Large-scale Optimization And Nonlinear Equations Problem Of Multivariate Spectral Gradient Algorithm And Its Application

Abstract： In this thesis, we first present a nonmonotone gradient-type method forunconstraint optimization problems. Based on nonmonotone line search, we establishthe global convergence of the proposed method. Furthermore, we extend this method tobound constrained optimization, and propose a new projected gradient method. Undermild conditions, we show that the method is globally convergent.In Chapter3, we develop a multivariate spectral gradient method for nonlinearsystems of monotone equations. Under very mild conditions, we prove that the methodis globally convergent even if the system of equations is not smooth. An attractive fea-ture of the method is that the distance between iterates and the solution set is decreasingmonotonically. Compared with the Gauss-Newton-type methods, moreover, it requiresless computer memory, hence it is very suitable for large scale problems. Taking thenumerical performance of the method into account, we proposed a modified multivari-ate spectral gradient method. Numerical experiments show that these methods is veryeffective and promising.In order to solve the nonlinear systems of monotone equations with convex con-straints, in Chapter4, we develop a multivariate spectral gradient projection method.Under suitable conditions, we get the global convergence of the method. The methodis an extension of the MSG method in Chapter3. Numerical experiments indicate thatthe method outperforms the Gauss-Newton based projection method greatly.Finally, we consider the application of spectral gradient method in image denois-ing. Combined the MSG method and Adaptive spectral gradient method, under mildconditions, we present a globally convergent hybrid spectral gradient (HSG) method.In the first phase of the two-phase method, adaptive median filter is used to detect noisypixels. In the second phase, a minimization problem is solved by HSG method. Com-paring the results of removing noise, our method is better than PRP conjugate gradientmethod with much less CPU time, especially when the impulse noise ratio is high.