We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical beha...We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.展开更多
For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers （ADMM） has exh...For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers （ADMM） has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function~ ADMM or its direct extended version may not converge. In this paper, we consider the multi-block sepa.rable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.展开更多
The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requ...The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.展开更多
The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization probl...The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11271195 and 11271196)the Project of Graduate Education Innovation of Jiangsu Province,China(Grant No.CXZZ12-0385)
文摘We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.
文摘For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers （ADMM） has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function~ ADMM or its direct extended version may not converge. In this paper, we consider the multi-block sepa.rable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.
基金supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20210267)supported by the National Natural Science Foundation of China (Grant No.11971239)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No.21KJA110002)supported by the National Natural Science Foundation of China (Grant Nos.12131004,11625105).
文摘The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.
文摘The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.
