Variable basic

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Yang, W., 1992, Electron Density as the Basic Variable A Divide-and-Conquer Approach to the Ab Initio Computation of Large Molecules , J. Mol. Struct. (Theochem), 255, 461. 

Yang, W. 1992. Electron Density as the Basic Variable a Divide-and-Conquer Approach... 

The Parsons function is defined through = 7 + it is the thermodynamic potential that has the charge density a as the basic variable instead of the potential (j>. Show that the surface excess of a... 

This energy functional attains its minimum for the true electronic density profile. This offers an attractive scheme of performing calculations, the density functional formalism. Instead of solving the Schrodinger equation for each electron, one can use the electronic density n(r) as the basic variable, and exploit the minimal properties of Eq. (17.8). Further, one can obtain approximate solutions for n(r) by choosing a suitable family of trial functions, and minimizing E[n(r)] within this family we will explore this variational method in the following. 

Results 1 and 2 imply that, in searching for an optimal solution, we need only consider vertices, hence only basic feasible solutions. Because a basic feasible solution has m basic variables, an upper bound to the number of basic feasible solutions is the number of ways m variables can be selected from a group of n variables, which is... 

The variables xl9. .., xm are associated with the columns of B and are called basic variables. They are also called dependent, because if values are assigned to the nonbasic, or independent variables, xm+l,..., , then xl9..., xm can be determined immediately. In particular, if w+1,. .., are all assigned zero values then we obtain the basic solution... 

Dependent (basic) variables Independent (nonbasic) variables Constants... 

Because the variables xm+l. . . xn are presently zero and are constrained to be nonnegative, the only way any one of them can change is for it to become positive. But if Cj > 0 for y = m + 1,. . . , n, then increasing any xj cannot decrease the objective function/because then CjXj 0. Because no feasible change in the non-basic variables can cause/to decrease, the present solution must be optimal. 

Assume that we know that jc5, xx, —/can be used as basic variables and that the basic solution will be feasible. We can thus reduce system (7.16) to feasible canonical form by pivoting successively on the terms jt5 (first equation) and xx (second equation) (—/already appears in the correct way). This yields... 

Note that an arbitrary pair of variables does not necessarily yield a basic solution to Equation (7.16) that is feasible. For example, had the variables xx and x2 been chosen as basic variables, the basic solution would have been... 

How large should x3 become It is reasonable to make it as large as possible, because the larger the value of jc3, the smaller the value off. The constraints place a limit on the maximum value x3 can attain, however. Note that, if x2 = x4 = 0, relations (7.18) state that the basic variables xx, x5 are related to x3 by... 

This solution reduces/from 28 to —8. The immediate objective is to see if it is optimal. This can be done if the system can be placed into feasible canonical form with x5, 3, —/ as basic variables. That is, 3 must replace xx as a basic variable. One reason that the simplex method is efficient is that this replacement can be accomplished by doing one pivot transformation. 

This gives the basic feasible solution (7.24), as predicted. It also indicates that the present solution although better, is still not optimal, because c2, the coefficient of 2 in the/equation, is —1. Thus we can again obtain a better solution by increasing 2 while keeping all other nonbasic variables at zero. From Equation (7.25), the current basic variables are then related to 2 by... 

Having decided on the variable xs to become basic, we increase it from zero, holding all other nonbasic variables zero, and observe the effects on the current basic variables. By Equation (7.12), these are related to xs by... 

The basic variable xr then becomes nonbasic, to be replaced by xs. We saw from the example in Equations (7.16)-(7.28) that a new canonical form with xs replacing xr as a basic variable is easily obtained by pivoting on the term arsxs. Note that the previous operations may be viewed as simply locating that pivot term. Finding cs = min Cj < 0 indicates that the pivot term was in column s, and finding that the minimum of the ratios bt/ais for ais> 0 occurred for i = r indicates that it was in row r. 

If, at some iteration, the basic feasible solution is degenerate, the possibility exists that/can remain constant for some number of subsequent iterations. It is then possible for a given set of basic variables to be repeated. An endless loop is then set up, the optimum is never attained, and the simplex algorithm is said to have cycled. Examples of cycling have been constructed [see Dantzig (1998), Chapter 10]. 

If xB is between its bounds, the basic solution is feasible and we begin phase 2, which optimizes the true objective. Otherwise, some components of xB violate their bounds. Let L and U be the sets of indices of basic variables that violate their bounds, that is... 

The minimum of these is bx/a 12 thus the basic variable with unity coefficient in row 1, jc3, leaves the basis. The pivot term is ai2x2 that is, the x2 term circled in Equation (b). Pivoting on this term yields... 

For the problem given in 7.9, find the next basis. Show the steps you take to calculate the improved solution, and indicate what the basic variables and nonbasic variables are in the new set of equations. (Just a single step from one vertex to the next is asked for in this problem.)... 

If dof(x) = n — act(x) = d > 0, then there are more problem variables than active constraints at x, so the (n-d) active constraints can be solved for n — d dependent or basic variables, each of which depends on the remaining d independent or nonbasic variables. Generalized reduced gradient (GRG) algorithms use the active constraints at a point to solve for an equal number of dependent or basic variables in terms of the remaining independent ones, as does the simplex method for LPs. 

Let the starting point be (1, 0), at which the objective value is 6.5 and the inequality is satisfied strictly, that is, its slack is positive (s = 1). At this point the bounds are also all satisfied, although y is at its lower bound. Because all of the constraints (except for bounds) are inactive at the starting point, there are no equalities that must be solved for values of dependent variables. Hence we proceed to minimize the objective subject only to the bounds on the nonbasic variables x and y. There are no basic variables. The reduced problem is simply the original problem ignoring the inequality constraint. In solving this reduced problem, we do keep track of the inequality. If it becomes active or violated, then the reduced problem changes. 

The reduced objective is obtained by substituting this expression into the objective function. The slack s will be fixed at its current zero value for the next iteration because moving into the interior of the circle from (0.697, 1.517) increases the objective. Thus, as in the linearly constrained example, y is again the only super-basic variable at this stage. 

Because analytic solution of the active constraints for the basic variables is rarely possible, especially when some of the constraints are nonlinear, a numerical procedure must be used. GRG uses a variation of Newton s method which, in this example, works as follows. With 5 = 0, the equation to be solved for x is... 

This variation on Newton s method usually requires more iterations than the pure version, but it takes much less work per iteration, especially when there are two or more basic variables. In the multivariable case the matrix Vg(x) (called the basis matrix, as in linear programming) replaces dg/dx in the Newton equation (8.85), and g(Xo) is the vector of active constraint values at x0. 

Value of objective function = 1268.75 /h BASIC = basic variable ZERO = 0... 

In the Bom like approaches to solvation energy, the electrostatic potential of the ion appears as the basic variable of the theory. From Eq (1), it may be seen that if we have accurate electron densities at hand, the electrostatic potential strongly depends on the ionic radius r. The choice of suitable ionic radii usually introduces some arbitrariness in the calculation of AESolv there is no a physical criterium to justify the use of empirical rA values coming from different sources [15-16],... 

---