METHOD FOR CALCULATING EXTREME VALUES

OF QUALITY INDICATORS OF CHEMICAL-TECHNOLOGICAL PROCESSES

A. I. Ruban

The application of statistical methods of experimental design makes it possible to find equations relating quality indicators to input variables in the form of first- and second-degree power polynomials. It is then necessary to find such a set of input variables that ensures an extremum of the quality indicator. The constraints imposed on the input variables often have the simplest form \(a \le x \le b\). In this case we have a problem of linear or quadratic programming. If the first of these is solved comparatively simply, then the solution of the second leads to complex algorithms whose implementation, even on medium-class digital computers, causes considerable difficulties.

In practice, problems are often encountered in which there are 2 or 3 independent variables and it is necessary, within a fairly short time, to find the optimal conditions for carrying out technological processes. For such problems of small dimension it is desirable to have an algorithm that allows any process engineer, unfamiliar with programming on a digital computer, to perform manual calculations. This would make it possible, first, to check calculations performed on a digital computer and, second, to control technological processes more promptly when there is no digital computer “at hand,” or when only small computers are available.

An algorithm possessing the indicated properties is presented in this paper.

Despite its simple conceptual basis, the author has not encountered a description of it in the existing literature on nonlinear programming. All calculations by this algorithm can be carried out using readily available computing aids (a slide rule, a calculating machine), and, once sufficient skill has been acquired, the solution is found in a short time (no more than 1–2 hours are required for one problem). In addition, for the number of variables 3 the algorithm can easily be implemented on a digital computer.

Problem statement. It is necessary to find the maximum of a function of second order.

\[ J(x_1,x_2,x_3)=p^T x + x^T Cx = p_1x_1+p_2x_2+p_3x_3+C_{11}x_1^2+ \]

\[ +\,C_{22}x_2^2+C_{33}x_3^2+2C_{12}x_1x_2+2C_{13}x_1x_3+2C_{23}x_2x_3 . \tag{1} \]

in the presence of constraints

\[ a_i \le x_i \le b_i, \qquad i = 1,2,3 \tag{2} \]

In what follows, for simplicity, we shall consider the case when

\[ -1 \le x_i \le 1, \qquad i = 1,2,3, \tag{3} \]

for the method does not thereby lose its generality.

In equation (1) the following notation is used:

\[ \rho = \begin{pmatrix} \rho_1\\ \rho_2\\ \rho_3 \end{pmatrix}, \quad x = \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}, \quad C = \begin{pmatrix} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{pmatrix}, \tag{4} \]

where \(C_{12}=C_{21}\), \(C_{13}=C_{31}\), \(C_{23}=C_{32}\).

Method of solution.

I. We determine the form of the surface \(\mathcal{Y}(x)\) in the coordinate space \((\mathcal{Y}, x_1, x_2, x_3)\). This can be done in the following two ways.

a) We use Sylvester’s theorem on the positive definiteness of the matrix \(C\).

If all Sylvester determinants are positive,

\[ \Delta_1 = |C_{11}| > 0,\quad \Delta_2 = \begin{vmatrix} C_{11} & C_{12}\\ C_{21} & C_{22} \end{vmatrix} >0,\quad \Delta_3 = \begin{vmatrix} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{vmatrix} >0, \tag{5} \]

then \(\mathcal{Y}(x)\) has a minimum.

If the odd Sylvester determinants are negative and the even ones are positive,

\[ \Delta_1 < 0,\qquad \Delta_2 > 0,\qquad \Delta_3 < 0, \tag{6} \]

then \(\mathcal{Y}(x)\) has a maximum.

In the remaining cases \(\mathcal{Y}(x)\) is a minimax surface (saddle-shaped), or of ridge (ravine) type.

b) We reduce \(\mathcal{Y}(x)\) to canonical form. Since we are interested only in the form of the surface, it is sufficient to find any canonical form from among all existing ones. Therefore we use a sufficiently simple and universal method—the Lagrange method.

We reduce the quadratic form \(x^T Cx\) to canonical form.

If in the matrix \(C\) the element \(C_{11} \ne 0\), then we pass to new coordinates in accordance with the linear transformation

\[ \begin{aligned} y_1 &= C_{11}x_1 + C_{12}x_2 + C_{13}x_3,\\ y_2 &= \hspace{4.5em} x_2,\\ y_3 &= \hspace{8.0em} x_3. \end{aligned} \tag{7} \]

Whence we obtain the relation between the old coordinates and the new ones

\[ \begin{aligned} x_1 &= \frac{1}{c_{11}}y_1-\frac{c_{12}}{c_{11}}y_2-\frac{c_{13}}{c_{11}}y_3,\\ x_2 &= y_2,\\ x_3 &= y_3, \end{aligned} \]

or, in matrix form,

\[ x=Ay,\quad A= \begin{pmatrix} \frac{1}{c_{11}} & -\frac{c_{12}}{c_{11}} & -\frac{c_{13}}{c_{11}}\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \tag{8} \]

In the new variables the quadratic form takes the form

\[ x^T Cx = y^T A^T C A y = y^T D y,\quad D=A^T C A= \begin{pmatrix} d_{11} & 0 & 0\\ 0 & d_{22} & d_{23}\\ 0 & d_{32} & d_{33} \end{pmatrix} \tag{9} \]

Since usually \(d_{22}\ne 0\), let us apply one more linear coordinate transformation, analogous to that described above:

\[ \begin{aligned} h_1 &= y_1,\\ h_2 &= d_{22}y_2+d_{23}y_3,\\ h_3 &= y_3, \end{aligned} \]

or

\[ y=Eh,\quad E= \begin{pmatrix} 1 & 0 & 0\\ 0 & 1/d_{22} & -d_{23}/d_{22}\\ 0 & 0 & 1 \end{pmatrix} \tag{10} \]

Let us write the quadratic form in the new coordinates \(h\):

\[ x^T Cx = y^T D y = h^T E^T D h = h^T F h,\quad F=E^T A^T C A E= \begin{pmatrix} f_{11} & 0 & 0\\ 0 & f_{22} & 0\\ 0 & 0 & f_{33} \end{pmatrix}, \tag{11} \]

i.e., we obtain the canonical form

\[ x^T Cx=f_{11}h_1^2+f_{22}h_2^2+f_{33}h_3^2. \tag{12} \]

Now the canonical form can also be written for the entire quadratic equation (if \(f_{ii}=0,\ i=1,2,3\)):

\[ J(x)=f_{11}z_1^2+f_{22}z_2^2+f_{33}z_3^2+f_0,\quad z_i=\left(h_i+\frac{q_i}{2f_{ii}}\right),\ i=1,2,3, \tag{13} \]

\[ f_0=-\sum_{i=1}^{3}\left(\frac{q_i}{2}\right)^2\frac{1}{f_{ii}},\quad q^T=p^T(AE)^{-1}=(q_1,q_2,q_3). \]

It follows from this that:

1) if \(f_{ii}>0,\ i=1,2,3\), then \(J(x)\) has a minimum;

2) if \(f_{ii}<0,\ i=1,2,3\), then it has a maximum;

3) If \(f_{ii}\) have different signs, then \(J(x)\) is a surface of the minimax type. Other types of second-order surfaces are possible. We do not consider them here.

1) The surface \(J(x)\) has its center at the point whose coordinates are found from the system of linear algebraic equations

\[ h_i = -\frac{q_i}{2 f_{ii}}, \qquad i = 1,2,3 \tag{14} \]

where

\[ h = E^{-1} A^{-1} x, \qquad (x = A E h). \tag{15} \]

2) If \(J(x)\) has a minimum, then the solution (the maximum of \(J(x)\)) will be at the vertices of the constraint cube. At these points \(J(x)\) is compared, and the point giving the largest value of \(J(x)\) will be the solution of the problem.

If \(J(x)\) has a maximum and it lies inside the specified region, then the point corresponding to the maximum of \(J(x)\) will be the solution of the problem. To calculate the coordinates \(x_1, x_2, x_3\), it is necessary to solve the system of linear nonhomogeneous algebraic equations

\[ p + 2 C x = 0, \tag{16} \]

or system (14).

In the remaining cases, the solution should be sought on the surface of the constraint cube. For this, it is necessary to successively investigate the response function for a maximum on all faces of the cube.

3) Search for the maximum of \(J(x)\) on the surface of the constraint cube.

a) Fix \(X_1 = +1\). Then \(J(x)\) becomes a polynomial of the second degree in two variables, \(x_2\) and \(x_3\), and \(J(x_2, x_3)\) should be investigated in the same way as for the case of two variables.

If \(J(x_2, x_3)\) has a minimum, then the maximum of \(J(x_2, x_3)\) should be sought at the vertices of the constraint square, i.e., at the points: \((1,1)\), \((1,-1)\), \((-1,+1)\), \((-1,-1)\). From the values of \(J(x_2, x_3)\) at these points, the largest is selected. This will be the maximum value of \(J(x_1, x_2, x_3)\) on the face \(x = +1\).

If \(J(x_2, x_3)\) has a maximum inside the region \(-1 \le x_i \le +1,\ i = 2,3\), then this maximum value will be the solution on the face \(X_1 = +1\).

In the remaining cases, it is necessary to seek the maximum of \(J(x_2, x_3)\) on the surface of the constraint square \(-1 \le x_i \le 1,\ i = 2,3\). Fix \(x_2 = +1\) and seek the extremum of \(J(x_3)\) subject to the presence of restric-

condition \(-I \le x_3 \le +I\). The solution of this one-dimensional problem is no longer difficult. We record the obtained maximum value of \(J\) and the corresponding coordinate values. Then, successively fixing \(x_2 = I\), \(x_3 = +I\), \(x_3 = -I\), we find and record the maximum value of \(J\). From these four maximum values of \(J\) we choose the greatest—the solution of the problem on the line \(x_1 = +I\).

b) We fix \(x_1 = -I\) and again repeat all the operations described in item “a.”

Then we successively examine the faces \(x_2 = +I\), \(x_2 = -I\), \(x_3 = +I\), \(x_3 = -I\). In the end, for each face we choose, from the solutions, the one that corresponds to the greatest value \((x)\).

Conclusion. On the basis of the algorithm described, the authors solved extremal problems for finding the optimal conditions for carrying out the hydrocracking process in obtaining pure benzene, naphthalene, and mesitylene /1/. The algorithm is simple computationally, is accessible to any process engineer, and can be implemented on a digital computer at low cost.

References

1. Ruban A. I., Elbert E. I. Calculation of extremal values of hydrocracking parameters in obtaining pure benzene, naphthalene, and mesitylene. Research report. Tomsk–Novokuznetsk, 1971.

STUDY OF THE SORPTION CAPACITY OF MAGNESIUM OXIDE WITH ADDITIVES

V. V. Nakhalov, N. F. Stas’, G. G. Savel’ev, T. S. Gordina

Studies on the sorption of carbon dioxide and water vapor were carried out on 13 samples of magnesium oxide obtained under various conditions, and on 11 samples with “homophase” additives.

Magnesium oxide without additives was obtained by thermovacuum decomposition of basic magnesium carbonates directly in the cell of a weighing apparatus. The basic carbonates were precipitated from solutions of magnesium nitrate or magnesium chloride of various concentrations by solutions of ammonium, potassium, or sodium bicarbonates. All syntheses were carried out at room temperature. The main data on the methods...