Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation

This paper includes a twofold result for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. First we consider a theoretical analysis, where preconditioning is embedded in a strong convergence framework of an NCG method from the literature. Mild conditions to be satisfied by the preconditioners are defined, in order to preserve NCG convergence. As a second task, we also detail the use of novel matrix-free preconditioners for NCG. Our proposals are based on quasi-Newton updates, and either satisfy the secant equation or a secant-like condition at some of the previous iterates. We show that, in some sense, the preconditioners we propose also approximate the inverse of the Hessian matrix. In particular, the structures of our preconditioners depend on low-rank updates used, along with different choices of specific parameters. The low-rank updates are obtained as by-product of NCG iterations. The results of an extended numerical experience using large scale CUTEst problems is reported, showing that our preconditioners can considerably improve the performance of NCG methods

Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table

Preconditioned subspace quasi-newton method for large scale optimization

Subspace quasi-Newton (SQN) method has been widely used in large scale unconstrained optimization problem. Its popularity is due to the fact that the method can construct subproblems in low dimensions so that storage requirement as well as the computation cost can be minimized. However, the main drawback of the SQN method is that it can be very slow on certain types of non-linear problem such as ill-conditioned problems. Hence, we proposed a preconditioned SQN method, which is generally more effective than the SQN method. In order to achieve this, we proposed that a diagonal updating matrix that was derived based on the weak secant relation be used instead of the identity matrix to approximate the initial inverse Hessian. Our numerical results show that the proposed preconditioned SQN method performs better than the SQN method which is without preconditioning

Some diagonal preconditioners for limited memory quasi-Newton method for large Scale optimization

One of the well-known methods in solving large scale unconstrained optimization is limited memory quasi-Newton (LMQN) method. This method is derived from a subproblem in low dimension so that the storage requirement as well as the computation cost can be reduced. In this paper, we propose a preconditioned LMQN method which is generally more effective than the LMQN method dueto the main defect of the LMQN method that it can be very slow on certain type of nonlinear problem such as ill-conditioned problems. In order to do this, we propose to use a diagonal updating matrix that has been derived based on the weak quasi-Newton relation to replace the identity matrix to approximate the initial inverse Hessian. The computational results show that the proposed preconditioned LMQN method performs better than LMQN method that without preconditioning

Exploiting damped techniques for nonlinear conjugate gradient methods

In this paper we propose the use of damped techniques within Nonlinear Conjugate Gradient (NCG) methods. Damped techniques were introduced by Powell and recently reproposed by Al-Baali and till now, only applied in the framework of quasi–Newton methods. We extend their use to NCG methods in large scale unconstrained optimization, aiming at possibly improving the efficiency and the robustness of the latter methods, especially when solving difficult problems. We consider both unpreconditioned and Preconditioned NCG (PNCG). In the latter case, we embed damped techniques within a class of preconditioners based on quasi–Newton updates. Our purpose is to possibly provide efficient preconditioners which approximate, in some sense, the inverse of the Hessian matrix, while still preserving information provided by the secant equation or some of its modifications. The results of an extensive numerical experience highlights that the proposed approach is quite promising

A novel Q-limit guided continuation power flow method for voltage stability analysis

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Voltage security assessment is becoming a more and more important issue due to the fact that electrical power systems are more prone to voltage instability under increased demand, and it can be time-consuming to determine the actual level of voltage security in large power systems. For this reason, this thesis presents a novel method for calculating the margin of voltage collapse that is based on the Continuation Power Flow (CPF) method. The method offers a flexible and reliable solution procedure without suffering from divergence problems even when near the bifurcation point. In addition, the new method accounts for reactive power limits. The algorithmic continuation steps are guided by the prediction of Q-limit breaking point. A Lagrange polynomial interpolation formula is used in this method in order to find the Q-limit breaking point indices that determine when the reactive power output of a generator has reached its limit. The algorithmic continuation steps will then be guided to the closest Q-limit breaking point, consequently reducing the number of continuation steps and saving computational time. The novel method is compared with alternative conventional and enhanced CPF methods. In order to improve CPF further, studies comparing the performance of using direct and iterative solvers in a power flow calculation have also been performed. I first attempt to employ the column approximate minimum degree (AMD) ordering scheme to reset the permutation of the coefficient matrix, which decreases the number of iterations required by iterative solvers. Finally, the novel method has been applied to a range of power system case studies including a 953 bus national grid transmission case study. The results are discussed in detail and compared against exiting CPF methods