An iterative algorithm for periodic sylvester matrix equations

Abstract

The problem of solving periodic Sylvester matrix equations is discussed in this paper. A new kind of iterative algorithm is proposed for constructing the least square solution for the equations. The basic idea is to develop the solution matrices in the least square sense. Two numerical examples are presented to illustrate the convergence and performance of the iterative method. 2010 Mathematics Subject Classification. 15A24.

Noise-tolerant continuous-time Zhang neural networks for time-varying Sylvester tensor equations

“…Remark 3.2 If γ 2 = 0, then NTCTZNN2 (16) reduces NTCTZNN1 (14). Though NTCTZNN2 ( 16) is more complicate than NTCTZNN1 ( 14) when γ 2 > 0, Theorems 3.1 and 3.2 indicate that NTCTZNN2 ( 16) is more stable than NTCTZNN1 (14) in this case.…”

“…Wang and Xu [12] developed some iterative algorithms for solving some tensor equations, which were generalized by Huang and Ma [13] to solve STEs. When the above equations are inconsistent, Lv and Zhang [14] designed an iterative algorithm to find the least squares solutions of SMEs. Sun and Wang [7] extended the conjugate gradient method to get the least squares solution with the least Frobenius norm of the generalized periodic SMEs.…”

Adaptive large neighborhood search Algorithm for route planning of freight buses with pickup and delivery

“…The neighborhood of a solution can be obtained by deleting several customer nodes from the current routes (solution) and re-inserting into them the customer nodes. In ALNS, a deletion operator and a re-insertion operator are dynamically selected in each iteration according to their past performance [20]; each operator is associated with a probability. If the operator improves the current solution, the probability will increase, otherwise the probability may decrease; The newly generated solution is accepted if it improves the current solution, otherwise it will be accepted with a probability depending on a temperature and defined according to a Simulation Annealing (SA) rule, the temperature will be gradually decreased with the progress of the algorithm; If the new generated solution is accepted, it will update the current solution for the next iteration.…”

“…The cooling rate of the simulated annealing is denoted by h, and hE (0, 1) is a given parameter. The algorithm returns the best solution after reaching the maximum number of iterations [23,20]. The remove operator d is applied to X current to obtain a partial solution X new .…”

“…The number of periods M is set to 3 or 5. since it is assumed that all freight buses have the same unit distance operating cost, we simply set γ to 1 for all instances. In order to further evaluate the impact of the joint distribution realized by freight buses in section 5.5, we generate the demand of each depot in each period by grouping the customer demands of two private third party logistics companies A and B at the depot in the period, where the demand (both the d'livery and pickup) of each company at each depot in each period is randomly generated from [1,20]. For the penalty coefficient α, because the ratio of α to γ and β to γ affect the tradeoff between the operating costs of the freight buses and the penalty costs for late delivery and pickup as well as the service level to customers, i.e., the percentage of customer demands delivered/picked up on-time, we cannot set α and β too big or too small.…”