Trust Region Method Of New Conic Model For Linearly Constrained Optimization

In 1996, the trust region methods with conic model for unconstrained optimization was proposed, and the new trust-region subproblem involving a conic model was proposed in 2005. The new subproblem eliminates the restriction on the horizon vector, so the conic model can well approximate to the original objective function. In this paper, we give the method to solve the linearly constrained optimization problems on the base of the new trust-region subproblem.First, we convert the linearly equality constrained optimization problem to the unconstrained optimization problem, and use the frame of the trust region algorithm with new conic model to solve the unconstrained optimization problem. Then, we return to the original linearly constrained optimization problem and give the solution. Based on these results a trust region algorithm with new conic model for the linearly equality constrained optimization problem is proposed. In the paper, the properties of the algorithm are analyzed and the convergence of the algorithm is proved. Then we use the technique of active set in the projected gradient method, and generalize the algorithm to solve the linearly constrained optimization problem. Finally, we give some numerical results of the algorithm proposed compared with the classic projected gradient method.These results show that the algorithm with new conic model to solve the linearly constrained optimization is worth to research.