Conjugate gradient search method

Recently, a semiclassical formulation of the optimal control theory has been derived [23, 24] by combining the conjugate gradient search method... 

Global Optimization Procedure and Conjugate Gradient Search Method... 

To find the optimal field, we employ the simplest form of global optimization procedure with the iterational conjugate gradient search method [7,8]. At each iteration of this algorithm, the correction to the optimized laser field is determined from the following equations ... 

The recurrence relations for the preconditioned conjugate gradient (PCG) method can be derived from Algorithm [A2] after substituting x = M 1/2x and r + M1/2r. New search vectors d = M 1/2d can be used to derive the iteration process, and then the tilde modifiers dropped. The PCG method becomes the following iterative process. 

One can use a parabolic line search also (Fletcher, 1985) to improve the convergence rate of the regularized conjugate gradient (RCG) method. 

The Estimates, Derivatives and Search parameters can be changed to optimize the solution process. The Search parameter specifies which gradient search method to use the Newton method requires more memory but fewer iterations, the Conjugate method requires less memory but more iterations. The Derivatives parameter specifies how the gradients for the search are calculated the Central derivatives method requires more calculations but may be helpful if the Solver reports that it is unable to find a solution. The Estimates parameter determines the method by which new estimates of the coefficients are obtained from previous values the Quadratic method may improve results if the system is highly nonlinear. 

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

The conjugate gradient method is one of the oldest in the Retcher-Reeves approach, the search direction is given by... 

There are different variants of the conjugate gradient method each of which corresponds to a different choice of the update parameter C - Some of these different methods and their convergence properties are discussed in Appendix D. The time has been discretized into N time steps (f, = / x 8f where i = 0,1, , N — 1) and the parameter space that is being searched in order to maximize the value of the objective functional is composed of the values of the electric field strength in each of the time intervals. 

In generating a third iterate for the conjugate-gradient method, we now estimate the search direction by VEfe) but insist that the search direction is orthogonal to both do and di. This idea is then repeated for subsequent iterations. 

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