Search numerical evaluation methods

Direct search methods use only function evaluations. They search for the minimum of an objective function without calculating derivatives analytically or numerically. Direct methods are based upon heuristic rules which make no a priori assumptions about the objective function. They tend to have much poorer convergence rates than gradient methods when applied to smooth functions. Several authors claim that direct search methods are not as efficient and robust as the indirect or gradient search methods (Bard, 1974 Edgar and Himmelblau, 1988 Scales, 1986). However, in many instances direct search methods have proved to be robust and reliable particularly for systems that exhibit local minima or have complex nonlinear constraints (Wang and Luus, 1978). 

Much of this progress has been centered around the discovery that energy derivatives can often be explicitly evaluated [3-7], freeing potential energy searches from inefficient minimization algorithms such as uniaxial methods, simplex methods and even numerical derivative methods [8]. 

Though it was already a hot topic 10-20 years ago, as evidenced in section 2 by the presentation of the numerous current methodological developments, the evaluation of the vibrational polarizabilities and hyperpolarizabilities continues to be a challenging task. In addition to the search of reliable methods to include as much as possible the different contributions to the (hyper)polarizabilities, a few works have addressed their amplitude in comparison with their electronic counterpart. 

The method of steepest descent uses only first-order derivatives to determine the search direction. Alternatively, Newton s method for single-variable optimization can be adapted to carry out multivariable optimization, taking advantage of both first- and second-order derivatives to obtain better search directions1. However, second-order derivatives must be evaluated, either analytically or numerically, and multimodal functions can make the method unstable. Therefore, while this method is potentially very powerful, it also has some practical difficulties. 

Let B be an approximation of the Jacobian matrix in x . As previously mentioned, the matrix Bo must be a good approximation of the Jacobian. Consequently, in the case of the quasi-Newton methods also, the Jacobian matrix must be evaluated either analytically or numerically. If the rate of convergence should decrease during the search for the solution, it is advisable to reevaluate the Jacobian. 

Number of Newton method applications 13 Number of Quasi Newton method applications 73 Number of analytical Jacobian evaluations 0 Number of numerical Jacobian evaluations 13 Number of Gradient searches 0 Number of Gauss factorizations 7 Number of LQ factorizations 13 Number of linear system solutions 172... 

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