Feasible direction vector

Definition 3.2.5 (Feasible direction vector) Let a feasible point S F. Then, any point jc in a ball of radius e around S which can be written as S + d is a nonzero vector if and only if jc S. A vector d 0 is called a feasible direction vector from S if there exists a ball of radius e ... 

The set of feasible direction vectors d 0 from x is called the cone of feasible directions of F at... 

Illustration 3.2.1 Figure 3.2 shows the feasible region, a point x, and feasible direction vectors from jc. 

Remark 2 If jc is a local minimum of (3.3) and d is a feasible direction vector from jc, then for sufficiently small A, we must have... 

Lemma 3.2.1 Let d be a non-zero feasible direction vector from jc. Then, x must satisfy the conditions ... 

Definition 3.2.6 (Improving feasible direction vector) A feasible direction vector d 0 at jc is called an improving feasible direction vector at x if... 

Figure 3.2 Feasible region and feasible direction vectors...

Now consider Figure 8.4, in which the gradient vectors Vgx and Vg2 are drawn. Note that — V/is contained in the cone generated by Vgx and Vg2. What if this were not so If —V/ were slightly above Vg2, it would make an angle of less than 90° with a feasible direction just below the line x + y — 2 = 0. If — V/ were slightly below Vg1 it would make an angle of less than 90° with a feasible direction just... 

Figure 5-15. Determination of the limits for the feasible solutions from the Lawton-Sylvestre method. Vector dir denotes the directional vector pointing to the spectral direction.

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

As is shown below for the solute, the linear term can be included in the quadratic term by redefining the fluctuation operator. G(m s) is a rectangular matrix obtained from the second cross-derivatives coupling the solute to the solvent l(s m) is a vector obtained by similar procedures from the (dGm(X)/dRs)(n) linear coupling. How to actually perfonn such operation is not directly important now. We assume that, in principle, such operations are feasible. 

While less complicated than the in vitro approach, direct in situ injection of vector into the immediate vicinity of target cells is not always feasible. This would be true, for example, if the... 

To obtain a meaningful extremum problem the number of experiments and the set of feasible vectors of the independent variables T are fixed. In most cases T is defined by inequalities x1- < x < x, i = l,2,...,k. Though introducing penalty functions such constrained extremum problems can be solved by the methods and modules described in Section 2.4, this direct approach is usually very inefficient. In fact, experiment design is not easy. The dimensionality of the extremum problem is high, the extrema are partly on the boundaries of the feasible region T, and since the objective functions are... 

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