Constraints vector projection

Fig. 34.20. ITTFA projection of the input vector iii in the PC-space gives out]. A new input target in2 is obtained by adapting outi to specific constraints. in2 projected in the PC-space gives out2.

The difference between vector projection on a set of linear constraints and the use of null space of constraints to satisfy the constraints was illustrated earlier in Chapter 8. In both cases, the original direction is transformed into a new direction that satisfies the selected linear constraints. 

The difference between the vector projection and the use of null space of the constraints to obtain a vector that satisfies them was discussed in Chapter 8 and in Section 11.3.3. We have also seen that LQ factorization can be adopted to get the projection for small dimension problems, whereas it is preferable to use the null space obtained with stable Gauss factorization for large-scale systems. 

The hypothetical enantiophore queries are constructed from the CSP receptor interaction sites as listed above. They are defined in terms of geometric objects (points, lines, planes, centroids, normal vectors) and constraints (distances, angles, dihedral angles, exclusion sphere) which are directly inferred from projected CSP receptor-site points. For instance, the enantiophore in Fig. 4-7 contains three point attachments obtained by ... 

Rotated axes are characterized by their position in the original space, given by the vectors = [cos0 -sin0] and F = [sin0 cos0] (see Fig. 34.8). In PCA or FA, these axes fulfil specific constraints (see Chapter 17). For instance, in PCA the direction of k is the direction of the maximum variance of all points projected on this axis. A possible constraint in FA is maximum simplicity of k, which is explained in Section 34.2.3. The new axes (k,l) define another basis of the same space. The position of the vector [x,- y,] is now [fc, /,] relative to these axes. 

From the mathematical restrictions on the solution of the equations comes a set of constraints known as quantum numbers. The first of these is n, the principal quantum number, which is restricted to integer values (1, 2, 3,. ..). The second quantum number is 1, the orbital angular momentum quantum number, and it must also be an integer such that it can be at most (n — 1). The third quantum number is m, the magnetic quantum number, which gives the projection of the 1 vector on the z axis as shown in Figure 2.2. 

We now show, conversely, that for each projection tensor P j, there exists a unique set of corresponding reciprocal basis vectors that are related to P j, by Eq. (2.195). To show this, we show that the set of arbitrary numbers required to uniquely define such a projection tensor at a point on the constraint surface is linearly related to the set of fK arbitrary numbers required to uniquely specify a system of reciprocal vectors. A total of (3A) coefficients are required to specify a tensor P v- Equation (2.193) yields a set of 3NK scalar equations that require vanishing values of both the hard-hard components, which are given by the quantities n P = 0, and of the fK mixed hard-soft ... 

Note that the soft reciprocal vectors b are expanded in a basis of tangent vectors, and so are manifestly parallel to the constraint surface (as indicated by the use of a tilde), while the hard reciprocal vectors ihi are expanded in normal vectors, and so lie entirely normal to the constraint surface (as indicated by the use of a caret). These basis vectors may be used to construct a geometric projection tensor... 

Such an a posteriori approach can be implemented quite easily and naturally using the machinery of constrained dynamics. The point is in using a proper constraint that freezes the motion along the predetermined, reference reaction path. Such a constraint was defined,33 based on the fact that in order to freeze the motion in a direction given by a vector rref, the projection of the displacement vector r on must be zero, rmrre = 0. 

To construct the projection operators corresponding to the constraints, the subspace unit vectors representing different constraints must be independent. As shown by Miller et ah, this can be affected by Gram-Schmidt orthogonalization that yields a set of orthogonal unit vectors ... 

Chemical functionality (H-bond acceptor, H-bond donor, positive ionizable, negative ionizable, hydrophobic) with geometric constraint, e.g., an H-bond acceptor vector including an acceptor point as well as a projected donor point aromatic ring including a ring plane + + +... 

In the large majority of cases, however, the object and constraint functions are so complicated that an analytical elimination of a variable is essentially intractable, and this is especially true when there is more than one constraint. The main exception is when the constraint equation is linear, in which case it can be considered as a vector in the coordinate space. Instead of eliminating a variable explicitly, the constraint condition can be fulfilled by removing the corresponding component of the object function by projection (Section 16.4), and performing the optimization on f. ... 

Here is a vector which needs to be computed to define the projection onto M. We would typically assume that qo lies on M so that (4.14)-(4.16) defines a map on M. The hidden constraint g (qo)M po = 0 will be violated, but one hopes it remains approximately satisfied. The justification for the latter assumption is that if we expand g in a Taylor series, we may write... 

One problem, related to the search for a point that fulfills a set of equations, is projecting a vector in the space of some constraints. As will be discussed in the following chapters, many optimization methods adopt this technique. To understand what projecting a vector means, it is useful to consider a simple problem with one single linear equality constraint. Suppose we know a point, Xj, that fulfills the constraint and we have a direction d that passes through the point x such that along it a certain function decreases. If d does not lie on the plane of the constraint, the step... 

Note that when a constraint is inserted into the working matrix, the KKT system is not solved to identify the search vector d, but the reduced or projected gradient is exploited. 

In Chapter 8, we demonstrated how, by knowing a point, xj, that firlfills a set of linear constraints and a direction, dj, along which the function decreases, it is possible to orthogonally project the vector d using an opportune matrix P so as to have a step along the vector p = Pd that satisfies all the constraints while simultaneously improving the objective function. 

Methods that use the direction of the function gradient as the projection vector on the active constraints are known as Gradient Projection Methods. 

In this case, it is opportune to use the null space of constraints rather than projecting the search vector on the constraints. This will be shown in the following section. 

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