Generalized Positive Semidefinite Matrix Least Squares Problem And Its Applications

Based on many applications,we present a new class of the generalized positive semidefinite least squares problem(abbreviated as GPSLSP).With the help of the framework of the iterative algorithm in[1],we design a new iterative algorithm for GPSLSP in this paper.In addition,in order to illustrate the common use of GPSLSP,a least squares problem occurred in the working process of Kalman filter,which is selected from the optimization control field,is analyzed and solved.The article is divided into four parts:In the introduction,we mainly introduce the development and applications of GPSLSP.In the first chapter,we construct GPSLSP,and GPSLSP can be found in a wide range of application fields of real world.For GPSLSP,we present an iterative algorithm related to corrected approximate point,and this algorithm can be regard?ed as the generalization of the algorithm in[1].Finally,we give the convergence analysis and the preliminary numerical results for this algorithm.In the second chapter,we introduce a least squares problem related to gener-alized autocovariance which exist in the process of optimizing the Kalman filter,and this problem is transformed into GPSLSP in chapter 1.Then we use the projection-based iterative algorithm in[2]to solve this least square problem.Fi?nally,by considering Van der Vusse reaction system in chemical engineering,we give the corresponding experimental results.The experimental results show that,for the case of Van der Vusse reaction system,the projection-based iterative algo-rithm does not have a higher efficiency than the interior point method in[3].But the implement of interior point method is strongly dependent on the selection of the initial point and parameter,whereas the projection-based iterative algorithm does not have these defects.More precisely,when the initial covariance is unknown in the Kalman filter,the numerical result obtained by projection-based iterative algorith-m with the corresponding initial point chosen as any randomly-generated positive semidefinite matrix is satisfied.Furthermore,when the least squares problem is generalized to a more general case,i.e.,the least squares problem with its partial parameters axe selected randomly,the corresponding experimental results show that the projection-based iterative algorithm is more stable.The corresponding conclusions and some questions studied in the future are summarized in the third chapter.