Global minimum function

The most ambitious approaches to the protein folding problem attempt to solve it from firs principles (ab initio). As such, the problem is to explore the coirformational space of th molecule in order to identify the most appropriate structure. The total number of possibl conformations is invariably very large and so it is usual to try to find only the very lowes energy structure(s). Some form of empirical force field is usually used, often augmente with a solvation term (see Section 11.12). The global minimum in the energy function i assumed to correspond to the naturally occurring structure of the molecule. 

The parameterization process may be done sequentially or in a combined fashion. In the sequential method a certain class of compound, such as hydrocarbons, is parameterized first. These parameters are held fixed, and a new class of compound, for example alcohols and ethers, is then parameterized. Tins method is in line with the basic assumption of force fields parameters are transferable. The advantage is that only a fairly small number of parameters are fitted at a time. The ErrF is therefore a relatively low-dimensional function, and one can be reasonably certain that a good minimum has been found (although it may not be the global minimum). The disadvantage is that the final set of parameters necessarily provides a poorer fit (as defined from the value of the ErrF) than if all the parameters are fitted simultaneously. 

An error function depending on parameters. Only minima are of interest, and the global minimum is usually (but not always) desired. This may for example be determination of parameters in a force field, a set of atomic charges, or a set of localized Molecular Orbitals. 

Ab initio calculations reproduce the experimentally observed geometrical changes. Ethylene is planar (absolute true minimum global minimum) at all theoretical levels. The computed geometry of disilene depends strongly on the basis functions and on electron... 

Neither of the problems illustrated in Figures 4.5 and 4.6 had more than one optimum. It is easy, however, to construct nonlinear programs in which local optima occur. For example, if the objective function / had two minima and at least one was interior to the feasible region, then the constrained problem would have two local minima. Contours of such a function are shown in Figure 4.7. Note that the minimum at the boundary point x1 = 3, x2 = 2 is the global minimum at / = 3 the feasible local minimum in the interior of the constraints is at / = 4. 

Figure 4.17b shows contours for the objective function in this example. Note that the global minimum can only be identified by evaluating/(x) for all the local minima. 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

The KTC comprise both the necessary and sufficient conditions for optimality for smooth convex problems. In the problem (8.25)-(8.26), if the objective fix) and inequality constraint functions gj are convex, and the equality constraint functions hj are linear, then the feasible region of the problem is convex, and any local minimum is a global minimum. Further, if x is a feasible solution, if all the problem functions have continuous first derivatives at x, and if the gradients of the active constraints at x are independent, then x is optimal if and only if the KTC are satisfied at x. ... 

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