Construction of the maximum cube inscribed in a given domain

U.S.S.R. Compur. Muths. Math. pt?vs. Vol. 20, No. 2,pp. 245-249,198O

Printed in Great Britain.

0041~5553/80/020245-05$07.50/O @ 1981 Pergamon Press Ltd.

CONSTRUCTIONOFTHEMAXIMUMCUBEINSCRIBED I.NAGIVENDOMAIN* L. T. ASHCHEPKOV Irkutsk. (Received 26 Febncagl 1979)

A METHOD of reducing the geometrical problem of inscribing the maximum cube (sphere) in a given domain to a mathematical programming problem is described. The connection between the solutions of these problems is explained. A practical example is given. The problem mentioned in the title arises in applications of mathematical programming when it is required to find not one optimal plan, but a set of plans, close to optimal in the value of the target function. If the degree of closeness to the optimal plan is specified, (the problem reduces to some subset D of the original set of plans. For practical workers there is great interest not in the construction of the subset D itself, which may be fairly complex to construct, but in a set B, simpler from the point of view of operative technological control, which on the one hand would be contained in D, and on the other hand would be a suitable approximation of it. If a cube is taken as B, then the formalization of the requirements enumerated leads to the problem of constructing the maximum cube inscribed in the given domain. 1. Formulation of the problem. The linear case. We consider the system of linear equations n

c

aijxj

< bi,

i-l,

2,. . . , m,

l-1

with real coefficients ov, b, and unknowns X~ i = 1,2, . . . , m, J = 1,2, . . . , n. Having formed as usual from the coefficients and unknowns the (m X n)matrix A, the m-vector b and the n-vector x, we write the system of inequalities (1) in the matrix form Az
The set of all solutions of the inequality (2) is denoted by D. It is obvious that D C R”. For the vectors x = (xl, . . . , x,,) of R” we introduce the norm

The closed sphere IIy-x II < r with centre at the point x of radius r 2 0 is denoted by B(x, r). Geometrically B(x, r) represents a cube symmetric about x; the sides of the cube of length 2r are parallel to the vectors e1 , . . . , en of the standard orthonormed basis R”. *Zh. vjkhid. Mat. mat. Fir., 20, 2, 510-513,

1980.

245

(2)

L. T. Ashchepkov

246

We say that B(x*, r*) is the maximum cube inscribed in D, ifB(x*, r*) C D and for any cube B(x, r) C D we always have I < T*. We explain the conditions for which a maximum cube inscribed in D exists. 2. Results. We put n

N, =

c

i=1,2 ,...,

I”ijlv

m,

N-(N

,,...,

N,,,).

(3)

>==I

Lemma

The cube B(x, r) is contained in D if and only if Ax + rN < b, t Z 0. Proof: Let B(x, r) C D. Obviously, r > 0. The points n rir

CZjej,

c

\a,)=&

.j=1,2

,...,

n,

1-l

being boundary points of the cube B(x, r) C D, satisfy the inequalities (1):

From this, because of the arbitrariness of the signs of the c$ and the notation (3), we have Ax + rN < b. Conversely, if the latter inequality is satisfied for r > 0, then for any y EB (x, r) we obtain Ay-Az=A(y-z)
from which Ay < b. Accordingly, y E D. The lemma is proved. By conditions (2) we construct the linear programming problem rdmax,

Az+rN<

b.

The connection between conditions (2) and problem (4) is described by the following theorem. 7heorem

Let problem (4) have a solution x*, r*. Then the following statements hold: 1) if r* < 0, inequalities (2) are inconsistent; 2) if r * = 0, then there does not exist an x E R” such that Ax < b; 3) if r* > 0, then B(x*, r*) is the maximum cube inscribed in D; 4) if problem (4) has no solution and N > 0, then D is an unbounded polyhedral set [l] with internal points.

(4)

247

Short communications

Proof: We assume the opposite statement to 1): r* < 0, but A; Q b for some x^E R”. Then the pair ;, i = 0 forms a plan of problem (4) where ; > r*. The latter inequality contradicts the optirnality of x*, r *. Statement 2) is established similarly. By condition 3) At’+r’NG b, r’>O, which by virtue of the lemma means that B(x*, r*) C D. If B(x, r) is an arbitrary cube inscribed in D, then the pair x, r, by the lemma, satisfies the conditions of problem (4); consequently, r Q r*. We will prove the last statement. A solution of problem (4) does not exist for two reasons: either because of the inconsistency of the conditions, or because of the unboundedness of the linear form on the set of plans. The condition N > 0 of the theorem excludes the first reason; therefore only the second remains. Therefore, for any I > 0 and x is found satisfying the inequality Ax t rN < b. By the lemma B(x, r) CD. The theorem is proved. The application of the theorem to solve a practical problem is illustrated below. We merely note briefly to what extent the results of this section transfer to the non-linear case. 3. Generalizations. We consider the continuous mapping g: R” + Rm and the domain G C R”, defined by the vector inequality g(x) < 0. Fixing some norm in R”, we denote as before by B(x, r) the set of points y E R” satisfying the identity /Iy - xl1 < T. Also, for each i-th coordinate gi of the mapping g we put

N,(z,r)=

sup

Pfl(X 7)

N(z, r)=(Ni(r,

i=1,2 ,...,

[gi(Y)-glwl!

m.

(5)

r), . . ., Nm(z, 4).

It is easy to see that the function N is defined and continuous at the points x E R”, r > 0 and possesses the properties N(x, r) > 0, N(x, 0) = 0. By analogy with section 2 we formulate an auxiliary linear programming problem r-rmax.

g(s)+N(z,

r)GO.

rZ0.

We denote the solution of the latter by x*, r*. It is found that for r* = 0 the system of inequalities g(x) < 0 does not satisfy Slater’s condition, that is, there does not exist an x E R” such that g(x) < 0. If r* > 0 the sphere B(x*, r*), corresponding to the metric R” fured above, is the maximum sphere inscribed in the domain G. The proofs of the statements are based on a non-linear analog of the lemma of section 2: for the inclusion E(x, r) C G to hold it is necessary and sufficient that g(x) t N(x, r) < 0, r > 0. These results reduce the problem of constructing the maximal sphere inscribed in the domain g(x) < 0 to problem (6). If there is an efficient method of calculating the functions (5), then to solve problem (6) the standard methods of non-linear programming can be used. 4. Example. We will illustrate the results of section 2 by the model of the production of a cryolite by an acid-alkaline method from aluminium production waste (the model was constructed under the leadership of N. P. Mokretskii). In statics the dependence of the technological indices yi of the quality of the cryolite on the main production factors Xj GUI be described in a linear approximation by the equations

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L. T. Ashchepkov

248

,

Y,=aio

+

z

i=i, 2,3.

@jzjv

(7)

e-1

The quantities yi, xi have a deftite physical meaning which we will not go into. The variables Xj in El+ (7) are normed, that is, reduced by a linear transformation to the segment [-I, I] . The coefficients aii are indicated in the table. TABLE

I

I

I

I

I

I

I

I

I

The factors ensuring production of cryolite of the required quality are defined by the conditions Y2+2,

Ylqil,

YaCP39 -1cx,<1,

j=l,.

.. 7 )

(8)

(the values of jr r, jr 2, jr 3 are shown in the table). For simplicity and effective operation of technological control it is desirable to know not the whole set D of solutions of the inequalities (8), but the ranges of permissible variations in each factor separately. Since the factors are normed. it is required, in other words, to inscribe in D the maximum cube. By the results of section 2 this reduces to solving an auxiliary linear progr amming problem (4) which here takes the form r+mas,

X;+fGl,

--I>+

rc 1,

j=l.....i.

The auxiliary problem was solved by a modified simplex method (the calculations were carried out by A. I. Benikov). The coordinates xi* of the centre of the maximum cube and its radius r* obtained were z,'=-0.8506. i=i, 2,3.7,

x,'=0.8506,j=4, 5,

Zr'=O.i53;.

r'=0.1494.

On passing to unnonned (natural) values of the production factors the cube is transformed into a parallellepiped.

Short communications

249

In conclusion the author acknowledges his indebtness to N. P. Mokretskii for the interesting example and to A. I. Benikov for useful discussions. Translated by J. Berry. REFERENCES

1.

YUDIN, D. B. and GOL’SHTEIN, E. G. Linear programming (Lineinoe programmirovanie),

Fizmatgiz, Moscow, 1963.

U.S.S.R. Comput. Maths. Math. Phys. Vol. 20, No. 2,pp. 249-253,198O Printed in Great Britam.

0041-5553/80/020249-06$07.50/O 0 1981 Pergamon Press Ltd.

A COMPLETELY CONSERVATIVE DIFFERENCE SCHEME FOR THE TWO-DIMENSIONAL LANDAU EQUATION* I. F. POTAPENKO and V. A. CHUYANOV Moscow (Received 26 Febnraly 1979)

AN ECONOMICAL difference scheme, completely conservative (preserving the number of particles and the energy of the system), is constructed for the two-dimensional multicomponent Landau equation with isotropic Rozenblyut potentials. 1. A considerable number of papers devoted to the numerical solution of Landau’s equation have appeared in connection with the study of the dynamics of a plasma confined in open magnetic traps [ 11. The principle of complete conservativeness, advanced in [ 21, here acquires great practical value, since the use of incompletely conservative schemes for the kinetic equation, that is, schemes conserving either the number of particles or the energy of the system, do not give an equilibrium solution of the difference problem over the entire velocity space [3] . In [4] it is shown that a calculation by a scheme where the number of particles is not conserved leads to physically absurd results for a magnetic trap with a radial electric field. Completely conservative schemes for a spatially homogeneous and isotropic gas were constructed in [3,5]. In this paper a completely conservative difference scheme is presented for the two-dimensional multicomponent Landau equation with isotropic Rozenblyut potentials (in this formulation the problem is of fundamental practical interest for the design of magnetic traps [ 1,4] ). 2. In the case considered the Rozenblyut potentials are of the form

( $-)*JI=~~ LJ’*~v’~ dp b(u,u’)fp(u’,p,t)

g.(V)+~ B

lZh. vjkhisl. Mat. mat. Fir., 20, 2, 513-517,

Q

1980.

-i