In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo (DGALs) was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discus...In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo (DGALs) was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discussed between the uneonstrained minimizers of DGALs on the product space of problem variables and multipliers, and the solutions of the eonstrained problem and the corresponding values of the Lagrange multipliers. The resulting properties indicate more precisely that this class of DGALs is exact multiplier penalty functions. Therefore, a solution of the equslity-constralned problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of a DGAL on the product space of problem variables and multipliers.展开更多
An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstr...An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.展开更多
This paper formulates a two-dimensional strip packing problem as a non- linear programming (NLP) problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving t...This paper formulates a two-dimensional strip packing problem as a non- linear programming (NLP) problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving this NLP problem is given to find exact solutions to strip-packing problems involving up to 10 items. Approximate solutions can be found for big-sized problems by decomposing the set of items into small-sized blocks of which each block adopts the proposed numerical algorithm. Numerical results show that the approximate solutions to big-sized problems obtained by this method are superior to those by NFDH, FFDH and BFDH approaches.展开更多
In this paper, we propose and analyze an accelerated augmented Lagrangian method(denoted by AALM) for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O(1/k^2) whil...In this paper, we propose and analyze an accelerated augmented Lagrangian method(denoted by AALM) for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O(1/k^2) while the convergence rate of the classical augmented Lagrangian method(ALM) is O1 k. Numerical experiments on the linearly constrained 1-2minimization problem are presented to demonstrate the effectiveness of AALM.展开更多
In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity ar...In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.展开更多
In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and...In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.展开更多
Recently, dictionary learning(DL) based methods have been introduced to compressed sensing magnetic resonance imaging(CS-MRI), which outperforms pre-defined analytic sparse priors. However, single-scale trained dictio...Recently, dictionary learning(DL) based methods have been introduced to compressed sensing magnetic resonance imaging(CS-MRI), which outperforms pre-defined analytic sparse priors. However, single-scale trained dictionary directly from image patches is incapable of representing image features from multi-scale, multi-directional perspective, which influences the reconstruction performance. In this paper, incorporating the superior multi-scale properties of uniform discrete curvelet transform(UDCT) with the data matching adaptability of trained dictionaries, we propose a flexible sparsity framework to allow sparser representation and prominent hierarchical essential features capture for magnetic resonance(MR) images. Multi-scale decomposition is implemented by using UDCT due to its prominent properties of lower redundancy ratio, hierarchical data structure, and ease of implementation. Each sub-dictionary of different sub-bands is trained independently to form the multi-scale dictionaries. Corresponding to this brand-new sparsity model, we modify the constraint splitting augmented Lagrangian shrinkage algorithm(C-SALSA) as patch-based C-SALSA(PB C-SALSA) to solve the constraint optimization problem of regularized image reconstruction. Experimental results demonstrate that the trained sub-dictionaries at different scales, enforcing sparsity at multiple scales, can then be efficiently used for MRI reconstruction to obtain satisfactory results with further reduced undersampling rate. Multi-scale UDCT dictionaries potentially outperform both single-scale trained dictionaries and multi-scale analytic transforms. Our proposed sparsity model achieves sparser representation for reconstructed data, which results in fast convergence of reconstruction exploiting PB C-SALSA. Simulation results demonstrate that the proposed method outperforms conventional CS-MRI methods in maintaining intrinsic properties, eliminating aliasing, reducing unexpected artifacts, and removing noise. It can achieve comparable performance of reconstruction with the state-of-the-art methods even under substantially high undersampling factors.展开更多
Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construc...Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.展开更多
An augmented Lagrangian trust region method with a bi=object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each ite...An augmented Lagrangian trust region method with a bi=object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.展开更多
In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–22...In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–2261].We use fast algorithms to solve the subproblems,such as the dual projection methods,primal-dual methods and augmented Lagrangian methods.With a special penalization term,some special algorithms are presented.Numerical experiments are given to demonstrate the performance of the proposed methods.We successfully show that our algorithms are effective and efficient due to two reasons:the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately(even 2 inner iterations of the subproblem are enough).It is also observed that better PSNR values are produced using the new algorithms.展开更多
In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficie...In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.展开更多
In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final t...In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examlales demonstrate the effectiveness of the algorithm.展开更多
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feas...This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feasible augmented La-grangian,which can be solved by the augmented Lagrangian method in an infinite dimensional setting.Based on this,by expressing the primal and dual variables with two individual deep neural network functions,we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimiza-tion method together with a projection technique.Compared to the traditional penalty method,the new method admits two main advantages:i)the choice of the penalty parameter isflexible and robust,and ii)the numerical solution is more accurate in the same magnitude of computational cost.As typical applications,we apply the new ap-proach to solve elliptic problems and(nonlinear)eigenvalue problems with essential boundary conditions,and numerical experiments are presented to show the effective-ness of the new method.展开更多
.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under....In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.展开更多
This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials.The mathematical optimization formulation is established under the constraints of individual volu...This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials.The mathematical optimization formulation is established under the constraints of individual volume fraction of constituent phase or total mass,as well as the local volume fraction of all phases.The original optimization problem with numerous constraints is converted into a box-constrained optimization problem by incorporating all constraints to the augmented Lagrangian function,avoiding the parameter dependence in the conventional aggregation process.Furthermore,the local volume percentage can be precisely satisfied.The effects including the globalmass bound,the influence radius and local volume percentage on final designs are exploited through numerical examples.The numerical results also reveal that porous structures keep a balance between the bulk design and periodic design in terms of the resulting compliance.All results,including those for irregular structures andmultiple volume fraction constraints,demonstrate that the proposedmethod can provide an efficient solution for multiple material infill structures.展开更多
The manuscript presents an augmented Lagrangian—fast projected gradient method (ALFPGM) with an improved scheme of working set selection, pWSS, a decomposition based algorithm for training support vector classificati...The manuscript presents an augmented Lagrangian—fast projected gradient method (ALFPGM) with an improved scheme of working set selection, pWSS, a decomposition based algorithm for training support vector classification machines (SVM). The manuscript describes the ALFPGM algorithm, provides numerical results for training SVM on large data sets, and compares the training times of ALFPGM and Sequential Minimal Minimization algorithms (SMO) from Scikit-learn library. The numerical results demonstrate that ALFPGM with the improved working selection scheme is capable of training SVM with tens of thousands of training examples in a fraction of the training time of some widely adopted SVM tools.展开更多
In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augment...In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.展开更多
In this paper,we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential.We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued f...In this paper,we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential.We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency,which used by the concepts of supremum and infimum of a set.We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map.Finally,we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem.In addition,we consider a generalized vector variational inequality as an application of abstract subdifferential.展开更多
Orthogonal nonnegative matrix factorization(ONMF)is widely used in blind image separation problem,document classification,and human face recognition.The model of ONMF can be efficiently solved by the alternating direc...Orthogonal nonnegative matrix factorization(ONMF)is widely used in blind image separation problem,document classification,and human face recognition.The model of ONMF can be efficiently solved by the alternating direction method of multipliers and hierarchical alternating least squares method.When the given matrix is huge,the cost of computation and communication is too high.Therefore,ONMF becomes challenging in the large-scale setting.The random projection is an efficient method of dimensionality reduction.In this paper,we apply the random projection to ONMF and propose two randomized algorithms.Numerical experiments show that our proposed algorithms perform well on both simulated and real data.展开更多
This paper proposes a new multi-area framework for unbalanced active distribution network(ADN) state estimation. Firstly, an innovative three-phase distributed generator(DG) model is presented to take the asymmetric c...This paper proposes a new multi-area framework for unbalanced active distribution network(ADN) state estimation. Firstly, an innovative three-phase distributed generator(DG) model is presented to take the asymmetric characteristics of DG three-phase outputs into consideration. Then a feasible method to set pseudo-measurements for unmonitored DGs is introduced. The states of DGs,together with the states of alternating current(AC) buses in ADNs, were estimated by using the weighted least squares(WLS) method. After that, the ADN was divided into several independent subareas. Based on the augmented Lagrangian method, this work proposes a fully distributed three-phase state estimator for the multi-area ADN.Finally, from the simulation results on the modified IEEE123-bus system, the effectiveness and applicability of the proposed methodology have been investigated and discussed.展开更多
文摘In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo (DGALs) was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discussed between the uneonstrained minimizers of DGALs on the product space of problem variables and multipliers, and the solutions of the eonstrained problem and the corresponding values of the Lagrange multipliers. The resulting properties indicate more precisely that this class of DGALs is exact multiplier penalty functions. Therefore, a solution of the equslity-constralned problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of a DGAL on the product space of problem variables and multipliers.
文摘An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.
基金State Foundstion of Ph.D Units of China(2003-05)under Grant 20020141013the NNSF(10471015)of Liaoning Province,China.
文摘This paper formulates a two-dimensional strip packing problem as a non- linear programming (NLP) problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving this NLP problem is given to find exact solutions to strip-packing problems involving up to 10 items. Approximate solutions can be found for big-sized problems by decomposing the set of items into small-sized blocks of which each block adopts the proposed numerical algorithm. Numerical results show that the approximate solutions to big-sized problems obtained by this method are superior to those by NFDH, FFDH and BFDH approaches.
基金Supported by Fujian Natural Science Foundation(2016J01005)Strategic Priority Research Program of the Chinese Academy of Sciences(XDB18010202)
文摘In this paper, we propose and analyze an accelerated augmented Lagrangian method(denoted by AALM) for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O(1/k^2) while the convergence rate of the classical augmented Lagrangian method(ALM) is O1 k. Numerical experiments on the linearly constrained 1-2minimization problem are presented to demonstrate the effectiveness of AALM.
基金This work is partially supported by the Postdoctoral Fellowship of The Hong Kong Polytechnic Universitythe Research Grants Council of Hong Kong(PolyU B-Q890)
文摘In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.
文摘In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.
基金Project supported by the National Natural Science Foundation of China(Nos.61175012 and 61201422)the Natural Science Foundation of Gansu Province of China(No.1208RJ-ZA265)+1 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.2011021111-0026)the Fundamental Research Funds for the Central Universities of China(Nos.lzujbky-2015-108 and lzujbky-2015-197)
文摘Recently, dictionary learning(DL) based methods have been introduced to compressed sensing magnetic resonance imaging(CS-MRI), which outperforms pre-defined analytic sparse priors. However, single-scale trained dictionary directly from image patches is incapable of representing image features from multi-scale, multi-directional perspective, which influences the reconstruction performance. In this paper, incorporating the superior multi-scale properties of uniform discrete curvelet transform(UDCT) with the data matching adaptability of trained dictionaries, we propose a flexible sparsity framework to allow sparser representation and prominent hierarchical essential features capture for magnetic resonance(MR) images. Multi-scale decomposition is implemented by using UDCT due to its prominent properties of lower redundancy ratio, hierarchical data structure, and ease of implementation. Each sub-dictionary of different sub-bands is trained independently to form the multi-scale dictionaries. Corresponding to this brand-new sparsity model, we modify the constraint splitting augmented Lagrangian shrinkage algorithm(C-SALSA) as patch-based C-SALSA(PB C-SALSA) to solve the constraint optimization problem of regularized image reconstruction. Experimental results demonstrate that the trained sub-dictionaries at different scales, enforcing sparsity at multiple scales, can then be efficiently used for MRI reconstruction to obtain satisfactory results with further reduced undersampling rate. Multi-scale UDCT dictionaries potentially outperform both single-scale trained dictionaries and multi-scale analytic transforms. Our proposed sparsity model achieves sparser representation for reconstructed data, which results in fast convergence of reconstruction exploiting PB C-SALSA. Simulation results demonstrate that the proposed method outperforms conventional CS-MRI methods in maintaining intrinsic properties, eliminating aliasing, reducing unexpected artifacts, and removing noise. It can achieve comparable performance of reconstruction with the state-of-the-art methods even under substantially high undersampling factors.
基金supported by the NNSF of China grants 11526110,11271069,61362036 and 61461032,the 863 Program of China grant 2015AA01A302the Open Research Fund of Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing(2016WICSIP013)the Youth Foundation of Nanchang Institute of Technology(2014KJ021).
文摘Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.
文摘An augmented Lagrangian trust region method with a bi=object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.
基金The authors’research was supported by MOE IDM project NRF2007IDM-IDM002-010,SingaporeThe first author was partially supported by PHD Program Scholarship Fund of ECNU with Grant No.2010026Overseas Research Fund of East China Normal University,China.Discussions with Dr.Zhifeng Pang,Dr.Haixia Liang and Dr.Yuping Duan are helpful.
文摘In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–2261].We use fast algorithms to solve the subproblems,such as the dual projection methods,primal-dual methods and augmented Lagrangian methods.With a special penalization term,some special algorithms are presented.Numerical experiments are given to demonstrate the performance of the proposed methods.We successfully show that our algorithms are effective and efficient due to two reasons:the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately(even 2 inner iterations of the subproblem are enough).It is also observed that better PSNR values are produced using the new algorithms.
基金Chao Ding’s research was supported by the National Natural Science Foundation of China(Nos.11671387,11531014,and 11688101)Beijing Natural Science Foundation(No.Z190002)+6 种基金Xu-Dong Li’s research was supported by the National Key R&D Program of China(No.2020YFA0711900)the National Natural Science Foundation of China(No.11901107)the Young Elite Scientists Sponsorship Program by CAST(No.2019QNRC001)the Shanghai Sailing Program(No.19YF1402600)the Science and Technology Commission of Shanghai Municipality Project(No.19511120700)Xin-Yuan Zhao’s research was supported by the National Natural Science Foundation of China(No.11871002)the General Program of Science and Technology of Beijing Municipal Education Commission(No.KM201810005004).
文摘In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.
基金supported by the Natural Science Foundation of China (No. 60974039)the National Science and Technology Major Project (No.2008ZX05011)
文摘In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examlales demonstrate the effectiveness of the algorithm.
基金supported by the National Key Research and Development Project(Grant No.2020YFA0709800)NSFC(Grant No.12071289)+4 种基金Shanghai Municipal Science and Technology Major Project(2021SHZDZX0102)supported by the National Key R&D Program of China(2020YFA0712000)NSFC(under grant numbers 11822111,11688101)the science challenge project(No.TZ2018001)youth innovation promotion association(CAS).
文摘This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions.To this end,wefirst reformulate the original problem into a minimax problem corresponding to a feasible augmented La-grangian,which can be solved by the augmented Lagrangian method in an infinite dimensional setting.Based on this,by expressing the primal and dual variables with two individual deep neural network functions,we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimiza-tion method together with a projection technique.Compared to the traditional penalty method,the new method admits two main advantages:i)the choice of the penalty parameter isflexible and robust,and ii)the numerical solution is more accurate in the same magnitude of computational cost.As typical applications,we apply the new ap-proach to solve elliptic problems and(nonlinear)eigenvalue problems with essential boundary conditions,and numerical experiments are presented to show the effective-ness of the new method.
基金supported by the 10 plus 10 project of Tongji University(No.4260141304/004/010).
文摘.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.
基金This study is financially supported by StateKey Laboratory of Alternate Electrical Power System with Renewable Energy Sources(Grant No.LAPS22012).
文摘This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials.The mathematical optimization formulation is established under the constraints of individual volume fraction of constituent phase or total mass,as well as the local volume fraction of all phases.The original optimization problem with numerous constraints is converted into a box-constrained optimization problem by incorporating all constraints to the augmented Lagrangian function,avoiding the parameter dependence in the conventional aggregation process.Furthermore,the local volume percentage can be precisely satisfied.The effects including the globalmass bound,the influence radius and local volume percentage on final designs are exploited through numerical examples.The numerical results also reveal that porous structures keep a balance between the bulk design and periodic design in terms of the resulting compliance.All results,including those for irregular structures andmultiple volume fraction constraints,demonstrate that the proposedmethod can provide an efficient solution for multiple material infill structures.
文摘The manuscript presents an augmented Lagrangian—fast projected gradient method (ALFPGM) with an improved scheme of working set selection, pWSS, a decomposition based algorithm for training support vector classification machines (SVM). The manuscript describes the ALFPGM algorithm, provides numerical results for training SVM on large data sets, and compares the training times of ALFPGM and Sequential Minimal Minimization algorithms (SMO) from Scikit-learn library. The numerical results demonstrate that ALFPGM with the improved working selection scheme is capable of training SVM with tens of thousands of training examples in a fraction of the training time of some widely adopted SVM tools.
基金partially supported by the NSFC Project Nos.12001529,12025104,11871298,81930119.
文摘In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.
基金supported by National Science Foundation of China(No.11401487)the Education Department of Shaanxi Province(No.17JK0330)+1 种基金the Fundamental Research Funds for the Central Universities(No.300102341101)State Key Laboratory of Rail Transit Engineering Informatization(No.211934210083)。
文摘In this paper,we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential.We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency,which used by the concepts of supremum and infimum of a set.We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map.Finally,we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem.In addition,we consider a generalized vector variational inequality as an application of abstract subdifferential.
基金the National Natural Science Foundation of China(No.11901359)Shandong Provincial Natural Science Foundation(No.ZR2019QA017)。
文摘Orthogonal nonnegative matrix factorization(ONMF)is widely used in blind image separation problem,document classification,and human face recognition.The model of ONMF can be efficiently solved by the alternating direction method of multipliers and hierarchical alternating least squares method.When the given matrix is huge,the cost of computation and communication is too high.Therefore,ONMF becomes challenging in the large-scale setting.The random projection is an efficient method of dimensionality reduction.In this paper,we apply the random projection to ONMF and propose two randomized algorithms.Numerical experiments show that our proposed algorithms perform well on both simulated and real data.
基金supported by National Natural Science Foundation of China(No.51277052)
文摘This paper proposes a new multi-area framework for unbalanced active distribution network(ADN) state estimation. Firstly, an innovative three-phase distributed generator(DG) model is presented to take the asymmetric characteristics of DG three-phase outputs into consideration. Then a feasible method to set pseudo-measurements for unmonitored DGs is introduced. The states of DGs,together with the states of alternating current(AC) buses in ADNs, were estimated by using the weighted least squares(WLS) method. After that, the ADN was divided into several independent subareas. Based on the augmented Lagrangian method, this work proposes a fully distributed three-phase state estimator for the multi-area ADN.Finally, from the simulation results on the modified IEEE123-bus system, the effectiveness and applicability of the proposed methodology have been investigated and discussed.