Optimization Algorithms conjugate gradients

We call this algorithm conjugate gradient re-weighted optimization because the weighting matrix is updated on every iteration (Portniaguine and Zhdanov,... 

HyperChem supplies three types of optimizers or algorithms steepest descent, conjugate gradient (Hetcher-Reeves and Polak-Ribiere), and block diagonal (Newton-Raphson). 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

To determine the optimal parameters, traditional methods, such as conjugate gradient and simplex are often not adequate, because they tend to get trapped in local minima. To overcome this difficulty, higher-order methods, such as the genetic algorithm (GA) can be employed [31,32]. The GA is a general purpose functional minimization procedure that requires as input an evaluation, or test function to express how well a particular laser pulse achieves the target. Tests have shown that several thousand evaluations of the test function may be required to determine the parameters of the optimal fields [17]. This presents no difficulty in the simple, pure-state model discussed above. 

Although the overall cost of the conjugate gradient algorithm may be higher than that of some of the iterative algorithms described in Section III.B, the algorithm allows us easily to restrict the spectral and temporal stmcture of optimal pulses and enables us to incorporate the exact form of the laser-molecule interactions. 

A large number of different algorithms of this type are described in M. R. Hestenes, Conjugate Gradient Methods in Optimization ,... 

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 ... 

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