GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieve...GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieved by several parameters,such as polynomial terms,periodic terms,offsets,and post-seismic models.The latter contains some stochastic noises,which can be affected by detecting the former parameters.If there are not enough parameters assumed,modeling errors will occur and adversely affect the analysis results.In this study,we propose a processing strategy in which the commonly-used 1-order of the polynomial term can be replaced with different orders for better fitting GNSS time series of the Crustal Movement Network of China(CMONOC)stations.Initially,we use the Bayesian Information Criterion(BIC)to identify the best order within the range of 1-4 during the fitting process using the white noise plus power-law noise(WN+PL)model.Then,we compare the 1-order and the optimal order on the effect of deterministic models in GNSS time series,including the velocity and its uncertainty,amplitudes,and initial phases of the annual signals.The results indicate that the first-order polynomial in the GNSS time series is not the primary factor.The root mean square(RMS)reduction rates of almost all station components are positive,which means the new fitting of optimal-order polynomial helps to reduce the RMS of residual series.Most stations maintain the velocity difference(VD)within ±1 mm/yr,with percentages of 85.6%,81.9%and 63.4%in the North,East,and Up components,respectively.As for annual signals,the numbers of amplitude difference(AD)remained at ±0.2 mm are 242,239,and 200 in three components,accounting for 99.6%,98.4%,and 82.3%,respectively.This finding reminds us that the detection of the optimal-order polynomial is necessary when we aim to acquire an accurate understanding of the crustal movement features.展开更多
In this paper, an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed algorithm is based on the Bernstein poly...In this paper, an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed algorithm is based on the Bernstein polynomial approach. Novel features of the proposed algorithm are that it uses a new rule for the selection of the subdivision point, modified rules for the selection of the subdivision direction, and a new acceleration device to avoid some unnecessary subdivisions. The performance of the proposed algorithm is numerically tested on a collection of 16 test problems. The results of the tests show the proposed algorithm to be superior to the existing Bernstein algorithm in terms of the chosen performance metrics.展开更多
The path planning of autonomous mobile robots(PPoAMR)is a very complex multi-constraint problem.The main goal is to find the shortest collision-free path from the starting point to the target point.By the fact that th...The path planning of autonomous mobile robots(PPoAMR)is a very complex multi-constraint problem.The main goal is to find the shortest collision-free path from the starting point to the target point.By the fact that the PPoAMR problem has the prior knowledge that the straight path between the starting point and the target point is the optimum solution when obstacles are not considered.This paper proposes a new path planning algorithm based on the prior knowledge of PPoAMR,which includes the fitness value calculation method and the prior knowledge particle swarm optimization(PKPSO)algorithm.The new fitness calculation method can preserve the information carried by each individual as much as possible by adding an adaptive coefficient.The PKPSO algorithm modifies the particle velocity update method by adding a prior particle calculated from the prior knowledge of PPoAMR and also implemented an elite retention strategy,which improves the local optima evasion capability.In addition,the quintic polynomial trajectory optimization approach is devised to generate a smooth path.Finally,some experimental comparisons with those state-of-the-arts are carried out to demonstrate the effectiveness of the proposed path planning algorithm.展开更多
This paper presents an approach for the optimal design of a new retrofit technique called weakening and damping that is valid for civil engineering inelastic structures. An alternative design methodology is developed ...This paper presents an approach for the optimal design of a new retrofit technique called weakening and damping that is valid for civil engineering inelastic structures. An alternative design methodology is developed with respect to the existing ones that is able to determine the locations and the magnitude of weakening and/or softening of structural elements and adding damping while insuring structural stability. An optimal polynomial controller that is a summation of polynomials in nonlinear states is used in Phase I of the method to reduce the peak response quantities of seismically excited nonlinear or hysteretic systems. The main advantage of the optimal polynomial controller is that it is able to automatically stabilize the structural system. The optimal design of a shear-type structure is used as an example to illustrate the feasibility of the proposed approach, which leads to a reduction of both peak inter-story drifts and peak total accelerations.展开更多
In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure betwe...In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.展开更多
To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection techniq...To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).展开更多
We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations...We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates(BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.展开更多
In this paper,we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete(typically binary)variables.Such models have natural applications in graph theory,neural networks...In this paper,we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete(typically binary)variables.Such models have natural applications in graph theory,neural networks,error-correcting codes,among many others.In particular,we focus on three types of optimization models:(1)maximizing a homogeneous polynomial function in binary variables;(2)maximizing a homogeneous polynomial function in binary variables,mixed with variables under spherical constraints;(3)maximizing an inhomogeneous polynomial function in binary variables.We propose polynomial-time randomized approximation algorithms for such polynomial optimizationmodels,and establish the approximation ratios(or relative approximation ratios whenever appropriate)for the proposed algorithms.Some examples of applications for these models and algorithms are discussed as well.展开更多
A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Bul′o and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by ...A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Bul′o and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local（global） minimizers of a homogeneous polynomial function of degree r and the maximal（maximum） cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Tur′an densities of complete non-uniform hypergraphs.展开更多
In this paper, we introduce a novel class of coplanar conics, the pencil of which can doubly contact to calibrate camera and estimate pose. We first analyze the properties of con-axes and con-eccentricity ellipses, wh...In this paper, we introduce a novel class of coplanar conics, the pencil of which can doubly contact to calibrate camera and estimate pose. We first analyze the properties of con-axes and con-eccentricity ellipses, which consist of a naturM extending pattern of concentric circles. Then the general case that two ellipses have two repeated complex intersection points is presented. This degenerate configuration results in a one-parameter family of homographies which map the planar pattern to its image. Although it is unable to compute the complete homography, an indirect 3-degree polynomial or 5-degree polynomial constraint on intrinsic parameters from one image can also be used for camera calibration and pose estimation under the minimal conditions. Furthermore, this nonlinear problem can be treated as a polynomial optimization problem （POP） and the global optimization solution can be also obtained by using SparsePOP （a sparse semidefinite programming relaxation of POPs）, Finally, the experiments with simulated data and real images are shown to verify the correctness and robustness of the proposed technique.展开更多
For a given set of data points in the plane, a new method is presented for computing a parameter value（knot） for each data point. Associated with each data point, a quadratic polynomial curve passing through three a...For a given set of data points in the plane, a new method is presented for computing a parameter value（knot） for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.展开更多
We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, ...We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.展开更多
We consider the problem of minimizing a fixed-degree polynomial over the standard simplex.This problem is well known to be NP-hard,since it contains the maximum stable set problem in combinatorial optimization as a sp...We consider the problem of minimizing a fixed-degree polynomial over the standard simplex.This problem is well known to be NP-hard,since it contains the maximum stable set problem in combinatorial optimization as a special case.In this paper,we revisit a known upper bound obtained by taking the minimum value on a regular grid,and a known lower bound based on Pólya’s representation theorem.More precisely,we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values.Our results refine the known upper bounds in the quadratic and cubic cases,and they asymptotically refine the known upper bound in the general case.展开更多
By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper...By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper is able to decide whether or not the global infimum of f is finite.In the case of f having a finite infimum,the global infimum of f can be accurately coded in the Interval Representation.Another usage of our algorithm to decide whether or not the infimum of f is attained when the global infimum of f is finite.In the design of our algorithm,Wu’s well-known method plays an important role.展开更多
Human motion prediction is a critical issue in human-robot collaboration(HRC)tasks.In order to reduce the local error caused by the limitation of the capture range and sampling frequency of the depth sensor,a hybrid h...Human motion prediction is a critical issue in human-robot collaboration(HRC)tasks.In order to reduce the local error caused by the limitation of the capture range and sampling frequency of the depth sensor,a hybrid human motion prediction algorithm,optimized sliding window polynomial fitting and recursive least squares(OSWPF-RLS)was proposed.The OSWPF-RLS algorithm uses the human body joint data obtained under the HRC task as input,and uses recursive least squares(RLS)to predict the human movement trajectories within the time window.Then,the optimized sliding window polynomial fitting(OSWPF)is used to calculate the multi-step prediction value,and the increment of multi-step prediction value was appropriately constrained.Experimental results show that compared with the existing benchmark algorithms,the OSWPF-RLS algorithm improved the multi-step prediction accuracy of human motion and enhanced the ability to respond to different human movements.展开更多
A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques.Applying it,they gave a new proof of Turán’s theorem.This aroused the interests in st...A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques.Applying it,they gave a new proof of Turán’s theorem.This aroused the interests in studying the connection between continuous optimization and extremal problems in combinatorics.In 2009,S.Rota Bulòand M.Pelillo extended the result of Motzkin-Straus to r-uniform hypergraphs.Recently,Johnston and Lu initiated the study of the Turán density of a non-uniform hypergraph.Polynomial optimization problems related to several types of non-uniform hypergraphs and its applications on Turán densities have also been studied.In this paper,we obtain a Motzkin-Straus type of results for all non-uniform hypergraphs.Applying it,we give an upper bound of the Turán density of a complete non-uniform hypergraph.展开更多
In a recent article, the authors provided an effective algorithm for both computing the global infimum of f and deciding whether or not the infimum of f is attained, where f is a multivariate polynomial over the field...In a recent article, the authors provided an effective algorithm for both computing the global infimum of f and deciding whether or not the infimum of f is attained, where f is a multivariate polynomial over the field R of real numbers. As a complement, the authors investigate the semi- algebraically connected components of minimum points of a polynomial function in this paper. For a given multivariate polynomial f over R, it is shown that the above-mentioned algorithm can find at least one point in each semi-algebraically connected component of minimum points of f whenever f has its global minimum.展开更多
In various Hilbert spaces of analytic functions on the unit disk,we charac-terize when a function has optimal polynomial approximants given by truncations of a single power series or,equivalently,when the approximants...In various Hilbert spaces of analytic functions on the unit disk,we charac-terize when a function has optimal polynomial approximants given by truncations of a single power series or,equivalently,when the approximants stabilize.We also intro-duce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto certain shift invariant subspaces.展开更多
The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tenso...The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.展开更多
In this paper,we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary.We proved the polynomial decay result with an estimation of the decay r...In this paper,we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary.We proved the polynomial decay result with an estimation of the decay rates.Our result is established using the frequency-domain method and Borichev-Tomilov theorem.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.42404017,42122025 and 42174030).
文摘GNSS time series analysis provides an effective method for research on the earth's surface deformation,and it can be divided into two parts,deterministic models and stochastic models.The former part can be achieved by several parameters,such as polynomial terms,periodic terms,offsets,and post-seismic models.The latter contains some stochastic noises,which can be affected by detecting the former parameters.If there are not enough parameters assumed,modeling errors will occur and adversely affect the analysis results.In this study,we propose a processing strategy in which the commonly-used 1-order of the polynomial term can be replaced with different orders for better fitting GNSS time series of the Crustal Movement Network of China(CMONOC)stations.Initially,we use the Bayesian Information Criterion(BIC)to identify the best order within the range of 1-4 during the fitting process using the white noise plus power-law noise(WN+PL)model.Then,we compare the 1-order and the optimal order on the effect of deterministic models in GNSS time series,including the velocity and its uncertainty,amplitudes,and initial phases of the annual signals.The results indicate that the first-order polynomial in the GNSS time series is not the primary factor.The root mean square(RMS)reduction rates of almost all station components are positive,which means the new fitting of optimal-order polynomial helps to reduce the RMS of residual series.Most stations maintain the velocity difference(VD)within ±1 mm/yr,with percentages of 85.6%,81.9%and 63.4%in the North,East,and Up components,respectively.As for annual signals,the numbers of amplitude difference(AD)remained at ±0.2 mm are 242,239,and 200 in three components,accounting for 99.6%,98.4%,and 82.3%,respectively.This finding reminds us that the detection of the optimal-order polynomial is necessary when we aim to acquire an accurate understanding of the crustal movement features.
文摘In this paper, an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed algorithm is based on the Bernstein polynomial approach. Novel features of the proposed algorithm are that it uses a new rule for the selection of the subdivision point, modified rules for the selection of the subdivision direction, and a new acceleration device to avoid some unnecessary subdivisions. The performance of the proposed algorithm is numerically tested on a collection of 16 test problems. The results of the tests show the proposed algorithm to be superior to the existing Bernstein algorithm in terms of the chosen performance metrics.
基金This work was supported by the National Key R&D Funding of China(No.2018YFB1403702)the Zhejiang Provincial Natural Science Foundation of China for Distinguished Young Scholars(No.LR22F030003).
文摘The path planning of autonomous mobile robots(PPoAMR)is a very complex multi-constraint problem.The main goal is to find the shortest collision-free path from the starting point to the target point.By the fact that the PPoAMR problem has the prior knowledge that the straight path between the starting point and the target point is the optimum solution when obstacles are not considered.This paper proposes a new path planning algorithm based on the prior knowledge of PPoAMR,which includes the fitness value calculation method and the prior knowledge particle swarm optimization(PKPSO)algorithm.The new fitness calculation method can preserve the information carried by each individual as much as possible by adding an adaptive coefficient.The PKPSO algorithm modifies the particle velocity update method by adding a prior particle calculated from the prior knowledge of PPoAMR and also implemented an elite retention strategy,which improves the local optima evasion capability.In addition,the quintic polynomial trajectory optimization approach is devised to generate a smooth path.Finally,some experimental comparisons with those state-of-the-arts are carried out to demonstrate the effectiveness of the proposed path planning algorithm.
文摘This paper presents an approach for the optimal design of a new retrofit technique called weakening and damping that is valid for civil engineering inelastic structures. An alternative design methodology is developed with respect to the existing ones that is able to determine the locations and the magnitude of weakening and/or softening of structural elements and adding damping while insuring structural stability. An optimal polynomial controller that is a summation of polynomials in nonlinear states is used in Phase I of the method to reduce the peak response quantities of seismically excited nonlinear or hysteretic systems. The main advantage of the optimal polynomial controller is that it is able to automatically stabilize the structural system. The optimal design of a shear-type structure is used as an example to illustrate the feasibility of the proposed approach, which leads to a reduction of both peak inter-story drifts and peak total accelerations.
基金Supported by the Natural Science Foundation of Hubei Province(2008CDZD47)
文摘In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.
基金supported by the the National Science and Technology Council(Grant Number:NSTC 112-2221-E239-022).
文摘To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).
基金supported by National Natural Science Foundation of China (Grant Nos. 11401364, 11322109, 11331012, 11471325 and 11461161005)the National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA122902)+1 种基金the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of SciencesNational Basic Research Program of China (973 Program) (Grant No. 2015CB856002)
文摘We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates(BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.
基金supported in part by Hong Kong General Research Fund(No.CityU143711)Zhening Li was supported in part by Natural Science Foundation of Shanghai(No.12ZR1410100)+1 种基金Ph.D.Programs Foundation of Chinese Ministry of Education(No.20123108120002)Shuzhong Zhang was supported in part by U.S.National Science Foundation(No.CMMI-1161242).
文摘In this paper,we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete(typically binary)variables.Such models have natural applications in graph theory,neural networks,error-correcting codes,among many others.In particular,we focus on three types of optimization models:(1)maximizing a homogeneous polynomial function in binary variables;(2)maximizing a homogeneous polynomial function in binary variables,mixed with variables under spherical constraints;(3)maximizing an inhomogeneous polynomial function in binary variables.We propose polynomial-time randomized approximation algorithms for such polynomial optimizationmodels,and establish the approximation ratios(or relative approximation ratios whenever appropriate)for the proposed algorithms.Some examples of applications for these models and algorithms are discussed as well.
基金Supported by the National Natural Science Foundation of China(No.11671124)
文摘A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Bul′o and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local（global） minimizers of a homogeneous polynomial function of degree r and the maximal（maximum） cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Tur′an densities of complete non-uniform hypergraphs.
基金the National Basic Research Program (973) of China(No.2011CB302203)the National Natural Science Foundation of China(No.60833009)
文摘In this paper, we introduce a novel class of coplanar conics, the pencil of which can doubly contact to calibrate camera and estimate pose. We first analyze the properties of con-axes and con-eccentricity ellipses, which consist of a naturM extending pattern of concentric circles. Then the general case that two ellipses have two repeated complex intersection points is presented. This degenerate configuration results in a one-parameter family of homographies which map the planar pattern to its image. Although it is unable to compute the complete homography, an indirect 3-degree polynomial or 5-degree polynomial constraint on intrinsic parameters from one image can also be used for camera calibration and pose estimation under the minimal conditions. Furthermore, this nonlinear problem can be treated as a polynomial optimization problem （POP） and the global optimization solution can be also obtained by using SparsePOP （a sparse semidefinite programming relaxation of POPs）, Finally, the experiments with simulated data and real images are shown to verify the correctness and robustness of the proposed technique.
基金Supported by the National Natural Science Foundation of China(61602277,61672327,61472227)the Shandong Provincial Natural Science Foundation,China(ZR2016FQ12)
文摘For a given set of data points in the plane, a new method is presented for computing a parameter value（knot） for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.
基金The authors would like to thank the reviewers for their insightful comments which help to improve the presentation of the paper. The first author＇s work was supported by the National Natural Science Foundation of China （Grant No. 11471242） and the work of the second author was supported by the National Natural Science Foundation of China （Grant No. 11601261）.
文摘We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.
文摘We consider the problem of minimizing a fixed-degree polynomial over the standard simplex.This problem is well known to be NP-hard,since it contains the maximum stable set problem in combinatorial optimization as a special case.In this paper,we revisit a known upper bound obtained by taking the minimum value on a regular grid,and a known lower bound based on Pólya’s representation theorem.More precisely,we consider the difference between these two bounds and we provide upper bounds for this difference in terms of the range of function values.Our results refine the known upper bounds in the quadratic and cubic cases,and they asymptotically refine the known upper bound in the general case.
基金partially supported by National Natural Science Foundation of China (Grant Nos. 10761006, 11161034)
文摘By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper is able to decide whether or not the global infimum of f is finite.In the case of f having a finite infimum,the global infimum of f can be accurately coded in the Interval Representation.Another usage of our algorithm to decide whether or not the infimum of f is attained when the global infimum of f is finite.In the design of our algorithm,Wu’s well-known method plays an important role.
基金supported by the National Natural Science Foundation of China(61701270)the Young Doctor Cooperation Foundation of Qilu University of Technology(Shandong Academy of Sciences)(2017BSHZ008)。
文摘Human motion prediction is a critical issue in human-robot collaboration(HRC)tasks.In order to reduce the local error caused by the limitation of the capture range and sampling frequency of the depth sensor,a hybrid human motion prediction algorithm,optimized sliding window polynomial fitting and recursive least squares(OSWPF-RLS)was proposed.The OSWPF-RLS algorithm uses the human body joint data obtained under the HRC task as input,and uses recursive least squares(RLS)to predict the human movement trajectories within the time window.Then,the optimized sliding window polynomial fitting(OSWPF)is used to calculate the multi-step prediction value,and the increment of multi-step prediction value was appropriately constrained.Experimental results show that compared with the existing benchmark algorithms,the OSWPF-RLS algorithm improved the multi-step prediction accuracy of human motion and enhanced the ability to respond to different human movements.
基金This paper is supported in part by National Natural Science Foundation of China(No.11931002).
文摘A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques.Applying it,they gave a new proof of Turán’s theorem.This aroused the interests in studying the connection between continuous optimization and extremal problems in combinatorics.In 2009,S.Rota Bulòand M.Pelillo extended the result of Motzkin-Straus to r-uniform hypergraphs.Recently,Johnston and Lu initiated the study of the Turán density of a non-uniform hypergraph.Polynomial optimization problems related to several types of non-uniform hypergraphs and its applications on Turán densities have also been studied.In this paper,we obtain a Motzkin-Straus type of results for all non-uniform hypergraphs.Applying it,we give an upper bound of the Turán density of a complete non-uniform hypergraph.
基金supported by the National Natural Science Foundation of China under Grant No.11161034the Science Foundation of the Education Department of Jiangxi Province under Grant No.Gjj12012
文摘In a recent article, the authors provided an effective algorithm for both computing the global infimum of f and deciding whether or not the infimum of f is attained, where f is a multivariate polynomial over the field R of real numbers. As a complement, the authors investigate the semi- algebraically connected components of minimum points of a polynomial function in this paper. For a given multivariate polynomial f over R, it is shown that the above-mentioned algorithm can find at least one point in each semi-algebraically connected component of minimum points of f whenever f has its global minimum.
文摘In various Hilbert spaces of analytic functions on the unit disk,we charac-terize when a function has optimal polynomial approximants given by truncations of a single power series or,equivalently,when the approximants stabilize.We also intro-duce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto certain shift invariant subspaces.
基金the Research Programme of National University of Defense Technology(No.ZK16-03-45).
文摘The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.
文摘In this paper,we study the energy decay rate for a one-dimensional nondegenerate wave equation under a fractional control applied at the boundary.We proved the polynomial decay result with an estimation of the decay rates.Our result is established using the frequency-domain method and Borichev-Tomilov theorem.
