such methods are necessary to determine the genesis of type ii symmetries of pdes as opposed to those that arise in ordinary differential equations odes

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

type ii hidden symmetries of the linear 2d and 3d wave equations type ii hidden symmetries of the linear two dimensional and three dimensional wave equations are analysed these hidden symmetries are lie point symmetries that appear in addition to the inherited point symmetries when the number of independent and dependent variables of a partial differential equation is reduced by a lie point symmetry the provenance of these hidden symmetries of partial differential equations is identified to be the same as found recently for some nonlinear partial differential equations the appearance of type ii hidden symmetries depends not only on the lie symmetries used but on the order in which the symmetries are applied the presence of type ii hidden symmetries of partial differential equations complicates the prediction of symmetry reductions based on the lie algebra associated with the original lie point symmetries

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation we consider a class of inhomogeneous nonlinear diffusion equations inde that arise in solute transport theory hidden nonlocal symmetries that seem not to be recorded in the literature are systematically determined by considering an integrated equation obtained using the general integral variable rather than a system of first order partial differential equations pdes associated with the concentration and flux of a conservation law reductions for the inde to ordinary differential equations odes are performed and some invariant solutions are constructed

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

hidden symmetries associated with the projective group of nonlinear first order ordinary differential equations hidden symmetries those not found by the classical lie group method for point symmetries are reported for nonlinear first order ordinary differential equations odes which arise frequently in physical problems these are for the special class of the eight nonabelian two parameter subgroups of the eight parameter projective group the first order odes can be transformed by non local transformations to new separable first order odes which then can be reduced to quadratures the first order odes include riccati equations and equations which in particular cases are of the form of abel s equation the procedure demonstrates the feasibility of integrating nonlinear odes that do not show any apparent lie group point symmetry applications to the vlasov characteristic equation and the reaction diffusion equation are given

solvable structures and hidden symmetries we show that some examples in the literature of non standard symmetry reductions of ordinary differential equations can be understood using the concept of a solvable structure of one forms

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems