Optimization conjugate gradient method

Note Because of its neglect of off-diagonal blocks, this optimizer can sometimes oscillate and fail to converge. In this case, use a conjugate gradient method. 

Hestenes, M. R. Conjugate Gradient Methods in Optimization, Springer-Verlag (1980). 

Energy minimization methods that exploit information about the second derivative of the potential are quite effective in the structural refinement of proteins. That is, in the process of X-ray structural determination one sometimes obtains bad steric interactions that can easily be relaxed by a small number of energy minimization cycles. The type of relaxation that can be obtained by energy minimization procedures is illustrated in Fig. 4.4. In fact, one can combine the potential U r) with the function which is usually optimized in X-ray structure determination (the R factor ) and minimize the sum of these functions (Ref. 4) by a conjugated gradient method, thus satisfying both the X-ray electron density constraints and steric constraint dictated by the molecular potential surface. 

Appendix B. Optimal Control Equations for Photodissociation Appendix C. Derivative of the Objective Functional Appendix D. Various Conjugate Gradient Methods... 

The basic difficulty with the steepest descent method is that it is too sensitive to the scaling of/(x), so that convergence is very slow and what amounts to oscillation in the x space can easily occur. For these reasons steepest descent or ascent is not a very effective optimization technique. Fortunately, conjugate gradient methods are much faster and more accurate. 

Lasdon LS, Mitter SK, Waren AD (1967) The conjugate gradient method for optimal control problem IEEE. Trans Automat Control 12 132-138... 

The Polak-Ribiere conjugate gradient method " was used in RSll to perform the non-linear mnlhvariate optimization of the objective function with the weighing factors, Wj = 1 and W2 = 10. 

On the other hand, conjugate gradient methods. are more effective in locating the minimum energy structure. In this approach previous optimization information is utilized. The second and all subsequent descent directions are linear combinations of the previous direction and the current negative gradient of the potential... 

---