Tensor Methods For Solving Unconstrained Optimization Problems

In this thesis, we consider the tensor method for solving unconstrained optimization. First, we construct a trust region method for solving an optimization problem with the objective function being a quartic tensor model, and then apply this method to an unconstrained optimization problem. A stronger convergence property of the method is established in this thesis. The thesis consists of five chapters.Chapter 1 is an introduction part. First, we present a tensor model used in this thesis and give the current development of tensor methods for solving such as systems of equations and (un)constrained optimization problems. Second, we introduce the main task based on the development. Third, we introduce how to form and solve the tensor model.In Chapter 2, we consider the methods on how to get a descent direction and the optimal step size of the tensor model at its non-critical point in trust region subproblem whose objective function is a quartic tensor model. First, we expand the tensor model using Taylor formula at its non-critical point so that it is converted into a new tensor model with respect to its descent direction and step size . Second, we proceed our discussion according to the cases that the (non-)critical point of the original tensor model belongs to the interior of the trust region or the bound of the trust region, and establish an equivalent reformulation of the constrained conditions. Third, we consider the methods to select a descent direction of the original tensor model at its non-critical point. Finally, we use the descentdirection obtained and equivalently reformulate the constrained conditions of the original constrained optimization simultaneously, so the original constrained optimization is converted into a constrained optimization of one-dimension function with respect to step size. Thus the optimal step size of the original tensor model can be obtained via solving the corresponding constrained optimization problem.In Chapter 3, we consider the methods on how to get a descent direction and the optimal step size of the tensor model at its critical point in trust region subproblem whose objective function is a quartic tensor model. First, we expand the tensor model using Taylor formula at its critical point so that it is converted into a new tensor model with respect to its descent direction and step size. Second, according to the cases that the Hessian matrix of the original tensor model having some negative eigenvalues and being some zero eigenvalues without negative eigenvalues, we consider the methods to select a descent direction of the original tensor model at its critical point. Third, we use the descent direction obtained and equivalently reformulate the constrained conditions of the original constrained optimization simultaneously, so the original constrained optimization is converted into a constrained optimization of one-dimension function with respect to step size. Thus the optimal step size of the original tensor model can be obtained via solving the corresponding constrained optimization problem.In Chapter 4, we propose an algorithm for solving the trust region sub-problem whose objective function is a quartic tensor model using the discussion of the preceding two chapters, furthermore, we analyze the convergence of the algorithm.In Chapter 5, we apply the algorithm obtained in Chapter 4 to the trust region model of the unconstrained optimization problem so that the tensor method of unconstrained optimization is established. The convergence analysis of the method is given. Compared with generic trust region algorithm, this method...