Gradient Minimization Methods

The gradient search methods require derivatives of the objective functions whereas the direct methods are derivative-free. The derivatives may be available analytically or otherwise they are approximated in some way. It is assumed that the objective function has continuous second derivatives, whether or not these are explicitly available. Gradient methods are still efficient if there are some discontinuities in the derivatives. On the other hand, direct search techniques, which use function values, are more efficient for highly discontinuous functions. 

The basic problem is to search for the parameter vector k that minimizes S(k) by following an iterative scheme, i.e., 

The need to utilize an iterative scheme stems from the fact that it is usually impossible to find the exact solutions of the equation that gives the stationary points of S(k) (Peressini et al. 1988), 

Vector VS(k) contains the first partial derivatives of the objective function S(k) with respect to k and it is often called the gradient vector. For simplicity, we denoted it as g(k) in this chapter. 

In order to find the solution, one starts from an initial guess for the parameters, k(0)=[ k ° kj0, .kj,° ]T. There is no general rule to follow in order to obtain an initial guess. However, some heuristic rules can be used and they are discussed in Chapter 8. 

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Example Compare the steps of a conjugate gradient minimization with the steepest descent method. Amolecular system can reach a potential minimum after the second step if the first step proceeds from A to B. If the first step is too large, placing the system at D, the second step still places the system near the minimum(E) because the optimizer remembers the penultimate step. 

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. 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

Conjugate gradient-type methods form a class of minimization procedures that accomplish two objectives ... 

The constrained minimization problem stated above may be transformed into a form well-suited to gradient projection methods of nonlinear programming by making the following substitution ... 

The minimal cost design problem formulated above was solved by Bickel et al. (B7) using the generalized reduced gradient (GRG) method of Abadie and Guigou (H4). If x and u are vectors of state and decision (independent) variables and u) is the objective function in a minimization subject to constraints [Eq. (90)], then the reduced gradient d/du is given by... 

In addition, the use of fast gradients elution mode has become the bioanalytical mainstream as a possible way to improve peak parameters (shape and symmetry) and to minimize method development time, especially for the multi-analytes methods. 

The CPR method is the most difficult method to implement, beeause of the complex rules for adding or rejecting points on the path. It is also the least efficient of the methods tested. It does, however, converge quickly to the saddle point once it is close, as is evident from comparing table 1 and 2. This is probably because of the use of the conjugate gradient minimization which is quite efficient. 

In a more recent study (Ref 16), Merzhanov s school used high dilution with inert materials to study the kinetics of exothermic reactions over a wider temp range than was previously feasible. Dilution prevents self-ignition of the studied sample and also minimizes temp gradients. The method used is an adaptation of DTA (differential thermal analysis) in which temp... 

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