This paper deals with extensions of higher-order optimality conditions for scalar optimization to multiobjective optimization.A type of directional derivatives for a multiobjective function is proposed,and with this n...This paper deals with extensions of higher-order optimality conditions for scalar optimization to multiobjective optimization.A type of directional derivatives for a multiobjective function is proposed,and with this notion characterizations of strict local minima of order k for a multiobjective optimization problem with a nonempty set constraint are established,generalizing the corresponding scalar case obtained by Studniarski[3].Also necessary not sufficient and sufficient not necessary optimality conditions for this minima are derived based on our directional derivatives,which are generalizations of some existing scalar results and equivalent to some existing multiobjective ones.Many examples are given to illustrate them there.展开更多
Relations of the 3D multi-directional derivatives are studied in this paper.These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained.These relations...Relations of the 3D multi-directional derivatives are studied in this paper.These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained.These relations and expressions play important roles in the meshless finite point method.展开更多
In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given seco...In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.展开更多
One of the most common image processing tasks involves the removal of noise from images.Noise can be introduced during image capture,during transmission,or during storage.For design purposes,noise sources are frequent...One of the most common image processing tasks involves the removal of noise from images.Noise can be introduced during image capture,during transmission,or during storage.For design purposes,noise sources are frequently approximated by random variables with a known probability distribution.One common noise model corrupts a signal by introducing impulses.And the surface of the image disturbed by impulse noise displays many peaks or vales.According to the characteristic of impulse noise,a novel algorithm is proposed to the detection of impulse noise point from images based on directional derivatives.First,the theory of calculus on directional derivatives is introduced in detail.Then it is applied to the field of image to removing noise with the discrete form derived from its continuous mathematical model.And a number of contrasting simulations illustrate that our algorithm not only can preserve the structure information while removing impulse noise but also can mostly save the gray value of the pixels undisturbed by noise.In addition,the comparisons of the filtering performance for removing impulse noise are analyzed in detail in the case of different noise densities,and also show that the algorithm suggested outperforms the conventional filter algorithms such as mean filter,median filter and so on in speed and impulse noise reduction,especially in random-valued impulse noise reduction.So it is a very good alternative to the existing schemes.展开更多
In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f - g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable...In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f - g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable functions on a closed convex set in a Euclidean space. As an application, we give a sufficient condition for f - g to have an error bound.展开更多
In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with...In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.展开更多
In this paper,we introduce a new directional derivative and subgradient of set-valued mappings by using a nonlinear scalarizing function.We obtain some properties of directional derivative and subgradient for cone-con...In this paper,we introduce a new directional derivative and subgradient of set-valued mappings by using a nonlinear scalarizing function.We obtain some properties of directional derivative and subgradient for cone-convex set-valued mappings.As applications,we present necessary and sufficient optimality conditions for set optimization problems and show that the local weak l-minimal solutions of set optimization problems are the global weak l-minimal solutions of set optimization problems under the assumption that the objective mapping is cone-convex.展开更多
Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this pap...Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this paper,we define the Gâteaux directional differentiability of Π_(C).We investigate some properties of the Gâteaux directional differentiability of Π_(C).In particular,if C is a closed ball,or a closed and convex cone(including proper closed subspaces),or a closed and convex cylinder,then,we give the exact representations of the directional derivatives of Π_(C).By comparing the results in[12]and this paper,we see the significant difference between the directional derivatives of the generalized metric projection operator Π_(C) and the Gâteaux directional derivatives of the standard metric projection operator PC.展开更多
Most image interpolation algorithms currently used suffer visually to some extent the effects of blurred edges and jagged artifacts in the image. This letter presents an adaptive feature preserving bidirectional flow ...Most image interpolation algorithms currently used suffer visually to some extent the effects of blurred edges and jagged artifacts in the image. This letter presents an adaptive feature preserving bidirectional flow process, where an inverse diffusion is performed to enhance edges along the normal directions to the iso-phote lines (edges), while a normal diffusion is done to remove artifacts ('jaggies') along the tangent directions. In order to preserve image features such as edges, angles and textures, the nonlinear diffusion coefficients are locally adjusted according to the first order and the second order directional derivatives of the image. Experimental results on the Lena image demonstrate that our interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional interpolations.展开更多
Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because th...Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because the multi-valued decision diagram( MDD) can reflect the relationship between the components and the system state bilaterally, it was introduced into the reliability calculation of the multi-state system( MSS). The building method,simplified criteria,and path search and probability algorithm of MSS structure function MDD were given,and the reliability of the system was calculated. The computing methods of importance based on MDD and direct partial logic derivatives( DPLD) were presented. The diesel engine fuel supply system was taken as an example to illustrate the proposed method. The results show that not only the probability of the system in each state can be easily obtained,but also the influence degree of each component and its state on the system reliability can be obtained,which is conducive to the condition monitoring and structure optimization of the system.展开更多
BrCF2CF2CH=CHCH2X(x=Cl, OAc, OH) reacted smoothly with alkynes in the presence of BrCo(dmgh)2Py/Zn, giving 4,4,5,5-tetrafluorocyclopentene derivatives in moderate yields.
The absorption spectra of 4f electron transitions of the complexes of neodymium and erbium with 8-hydroxyquinoline-5-sulphonic acid in the presence of diethylamine and ethanol have been measured by normal and third-de...The absorption spectra of 4f electron transitions of the complexes of neodymium and erbium with 8-hydroxyquinoline-5-sulphonic acid in the presence of diethylamine and ethanol have been measured by normal and third-derivative spectrophotometry. Their molar absorptivities are 70.7 l.mol^(-1).cm^(-1) for Nd and 62.5 l.mol^(-1).cm^(-1) for Er. They are 7.6 times and 14.9 times greater than those of corresponding chlorides, respectively. Use of the third-derivative spectra both eliminates the interference of Ce(Ⅳ) and increases the sensitivity for Nd and Er. Beer,s Law was obeyed from 0-10 ug/ml of Nd and Er. The method has been applied to the determination of neodymium and erbium in rare earth mixtures.展开更多
To improve the generalization performance and prediction accuracy of the stochastic configuration network(SCN)model,a novel SCN modeling method is proposed.First,the first-and second-order directional derivatives of t...To improve the generalization performance and prediction accuracy of the stochastic configuration network(SCN)model,a novel SCN modeling method is proposed.First,the first-and second-order directional derivatives of the hidden layer output matrix are calculated.The key factors extracted from the directional derivatives are linearly added to the original hidden layer output matrix to formulate a new hidden layer output matrix.Second,a spatial angle adaptive supervisory mechanism is established to improve the quality of the parameter configuration of the hidden layer nodes.The experimental results show that the proposed method improves the generalization performance and prediction accuracy.This work is a beneficial exploration of the standard SCN algorithm.展开更多
This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of c...This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived.展开更多
Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search d...Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.展开更多
This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers.A comparison relationship b...This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers.A comparison relationship between the excess misclassification error and the excess generalization error is provided;from this,along with the convex analysis theory,a kind of learning rate is derived.The results show that the performance of the classifier is effected by the outliers,and the extent of impact can be controlled by choosing the homotopy parameters properly.展开更多
This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems.Using the duality theory of the linear programmin...This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems.Using the duality theory of the linear programming and convex theory,the generalized directional derivative of the general multicommodity minimal cost flow problems is derived.The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.展开更多
This paper considers online classification learning algorithms for regularized classification schemes with generalized gradient. A novel capacity independent approach is presented. It verifies the strong convergence o...This paper considers online classification learning algorithms for regularized classification schemes with generalized gradient. A novel capacity independent approach is presented. It verifies the strong convergence of sizes and yields satisfactory convergence rates for polynomially decaying step sizes. Compared with the gradient schemes, this al- gorithm needs only less additional assumptions on the loss function and derives a stronger result with respect to the choice of step sizes and the regularization parameters.展开更多
In this paper, we establish a second-order sufficient condition for constrained optimization problems of a class of so called t-stable functions in terms of the first-order and the second-order Dini type directional d...In this paper, we establish a second-order sufficient condition for constrained optimization problems of a class of so called t-stable functions in terms of the first-order and the second-order Dini type directional derivatives. The result extends the corresponding result of [D. Bednarik and K. Pastor, Math. Program. Ser. A, 113（2008）, 283-298] to constrained optimization problems.展开更多
Through a precise recursion of B-spline bases and the resursive expression of the derivatives of rational surfaces, this paper presents an efficient algorithm for the calculation of NURBS surfaces and all their direct...Through a precise recursion of B-spline bases and the resursive expression of the derivatives of rational surfaces, this paper presents an efficient algorithm for the calculation of NURBS surfaces and all their directional derivatives. The algorithm requires less storage and proves to be stable.展开更多
文摘This paper deals with extensions of higher-order optimality conditions for scalar optimization to multiobjective optimization.A type of directional derivatives for a multiobjective function is proposed,and with this notion characterizations of strict local minima of order k for a multiobjective optimization problem with a nonempty set constraint are established,generalizing the corresponding scalar case obtained by Studniarski[3].Also necessary not sufficient and sufficient not necessary optimality conditions for this minima are derived based on our directional derivatives,which are generalizations of some existing scalar results and equivalent to some existing multiobjective ones.Many examples are given to illustrate them there.
基金Supported by the National Natural Science Foundation of China 1060100910701014+1 种基金10871029)the Foundation of China Academy of Engineering Physics(2007B09008)
文摘Relations of the 3D multi-directional derivatives are studied in this paper.These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained.These relations and expressions play important roles in the meshless finite point method.
基金Supported by the National Natural Science Foundation of China (10871029,11071025)the Foundation of CAEP (2010A0202010)the Foundation of National Key Laboratory of Science and Technology on Computational Physics
文摘In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.
基金Supported by National Natural Science Foundation of China(6067207260832003)Zhejiang Provincial Natural Science Foundation of China(Y106505)
文摘One of the most common image processing tasks involves the removal of noise from images.Noise can be introduced during image capture,during transmission,or during storage.For design purposes,noise sources are frequently approximated by random variables with a known probability distribution.One common noise model corrupts a signal by introducing impulses.And the surface of the image disturbed by impulse noise displays many peaks or vales.According to the characteristic of impulse noise,a novel algorithm is proposed to the detection of impulse noise point from images based on directional derivatives.First,the theory of calculus on directional derivatives is introduced in detail.Then it is applied to the field of image to removing noise with the discrete form derived from its continuous mathematical model.And a number of contrasting simulations illustrate that our algorithm not only can preserve the structure information while removing impulse noise but also can mostly save the gray value of the pixels undisturbed by noise.In addition,the comparisons of the filtering performance for removing impulse noise are analyzed in detail in the case of different noise densities,and also show that the algorithm suggested outperforms the conventional filter algorithms such as mean filter,median filter and so on in speed and impulse noise reduction,especially in random-valued impulse noise reduction.So it is a very good alternative to the existing schemes.
基金Supported by the National Natural Science Foundation of China (No. 10801137)
文摘In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f - g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable functions on a closed convex set in a Euclidean space. As an application, we give a sufficient condition for f - g to have an error bound.
基金Supported by the National Natural Science Foundation of China(No.11371066,11372050)
文摘In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.
基金supported by the National Natural Science Foundation of China(11801257).
文摘In this paper,we introduce a new directional derivative and subgradient of set-valued mappings by using a nonlinear scalarizing function.We obtain some properties of directional derivative and subgradient for cone-convex set-valued mappings.As applications,we present necessary and sufficient optimality conditions for set optimization problems and show that the local weak l-minimal solutions of set optimization problems are the global weak l-minimal solutions of set optimization problems under the assumption that the objective mapping is cone-convex.
文摘Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this paper,we define the Gâteaux directional differentiability of Π_(C).We investigate some properties of the Gâteaux directional differentiability of Π_(C).In particular,if C is a closed ball,or a closed and convex cone(including proper closed subspaces),or a closed and convex cylinder,then,we give the exact representations of the directional derivatives of Π_(C).By comparing the results in[12]and this paper,we see the significant difference between the directional derivatives of the generalized metric projection operator Π_(C) and the Gâteaux directional derivatives of the standard metric projection operator PC.
基金Supported by the National Natural Science Foundation of China(No.60472033)the Key Laboratory Project of Information Science & Engineering of Railway of National Ministry of Railways, China (No.tdxx0510)the Technological Innovation Fund of Excellent Doctorial Candidate of Beijing Jiaotong University,China(No.48007)
文摘Most image interpolation algorithms currently used suffer visually to some extent the effects of blurred edges and jagged artifacts in the image. This letter presents an adaptive feature preserving bidirectional flow process, where an inverse diffusion is performed to enhance edges along the normal directions to the iso-phote lines (edges), while a normal diffusion is done to remove artifacts ('jaggies') along the tangent directions. In order to preserve image features such as edges, angles and textures, the nonlinear diffusion coefficients are locally adjusted according to the first order and the second order directional derivatives of the image. Experimental results on the Lena image demonstrate that our interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional interpolations.
基金National Natural Science Foundation of China(No.61164009)the Science and Technology Research Project,Department of Education of Jiangxi Province,China(No.GJJ14420)Natural Science Foundation of Jiangxi Province,China(No.20132BAB206026)
文摘Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because the multi-valued decision diagram( MDD) can reflect the relationship between the components and the system state bilaterally, it was introduced into the reliability calculation of the multi-state system( MSS). The building method,simplified criteria,and path search and probability algorithm of MSS structure function MDD were given,and the reliability of the system was calculated. The computing methods of importance based on MDD and direct partial logic derivatives( DPLD) were presented. The diesel engine fuel supply system was taken as an example to illustrate the proposed method. The results show that not only the probability of the system in each state can be easily obtained,but also the influence degree of each component and its state on the system reliability can be obtained,which is conducive to the condition monitoring and structure optimization of the system.
文摘BrCF2CF2CH=CHCH2X(x=Cl, OAc, OH) reacted smoothly with alkynes in the presence of BrCo(dmgh)2Py/Zn, giving 4,4,5,5-tetrafluorocyclopentene derivatives in moderate yields.
文摘The absorption spectra of 4f electron transitions of the complexes of neodymium and erbium with 8-hydroxyquinoline-5-sulphonic acid in the presence of diethylamine and ethanol have been measured by normal and third-derivative spectrophotometry. Their molar absorptivities are 70.7 l.mol^(-1).cm^(-1) for Nd and 62.5 l.mol^(-1).cm^(-1) for Er. They are 7.6 times and 14.9 times greater than those of corresponding chlorides, respectively. Use of the third-derivative spectra both eliminates the interference of Ce(Ⅳ) and increases the sensitivity for Nd and Er. Beer,s Law was obeyed from 0-10 ug/ml of Nd and Er. The method has been applied to the determination of neodymium and erbium in rare earth mixtures.
基金supported by the National Natural Science Foundation of China(62373017).
文摘To improve the generalization performance and prediction accuracy of the stochastic configuration network(SCN)model,a novel SCN modeling method is proposed.First,the first-and second-order directional derivatives of the hidden layer output matrix are calculated.The key factors extracted from the directional derivatives are linearly added to the original hidden layer output matrix to formulate a new hidden layer output matrix.Second,a spatial angle adaptive supervisory mechanism is established to improve the quality of the parameter configuration of the hidden layer nodes.The experimental results show that the proposed method improves the generalization performance and prediction accuracy.This work is a beneficial exploration of the standard SCN algorithm.
基金the Natural Science Foundation ofFujian Province of China(S0650021,2006J0215)the National Natural Science Foundation of China(10771086)
文摘This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived.
基金supported in part by the Teaching Reform Research Foundation of Shengli College in China University of Petroleum(East China)(JG201725)the Natural Science Foundation Shandong Province of China(ZR2018PEE009)the Project of Science and Technology of Shandong Universities in China(J17KA044,J17KB061)。
文摘Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.
基金supported by the NSF(61877039)the NSFC/RGC Joint Research Scheme(12061160462 and N City U 102/20)of China+2 种基金the NSF(LY19F020013)of Zhejiang Provincethe Special Project for Scientific and Technological Cooperation(20212BDH80021)of Jiangxi Provincethe Science and Technology Project in Jiangxi Province Department of Education(GJJ211334)。
文摘This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers.A comparison relationship between the excess misclassification error and the excess generalization error is provided;from this,along with the convex analysis theory,a kind of learning rate is derived.The results show that the performance of the classifier is effected by the outliers,and the extent of impact can be controlled by choosing the homotopy parameters properly.
基金the National Natural Science Foundation of China ( 1 0 4 71 0 94) ,the ScienceFoundation of Shanghai Technical Sciences Committee ( 0 2 ZA1 40 70 ) and the Science Foundation ofShanghai Education Committee( 0 2 DK0 6)
文摘This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems.Using the duality theory of the linear programming and convex theory,the generalized directional derivative of the general multicommodity minimal cost flow problems is derived.The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.
文摘This paper considers online classification learning algorithms for regularized classification schemes with generalized gradient. A novel capacity independent approach is presented. It verifies the strong convergence of sizes and yields satisfactory convergence rates for polynomially decaying step sizes. Compared with the gradient schemes, this al- gorithm needs only less additional assumptions on the loss function and derives a stronger result with respect to the choice of step sizes and the regularization parameters.
基金The Graduate Students Innovate Scientific Research Program (YJSCX2008-158HLJ) of Heilongjiang Provincesupported by the Distinguished Young Scholar Foundation (JC200707) of Heilongjiang Province of China
文摘In this paper, we establish a second-order sufficient condition for constrained optimization problems of a class of so called t-stable functions in terms of the first-order and the second-order Dini type directional derivatives. The result extends the corresponding result of [D. Bednarik and K. Pastor, Math. Program. Ser. A, 113（2008）, 283-298] to constrained optimization problems.
基金Supported by National Science Foundation of China,China Postdoctral Science Foundation863 projects。
文摘Through a precise recursion of B-spline bases and the resursive expression of the derivatives of rational surfaces, this paper presents an efficient algorithm for the calculation of NURBS surfaces and all their directional derivatives. The algorithm requires less storage and proves to be stable.
