Quasi-Newton Method In Nonlinear Optimization Problems

The article has three parts, the first part we present a new inexact line search rule for Quasi-Newton method. The method restricts the Quasi-Newton matrix to a sparse matrix, and uses approximate Quasi-Newton condition to determine a search direction and uses a new line search rule to define a step-size at each iteration. Based on this, we present a new sparse Quasi-Newton method, under this method, we use more information about the objiect function. It avoids the storage and computation of some matrices in its iteration. Under some mild assumptions, we prove the global convergence and the super-linear convergence property of this method. Numerical results show that the new algorithms are efficient.The second part, we propose a new non-monotone step size rule and analyze the global convergence of a Lampariello modified diagonal-sparse Quasi-Newton method. The new step size rule is similar to the Grippo non-monotone step size rule and contains it as a special case. We can choose a larger stepsize in each line search procedure and maintain the global convergence property of our Lampariello modified diagonal-sparse Quasi-Newton method. Numerical results show that the new algorithms are efficient.Last part we present a new memoryless non-quasi-newton method for unconstrained optimization problems. We use more information about the objiect function, make the method convergences smoothly, we also prove the method with a Wolfe-type line search convergences globally if the function is uniformly convex. It avoids the storage and computation of some matrices in its iteration, so that it is suitable for solving large scale optimization problem.