Quasi-Newton type method via weak secant equations for unconstrained optimization

In this thesis, variants of quasi-Newton methods are developed for solving unconstrained optimization problems. These quasi-Newton type methods are differed from the standard quasi-Newton methods in the calculation and storage of the approximate Hessian matrix at every iteration. Using the concept of least change updating strategy, two updating formulas are derived by the mean of variational problem, via weak secant equation and some other non-secant equations. The convergence analysis for these methods are presented under inexact line search with Armijo condition. Motivated by the idea of memoryless scheme, the proposed formulas are further modified such that only vector computations and storage are required at every iteration. In other words, these memoryless updating formulas can provide some approximation to the quasi-Newton direction in matrix free setting. Armijo condition is implemented in the algorithms to generate monotone property in each iteration. The convergence analysis is presented for these memoryless methods under some standard assumptions. Numerical experiments are carried out using standard test problems and show that the proposed methods are superior to some existing conjugate gradient methods, in terms of iterations and function evaluations required to reach the optimal solutions. The possible variants of matrix free quasi-Newton methods are further explored, using the weak secant equation. A diagonal updating formula is derived by the mean of minimizing the magnitude of the updating matrix under the Frobenius norm. This formula is then generalized by using weighted Frobenius norm in the derivation, which gives a new version of diagonal updating formula where the previous diagonal matrix is chosen to be the weighting matrix, for the weighted diagonal updating formula. The convergence analysis is established for these formulas under two sets of line search strategies, namely monotone and non-monotone line search. Numerical experiments are conducted to test their effectiveness in solving standard test set. The results obtained indicate that the diagonal updating formula is superior to its weighted version and some conjugate gradient methods. The formula works best when monotone line search is implemented, which often requires fewer number of iterations and function evaluations to obtain the solutions. Overall, numerical results prove that these proposed methods are superior in terms of number of iterations and function evaluations. Furthermore, the diagonal updating formulas with non-monotone line search are more efficient than some classical conjugate gradient methods in all three performance measures, including the CPU time.

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