New DY-HS hybrid conjugate gradient algorithm for solving optimization problem of unsteady partial differential equations with convection term

Abstract

This paper studies an optimization problem for the unsteady partial differential equations (PDEs) with convection term, widely used in continuous casting process. Considering the change of casting speed, a dynamic optimization method based on new DY-HS hybrid conjugate gradient algorithm (DY-HSHCGA) is proposed. In the DY-HSHCGA, the Dai–Yuan and the Hestenes–Stiefel conjugate gradient algorithms are convex combined, and a new conjugate parameter θk is obtained through the condition of quasi-Newton direction. Moreover, Lipschitz continuity of the gradient of cost function, as an important conditions for convergence, is analyzed in this paper. On the basis on this condition, the global convergence of DY-HSHCGA is proved. Finally, the effectiveness of DY-HSHCGA is verified by some instances from the steel plant. Comparing with other algorithms DY-HSHCGA obviously accelerates the convergence rate and reduces the number of iteration. The optimizer based on the DY-HSHCGA shows a more stable results.