a new neural network for solving linear and quadratic programming problems a new neural network for solving linear and quadratic programming problems is presented and is shown to be globally convergent the new neural network improves existing neural networks for solving these problems it avoids the parameter turning problem it is capable of achieving the exact solutions and it uses only simple hardware in which no analog multipliers for variables are required furthermore the network solves both the primal problems and their dual problems simultaneously

a method to synchronise cameras using the direct linear transformation technique the aim of this paper is to present a method which enables the recordings of cameras that are not equipped with a synchronisation system to be synchronised a posteriori using the direct linear transformation technique this method estimates the phase difference between two cameras by minimising the reconstruction errors of a moving point once the phase difference value is known one of the recordings is chosen as a reference and the second one is synchronised to the first by cubic spline interpolation

statistical comparison of dlt versus ilssc in the calibration of a photogrammetric stereo system this paper compares the dlt and ilssc approaches in the geometrical calibration of a photogrammetric stereo system in terms of accuracy and speed to come up with an unbiased quantitative evaluation of the accuracy of the algorithms the concept of reliable estimate has been introduced the statistical distribution of the accuracy is assessed over different calibration experiments performed with the same data but with different noise distribution and different test sets results show that in the simulations where the only error on the two dimensional points was gaussian zero mean and on real data which were corrected for distortions through polynomial or linear interpolation the accuracy of the two methods was quite similar dlt showed more accurate than ilssc on simulated data with residual distortion errors and on real data which were not corrected for distortions as far as speed is concerned a fast triangulation algorithm is associated with ilssc while the simultaneous solution of two pairs of dlt equations is associated to dlt the first algorithm is much faster requiring 113 flops per point versus 259 of dlt the fast triangulation with dlt parameters does not achieve the same accuracy on the reconstructed three dimensional position taken all together the results suggest that ilssc can be theoretically considered the best approach to three dimensional reconstruction provided that distortions are corrected in advance the statistical evaluation of the accuracy allows a fair judgement of the performances of the algorithms to be obtained unbiased by particular distributions of measurement errors and test points

hermitian analogues of hilbert s 17 th problem abstract we pose and discuss several hermitian analogues of hilbert s 17 th problem we survey what is known offer many explicit examples and some proofs and give applications to cr geometry we prove one new algebraic theorem a non negative hermitian symmetric polynomial divides a non zero squared norm if and only if it is a quotient of squared norms we also discuss a new example of putinar scheiderer

improved rsa accuracy with dlt and balanced calibration marker distributions with an assessment of initial calibration roentgen stereophotogrammetric analysis rsa has been used for over 25 years for accurate micromotion measurement in a wide variety of orthopaedic applications this study investigated two possible improvements to the method first direct linear transformation dlt was compared to the traditional rsa reconstruction algorithm the two methods were considered with respect to standard extrapolation and interpolation calibration cages matlab simulations showed that reconstruction accuracy was greatly improved 60 by combining dlt with an even distribution of enclosing calibration markers second a benchtop study using phantoms translated at 0 0254 mm intervals showed initial calibration followed by removal of the interpolation cage for subsequent exposures was potentially twice as accurate as self calibration with an extrapolation cage these results showed optimizations for the application of rsa when unobstructed space is required

instantaneous helical axis estimation from 3 d video data in neck kinematics for whiplash diagnostics to date the diagnosis of whiplash injuries has been very difficult and largely based on subjective clinical assessment the work by winters and peles multiple muscle systems biomechanics and movement organization springer new york 1990 suggests that the use of finite helical axes fhas in the neck may provide an objective assessment tool for neck mobility thus the position of the fha describing head trunk motion may allow discrimination between normal and pathological cases such as decreased mobility in particular cervical joints for noisy unsmoothed data the fhas must be taken over rather large angular intervals if the fhas are to be reconstructed with sufficient accuracy in the winters and peles study these intervals were approximately 10 in order to study the movements microstructure the present investigation uses instantaneous helical axes ihas estimated from low pass smoothed video data here the small step noise sensitivity of the fha no longer applies and proper low pass filtering allows estimation of the iha even for small rotation velocity of the moving neck for marker clusters mounted on the head and trunk technical system validation showed that the ihas direction dispersions were on the order of one degree while their position dispersions were on the order of 1 mm for low pass cut off frequencies of a few hz the dispersions were calculated from weighted errors in order to account for the adverse effects of vanishing various simple planar models relating the instantaneous 2 d centre of rotation with the geometry and kinematics of a multi joint neck model are derived in order to gauge the utility of the fha and iha approaches some preliminary results on asymptomatic and pathological subjects are provided in terms of the ruled surface formed by sampled ihas and of their piercing points through the mid sagittal plane during a prescribed flexion extension movement of the neck

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

on convergence conditions of an extended projection neural network the output trajectory convergence of an extended projection neural network was developed under the positive definiteness condition of the jacobian matrix of nonlinear mapping this note offers several new convergence results the state trajectory convergence and the output trajectory convergence of the extended projection neural network are obtained under the positive semidefiniteness condition of the jacobian matrix comparison and illustrative examples demonstrate applied significance of these new results

the accuracy of dlt extrapolation in three dimensional film analysis an analysis of errors arising from the direct linear transformation dlt approach to three dimensional reconstructions from two dimensional images has been undertaken the principal factor studied being the number and distribution of control points used in the calibration procedure significantly increased error was found to be associated with extrapolation to unknown points outside the control point distribution space differences in accuracy between two camera position set ups and 11 vs 12 dlt parameter solutions were also examined

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

a method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming abstract a method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by n z shor j b lasserre and p a parrilo the aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables x and y satisfying y x x y 1 and x x y y global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators we also discuss how to exploit sparsity and symmetry several numerical experiments are included the last section explains how our theory fits into the framework of noncommutative real algebraic geometry

an efficient simplified neural network for solving linear and quadratic programming problems we present a high performance and efficiently simplified new neural network which improves the existing neural networks for solving general linear and quadratic programming problems the network having no need for parameter setting results in a simple hardware requiring no analog multipliers is shown to be stable and converges globally to the exact solution moreover using this network we can solve both linear and quadratic programming problems and their duals simultaneously high accuracy of the obtained solutions and low cost of implementation are among the features of this network we prove the global convergence of the network analytically and verify the results numerically

a novel neural network for variational inequalities with linear and nonlinear constraints variational inequality is a uniform approach for many important optimization and equilibrium problems based on the sufficient and necessary conditions of the solution this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of lyapunov and converges to an exact solution of the original problem meanwhile the proposed network with a gradient mapping is also proved to be stable in the sense of lyapunov and to have a finite time convergence under some mild conditions by using a new energy function compared with the existing neural networks the new model can be applied to solve some nonmonotone problems has no adjustable parameter and has lower complexity thus the structure of the proposed network is very simple since the proposed network can be used to solve a broad class of optimization problems it has great application potential the validity and transient behavior of the proposed neural network are demonstrated by several numerical examples

the simulation of aerial movement i the determination of orientation angles from film data abstract quantitative mechanical analyses of human movement require the time histories of the angles which specify body configuration and orientation when these angles are obtained from a filmed performance they may be used to evaluate the accuracy of a simulation model this paper presents a method of determining orientation angles and their rates of change from film data the stages used comprise the synchronization of data obtained from two camera views the determination of three dimensional coordinates of joint centres the calculation of an angle from a sequence of sine and cosine values and the curve fitting of angles using quintic splines for each state other possible approaches are discussed original procedures are presented for obtaining individual error estimates of both the film data and the calculated angles to permit the automatic fitting of quintic splines for interpolation and differentiation and for deriving the time history of an angle as a continuous function from a sequence of sine and cosine values the method is applied to a forward somersault with 1 1 2 twists and the average error estimate of 17 orientation angles is obtained as 2 1 degrees

impossibility of c variation or formal power series variation in solutions to hilbert s 17th problem abstract no matter how a positive semidefinite polynomial f r x 1 x n is represented according to e artin s 1926 solution to hilbert s 17th problem in the form f piri2 with 0 p i r and r i r x 1 x n the pi and the coefficients of the ri cannot be chosen to depend in a c i e infinitely differentiable manner upon the coefficients of f unless degf 2 formal powers series variation is also impossible this answers a question we had raised in 1990 contemp math vol 155 amer math soc 1994 pp 107 117 where we had already shown that real analytic variation was impossible and gonzalez vega and lombardi math z 225 3 1997 427 451 then showed that for every fixed finite r n cr variation is possible improving upon their and the author s result that continuous piecewise polynomial variation is possible

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

k winners take all circuit with o n complexity presents a k winners take all circuit that is an extension of the winner take all circuit by lazzaro et al 1989 the problem of selecting the largest k numbers is formulated as a mathematical programming problem whose solution scheme based on the lagrange multiplier method is directly implemented on an analog circuit the wire length in this circuit grows only linearly with the number of elements and the circuit is more suitable for real time processing than the hopfield networks because the present circuit produces the solution almost instantaneously in contrast to the hopfield network which requires transient convergence to the solution from a precise initial state the selection resolution in the present circuit is however only finite in contrast to the almost infinite resolution in the hopfield networks

a convex polynomial that is not sos convex a multivariate polynomial p x p x 1 x n is sos convex if its hessian h x can be factored as h x m t x m x with a possibly nonsquare polynomial matrix m x it is easy to see that sos convexity is a sufficient condition for convexity of p x moreover the problem of deciding sos convexity of a polynomial can be cast as the feasibility of a semidefinite program which can be solved efficiently motivated by this computational tractability it is natural to study whether sos convexity is also a necessary condition for convexity of polynomials in this paper we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos convex

algorithms for multidimensional spectral factorization and sum of squares in this paper algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems these algorithms are based on the calculus of 2n variable polynomial matrices and their associated quadratic differential forms and share the common feature that the problems are lifted from the original n variable polynomial context to a 2n variable polynomial context this allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities lmi s to the feasibility of a semialgebraic set or to a linear eigenvalue problem

a novel recurrent neural network with one neuron and finite time convergence for k winners take all operation in this paper based on a one neuron recurrent neural network a novel k winners take all k wta network is proposed finite time convergence of the proposed neural network is proved using the lyapunov method the k wta operation is first converted equivalently into a linear programming problem then a one neuron recurrent neural network is proposed to get the kth or k 1 th largest inputs of the k wta problem furthermore a k wta network is designed based on the proposed neural network to perform the k wta operation compared with the existing k wta networks the proposed network has simple structure and finite time convergence in addition simulation results on numerical examples show the effectiveness and performance of the proposed k wta network

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

a neural network for linear matrix inequality problems gradient type hopfield networks have been widely used in optimization problems solving the paper presents a novel application by developing a matrix oriented gradient approach to solve a class of linear matrix inequalities lmis which are commonly encountered in the robust control system analysis and design the solution process is parallel and distributed in neural computation the proposed networks are proven to be stable in the large representative lmis such as generalized lyapunov matrix inequalities simultaneous lyapunov matrix inequalities and algebraic riccati matrix inequalities are considered several examples are provided to demonstrate the proposed results to verify the proposed control scheme in real time applications a high speed digital signal processor is used to emulate the neural net based control scheme

dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities a novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities first the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function then a continuous time dynamic system and a discrete time dynamic system are proposed for solving it under some mild conditions the proposed dynamic systems are shown to be globally convergent to a solution of the problem the merits of the methods refer to their simple numerical implementations and capability for handling nonstrict lmis easily in addition the methods are promising in neural circuits realization and therefore have potential applications in many online control problems several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results

three dimensional videography of swimming with panning periscopes abstract the purpose of this paper is to describe a three dimensional videography method with panning periscopes for reconstructing swimming techniques two panning periscope systems developed in our laboratory were used for the data collection a control object 1 1 2 2 m 3 with 72 markers was placed in seven consecutive locations along the 1 5 8 4 2 m 3 calibration space the position of the control object was recorded while panning the periscope systems and for each of the seven locations the dlt camera parameters were determined for eight fields equally spaced through the panning motion each dlt camera parameter was then expressed as a function of the camera orientation by using a least squares second order polynomial regression equation so that the dlt camera parameters for any camera orientation could be predicted yu et al j biomechanics 26 741 751 1993 the three dimensional coordinates of desired body landmarks at an instant were determined from the digitized two dimensional coordinates for two cameras and the predicted dlt camera parameters at that instant the accuracy of the method was evaluated by a the resultant errors in computing three dimensional coordinates of precalibrated static points and b the errors in computing the length of a scale rod pulled through the calibration space the mean resultant errors ranged from 8 34 to 16 44 mm for the above and from 9 93 to 16 22 mm for the below water control volumes the mean error in computing lenghts on the scale rod ranged from 3 32 to 5 83 mm for three 0 5 m lengths and 9 97 mm for a 1 5 m length the method produced acceptable results in the reconstruction of three dimensional motions recorded from a large space above and below the water surface

3d kinematic and dynamic analysis of the front crawl tumble turn in elite male swimmers the aim of this study was to identify kinematic and dynamic variables related to the best tumble turn times 3mrtt the turn time from 3 m in to 3 m out independent variable in ten elite male swimmers using a three dimensional 3d underwater analysis protocol and the lasso least absolute shrinkage and selection operator as statistical method for each swimmer the best time turn was analyzed with five stationary and synchronized underwater cameras the 3d reconstruction was performed using the direct linear transformation algorithm an underwater piezoelectric 3d force platform completed the setup to compute dynamic variables data were smoothed by the savitzky golay filtering method three variables were considered relevant in the best lasso model 3mrtt 2 58 a 0 425 rd 0 204 vpe 0 0046 td the head wall distance where rotation starts rd the horizontal speed at the force peak vpe and the 3d length of the path covered during the turn td furthermore bivariate analysis showed that upper body cubei and lower limb extension indexes at first contact cllei were also linked to the turn time r a0 65 and po 0 05 for both variables thus the best turn times were associated with a long rd slower vpe and reduced td by an early transverse rotation male elite swimmers reach the wall with a slightly flexed posture that results in fast extension these swimmers opt for a movement that is oriented forward and they focus on reducing the distance covered

lmi approximations for cones of positive semidefinite forms an interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems in particular this trend has been successful in showing that under suitable circumstances polynomial optimization problems can be approximated via a sequence of semidefinite programs similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information we bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms homogeneous polynomials our approach enables us to extend the existing methodology to new approximation schemes in particular we derive a novel approximation to the cone of copositive forms that is the cone of forms that are positive semidefinite over the nonnegative orthant the format of our construction can be extended to forms that are positive semidefinite over more general conic domains we also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain this opens the possibility of using linear programming technology in optimization problems over these cones

solving quadratic programming problems by delayed projection neural network in this letter the delayed projection neural network for solving convex quadratic programming problems is proposed the neural network is proved to be globally exponentially stable and can converge to an optimal solution of the optimization problem three examples show the effectiveness of the proposed network

an investigation on the accuracy of three dimensional space reconstruction using the direct linear transformation technique an investigation on the three dimensional calibration errors associated with the direct linear transformation dlt technique was carried out using a two video camera system thirty different configurations in five groupings of different numbers of control points were tested the results showed that the accuracy improved as the number of control points increased from 8 to 24 further it was demonstrated that the best accuracy was achieved when the control points were evenly distributed throughout the control region for the system under investigation an accuracy of 1 2 mm in the x and y directions 4 6 mm in the z direction and 6 7 mm for the resultant was obtained in a control space of 2 10 1 35 1 00 m additionally the accuracy for points outside the control space and the systematic error associated with the control object were studied tests were done on a 32 point rectangular calibration frame and the results indicated that the calibration errors could be reduced by 20 40 when an appropriate quadratic function was used to modify the standard dlt method

a one layer projection neural network for nonsmooth optimization subject to linear equalities and bound constraints this paper presents a one layer projection neural network for solving nonsmooth optimization problems with generalized convex objective functions and subject to linear equalities and bound constraints the proposed neural network is designed based on two projection operators linear equality constraints and bound constraints the objective function in the optimization problem can be any nonsmooth function which is not restricted to be convex but is required to be convex pseudoconvex on a set defined by the constraints compared with existing recurrent neural networks for nonsmooth optimization the proposed model does not have any design parameter which is more convenient for design and implementation it is proved that the output variables of the proposed neural network are globally convergent to the optimal solutions provided that the objective function is at least pseudoconvex simulation results of numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural network

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

a generalized neural network for solving a class of minimax optimization problems with linear constraints abstract in this paper a generalized neural network was proposed based on projection method and differential inclusions which is contributed to solve a class of minimax optimization problems with linear constraints it is proved that the solution trajectory can converge to the feasible region in the finite time when the initial point is not in the feasible region once the solution trajectory reaches the feasible region it will stay therein thereafter in addition we investigate the global convergence and exponential convergence furthermore three illustrative examples are given to show the efficiency of the proposed neural network

a panning dlt procedure for three dimensional videography abstract the direct linear transformation dlt method abdel aziz and karara aps symposium on photogrammetry american society of photogrammetry falls church va 1971 is widely used in biomechanics to obtain three dimensional space coordinates from film and video records this method has some major shortcomings when used to analyze events which take place over large areas to overcome these shortcomings a three dimensional data collection method based on the dlt method and making use of panning cameras was developed several small single control volumes were combined to construct a large total control volume for each single control volume a regression equation calibration equation is developed to express each of the 11 dlt parameters as a function of camera orientation so that the dlt parameters can then be estimated from arbitrary camera orientations once the dlt parameters are known for at least two cameras and the associated two dimensional film or video coordinates of the event are obtained the desired three dimensional space coordinates can be computed in a laboratory test five single control volumes in a total control volume of 24 40 2 44 2 44 m3 were used to test the effect of the position of the single control volume on the accuracy of the computed three dimensional space coordinates linear and quadratic calibration equations were used to test the effect of the order of the equation on the accuracy of the computed three dimensional space coordinates for four of the five single control volumes tested the mean resultant errors associated with the use of the linear calibration equation were significantly larger than those associated with the use of the quadratic calibration equation the position of the single control volume had no significant effect on the mean resultant errors in computed three dimensional coordinates when the quadratic calibration equation was used under the same data collection conditions the mean resultant errors in the computed three dimensional coordinates associated with the panning and stationary dlt methods were 17 and 22 mm respectively the major advantages of the panning dlt method lie in the large image sizes obtained and in the ease with which the data can be collected the method also has potential for use in a wide variety of contexts the major shortcoming of the method is the large amount of digitizing necessary to calibrate the total control volume adaptations of the method to reduce the amount of digitizing required are being explored

an improved dual neural network for solving a class of quadratic programming problems and its k winners take all application this paper presents a novel recurrent neural network for solving a class of convex quadratic programming qp problems in which the quadratic term in the objective function is the square of the euclidean norm of the variable this special structure leads to a set of simple optimality conditions for the problem based on which the neural network model is formulated compared with existing neural networks for general convex qp the new model is simpler in structure and easier to implement the new model can be regarded as an improved version of the dual neural network in the literature based on the new model a simple neural network capable of solving the k winners take all k wta problem is formulated the stability and global convergence of the proposed neural network is proved rigorously and substantiated by simulation results

on new classes of nonnegative symmetric tensors in this paper we introduce three new classes of nonnegative forms or equivalently symmetric tensors and their extensions the newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors order six or above for the special case of quartic forms they collapse into the set of convex quartic homogeneous polynomial functions we discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented

a strict positivstellensatz for the weyl algebra let c be an element of the weyl algebra open image in new window which is given by a strictly positive operator in the schrodinger representation it is shown that under some conditions there exist certain elements b1 bd from open image in new window such that dj 1bjcb j is a finite sum of squares

linear and quadratic programming neural network analysis neural networks for linear and quadratic programming are analyzed the network proposed by m p kennedy and l o chua ieee trans circuits syst vol 35 pp 554 562 may 1988 is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems such as the least squares problem for quadratic programming the network converges either to an equilibrium or to an exact solution depending on whether the problem has constraints or not the results also suggest an analytical approach to solve the linear system bx b without calculating the matrix inverse the results are directly applicable to optimization problems with c sup 2 convex objective functions and linear constraints the dynamics and applicability of the networks are demonstrated by simulation the distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter

a strict positivstellensatz for enveloping algebras let g be a connected and simply connected real lie group with lie algebra open image in new window semialgebraic subsets of the unitary dual of g are defined and a strict positivstellensatz for positive elements of the universal enveloping algebra open image in new window of open image in new window is proved

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

products of positive forms linear matrix inequalities and hilbert 17th problem for ternary forms a form p on r n homogeneous n variate polynomial is called positive semidefinite p s d if it is nonnegative on r n in other words the zero vector is a global minimizer of p in this case the famous 17th conjecture of hilbert bull amer math soc n s 37 4 2000 407 later proven by artin the collected papers of emil artin addisonwesley publishing co inc reading ma london 1965 is that a form p is p s d if and only if it can be decomposed into a sum of squares of rational functions in this paper we give an algorithm to compute such a decomposition for ternary forms n 3 this algorithm involves the solution of a series of systems of linear matrix inequalities lmis in particular for a given p s d ternary form p of degree 2m we show that the abovementioned decomposition can be computed by solving at most m 4 systems of lmis of dimensions polynomial in m the underlying methodology is largely inspired by the original proof of hilbert who had been able to prove his conjecture for the case of ternary forms 2003 elsevier b v all rights reserved

the sum of squares hierarchy on the sphere and applications in quantum information theory we consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum of squares relaxations exploiting the polynomial kernel technique we obtain a quadratic improvement of the known convergence rate by reznick and doherty and wehner specifically we show that the rate of convergence is no worse than o d 2 ell 2 in the regime ell omega d where ell is the level of the hierarchy and d the dimension solving a problem left open in the recent paper by de klerk and laurent arxiv 1904 08828 importantly our analysis also works for matrix valued polynomials on the sphere which has applications in quantum information for the best separable state problem by exploiting the duality relation between sums of squares and the doherty parrilo spedalieri hierarchy in quantum information theory we show that our result generalizes to nonquadratic polynomials the convergence rates of navascues owari and plenio

a delayed neural network for solving linear projection equations and its analysis in this paper we present a delayed neural network approach to solve linear projection equations the lyapunov krasovskii theory for functional differential equations and the linear matrix inequality lmi approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network compared with the existing linear projection neural network theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems

a one layer recurrent neural network with a discontinuous hard limiting activation function for quadratic programming in this paper a one layer recurrent neural network with a discontinuous hard limiting activation function is proposed for quadratic programming this neural network is capable of solving a large class of quadratic programming problems the state variables of the neural network are proven to be globally stable and the output variables are proven to be convergent to optimal solutions as long as the objective function is strictly convex on a set defined by the equality constraints in addition a sequential quadratic programming approach based on the proposed recurrent neural network is developed for general nonlinear programming simulation results on numerical examples and support vector machine svm learning show the effectiveness and performance of the neural network

solving generally constrained generalized linear variational inequalities using the general projection neural networks generalized linear variational inequality glvi is an extension of the canonical linear variational inequality in recent years a recurrent neural network nn called general projection neural network gpnn was developed for solving glvis with simple bound often box type or sphere type constraints the aim of this paper is twofold first some further stability results of the gpnn are presented second the gpnn is extended for solving glvis with general linear equality and inequality constraints a new design methodology for the gpnn is then proposed furthermore in view of different types of constraints approaches for reducing the number of neurons of the gpnn are discussed which results in two specific gpnns moreover some distinct properties of the resulting gpnns are also explored based on their particular structures numerical simulation results are provided to validate the results

moments tensors hilbert s identity and k wise uncorrelated random variables in this paper we introduce a notion to be called k wise uncorrelated random variables which is similar but not identical to the so called k wise independent random variables in the literature we show how to construct k wise uncorrelated random variables by a simple procedure the constructed random variables can be applied e g to express the quartic polynomial xtqx 2 where q is an n n positive semidefinite matrix by a sum of fourth powered polynomial terms known as hilbert s identity by virtue of the proposed construction the number of required terms is no more than 2n4 n this implies that it is possible to find a 2n4 n point distribution whose fourth moments tensor is exactly the symmetrization of q q moreover we prove that the number of required fourth powered polynomial terms to express xtqx 2 is at least n n 1 2 the result is applied to prove that computing the matrix 2 4 norm is np hard extensions of the results to complex random variables are discussed as well

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

a simple and high performance neural network for quadratic programming problems in this paper a neural network for quadratic programming problems is simplified the simplicity is necessary for the high accuracy of solutions and low cost of implementation the proposed network is proved to be an extension of newton s optimal descent flow about constraints problems and is globally convergent the network dynamic behaviors are also discussed and these can get the feasible solution more easily the simulations demonstrate the reasonability of the theory and advantages of the network

positive definiteness of paired symmetric tensors and elasticity tensors abstract in this paper we consider higher order paired symmetric tensors and strongly paired symmetric tensors elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors a strongly paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics which plays an important role in nonlinear elasticity theory we mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional strongly paired symmetric tensors we first show that the concerned strongly paired symmetric tensor is positive definite if and only if its smallest m eigenvalue is positive second we propose several necessary and sufficient conditions under which the concerned strongly paired symmetric tensor is positive definite third we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional strongly paired symmetric tensor can be written as a sum of squares of polynomials and further propose several necessary and or sufficient conditions to judge whether the concerned strongly paired symmetric tensors are positive definite or not fourth by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest m eigenvalue of a fourth order three dimensional strongly paired symmetric tensor by which we can check positive definiteness of the concerned tensor the preliminary numerical results confirm our theoretical findings

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

three dimensional cinematography with control object of unknown shape a technique for reconstruction of three dimensional 3d motion which involves a simple filming procedure but allows the deduction of coordinates in large object volumes was developed internal camera parameters are calculated from measurements of the film images of two calibrated crosses while external camera parameters are calculated from the film images of points in a control object of unknown shape but at least one known length the control object which includes the volume in which the activity is to take place is formed by a series of poles placed at unknown locations each carrying two targets from the internal and external camera parameters and from locations of the images of point in the films of the two cameras 3d coordinates of the point can be calculated root mean square errors of the three coordinates of points in a large object volume 5m x 5m x 1 5m were 15 mm 13 mm 13 mm and 6 mm and relative errors in lengths averaged 0 5 0 7 and 0 5 respectively

relative entropy relaxations for signomial optimization signomial programs sps are optimization problems specified in terms of signomials which are weighted sums of exponentials composed with linear functionals of a decision variable sps are nonconvex optimization problems in general and families of np hard problems can be reduced to sps in this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of sps this sequence of lower bounds is computed by solving increasingly larger sized relative entropy optimization problems which are convex programs specified in terms of linear and relative entropy functions our approach relies crucially on the observation that the relative entropy function by virtue of its joint convexity with respect to both arguments provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic geometric mean inequality by appealing to representation theorems from real algeb

extending positive definiteness the main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of semigroups to positive definite ones by specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems like complex two sided complex and multidimensional trigonometric moment problems in addition unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme

high precision three dimensional photogrammetric calibration and object space reconstruction using a modified dlt approach two modified dlt algorithms are presented that improve the accuracy of three dimensional object space reconstruction by almost an order of magnitude when compared with conventional methods the improvement in the linear modified dlt mdlt algorithm is achieved by satisfying certain orthogonality conditions in the form of a non linear constraint thereby effectively eliminating a redundant dlt parameter in the non linear mdlt algorithm the improvement and computational stability results from the appropriate elimination of implicit variables from one side of the approximating relations and the corresponding reformulation of the objective function to be minimized the highest reconstruction accuracy of 0 733 mm rms mean error was obtained with the non linear mdlt algorithm this corresponds to a spatial resolution of about one part in 2860 or 0 035 overall accuracy the accuracy obtainable with the linear mdlt was found to be slightly less and about 0 041 0 833 mm rms mean error

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

a novel neural network for a class of convex quadratic minimax problems based on the inherent properties of convex quadratic minimax problems this article presents a new neural network model for a class of convex quadratic minimax problems we show that the new model is stable in the sense of lyapunov and will converge to an exact saddle point in finite time by defining a proper convex energy function furthermore global exponential stability of the new model is shown under mild conditions compared with the existing neural networks for the convex quadratic minimax problem the proposed neural network has finite time convergence a simpler structure and lower complexity thus the proposed neural network is more suitable for parallel implementation by using simple hardware units the validity and transient behavior of the proposed neural network are illustrated by some simulation results

nlt and extrapolated dlt 3 d cinematography alternatives for enlarging the volume of calibration abstract this study investigated the accuracy of the direct linear transformation dlt and non linear transformation nlt methods of 3 d cinematography videography a comparison of standard dlt extrapolated dlt and nlt calibrations showed the standard non extrapolated dlt to be the most accurate especially when a large number of control points 40 60 were used the nlt was more accurate than the extrapolated dlt when the level of extrapolation exceeded 100 the results indicated that when possible one should use the dlt with a control object sufficiently large as to encompass the entire activity being studied however in situations where the activity volume exceeds the size of one s dlt control object the nlt method should be considered

on the real multidimensional rational k moment problem we present a solution to the real multidimensional rational k moment problem where k is defined by finitely many polynomial inequalities more precisely let s be a finite set of real polynomials in x x1 xn such that the corresponding basic closed semialgebraic set ks is nonempty let e d 1r x be a localization of the real polynomial algebra and t s the preordering on e generated by s we show that every linear functional l on e such that l t s 0 is represented by a positive measure on a certain subset of ks provided d contains an element that grows fast enough on ks

camera calibration with distortion models and accuracy evaluation a camera model that accounts for major sources of camera distortion namely radial decentering and thin prism distortions is presented the proposed calibration procedure consists of two steps 1 the calibration parameters are estimated using a closed form solution based on a distribution free camera model and 2 the parameters estimated in the first step are improved iteratively through a nonlinear optimization taking into account camera distortions according to minimum variance estimation the objective function to be minimized is the mean square discrepancy between the observed image points and their inferred image projections computed with the estimated calibration parameters the authors introduce a type of measure that can be used to directly evaluate the performance of calibration and compare calibrations among different systems the validity and performance of the calibration procedure are tested with both synthetic data and real images taken by tele and wide angle lenses

a conic programming approach to generalized tchebycheff inequalities consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments this is a special class ofgeneralized tchebycheff inequalities in probability theory we study this problem within the framework of conic programming relying on a general approximation scheme for conic programming we show that these bounds can be numerically computed or approximated via semidefinite programming we also describe some applications of this class of generalized tchebycheff inequalities in probability finance and inventory theory

a high performance neural network for solving linear and quadratic programming problems two classes of high performance neural networks for solving linear and quadratic programming problems are given we prove that the new system converges globally to the solutions of the linear and quadratic programming problems in a neural network network parameters are usually not specified the proposed models can overcome numerical difficulty caused by neural networks with network parameters and obtain desired approximate solutions of the linear and quadratic programming problems

a three dimensional kinematic method for determining the effectiveness of arm segment rotations in producing racquet head speed the contribution that a segment s anatomical rotations make to racquet head speed depends on both the segment s angular velocity and the instantaneous position of the head of the racquet with respect to the segment s axes of rotation any analysis of racquet swing technique that does not consider both of these factors simultaneously is at best incomplete with this in mind a three dimensional kinematic method was developed to determine the effectiveness of the anatomical rotations of the upper arm forearm and hand in producing racquet head speed the method entailed developing a system of vector equations for three dimensional upper limb rotations that used displacement histories of 10 selected landmarks as input the required three dimensional displacement histories were obtained using three cine cameras and the dlt approach to test the diagnostic capabilities of the method a tennis serve was selected for analysis for the player and serve analyzed the greatest contribution to racquet head speed at impact was produced by internal rotation of the upper arm 8 m s 1 forearm pronation although exhibiting the fastest rotation at impact 24 rad s 1 ranked only fourth in terms of its contribution 4 m s 1 to racquet head speed to test the performance of the method a comparison was made between the racquet head speed measured directly from film and the racquet head speed computed by summing all of the individual segment contributions to speed commencing at the start of forward swing and ending at ball contact the results indicate that the method can successfully determine the individual contributions that the different anatomical rotational velocities of the arm segments make to the measured instantaneous racquet head speed

accuracy assessment and control point configuration when using the dlt for photogrammetry the direct linear transformation dlt is a common technique used to calibrate cameras and subsequently reconstruct points filmed with two or more cameras in a three dimensional object space the assessment of the accuracy of this technique and of the influence of the distribution of control points on accuracy were examined it was concluded that to obtain a true estimation of reconstruction accuracy an independent assessment criterion is required and that the use of control points distributed around the outside rather than within the space to be calibrated is preferred

neural network for solving linear programming problems with bounded variables a new neural network for solving linear programming problems with bounded variables is presented the network is shown to be completely stable and globally convergent to the solutions to the linear programming problems the proposed new network is capable of achieving the exact solutions in contrast to existing optimization neural networks which need a suitable choice of the network parameters and thus can obtain only approximate solutions furthermore both the primal problems and their dual problems are solved simultaneously by the new network