Stochastic Linear-Quadratic Control Problem With Dual Interior Point Algorithms

The stochastic linear-quadratic (LQ) control problem is considered in this paper.Under the condition that the stochastic LQ control problem is mean-square stabilizable, it gives rise to a stochastic algebraic Riccati equation (SARE) which can be transformed into a semidefinite programming (SDP). In this paper, by exploiting the inherent structure of the SDP problem, an effective solving method is proposed for it.Firstly, the background and development of the stochastic LQ control problem are introduced. Then the semidefinite programming problem is described and dual logarithmic barrier methods for solving it are introduced.Then, the dual interior point algorithm for stochastic linear-quadratic control problem is stated in detail. Firstly, the SDP is transformed into the standard form, and it is proved that this problem can be solved by the interior point algorithm. However, large-scale and dense linear equations need to be solved during each step of iteration. For small sized problem, it can be solved directly. However, for the much larger dimension, it is difficult to solve it efficiently, because it requires much more storage for the data and computation. Here, the symmetric positive definiteness of the coefficient matrix is proved, so it can be solved by the conjugate gradient methods. By taking advantage of particular sparsity of matrices in the SDP, an effective method without using too much storage is given.Finally, some numerical examples which are mean-square stabilizable are constructed and the feasibility of the proposed methods is proved by doing the numerical tests.