UDC 519.95

CYBERNETICS AND CONTROL THEORY

V. A. EMELICHEV

ON PROBLEMS OF DISCRETE OPTIMIZATION

(Presented by Academician L. V. Kantorovich, 15 XII 1969)

The idea of the method, whose basic scheme will be set forth below, has recently been used for solving many problems of discrete optimization: variants of problems of production allocation ((^{1-6})), integer problems of linear programming ((^7,{}^8)), problems of discrete programming with separable ((^9)) and nonseparable ((^{10})) objective functions, as well as problems of concave programming ((^{11})) and problems of device reliability ((^{12})).

In the present note this method (\psi), based on the construction of a sequence of plans, will be formulated in general form.

Let it be required to find the minimum of a function (F(x)) defined on a finite set (P={p_1,p_2,\ldots,p_n}). Thus it is required to determine an optimal element (plan) (p^*\in P), i.e., one such that

[
F(p^*)=\min {F(p_i)\mid p_i\in P}.
]

The method (\psi) is applicable to this problem (A) if the following conditions are satisfied:

1) it is possible to find a finite extension (R={r_1,r_2,\ldots,r_m}) of the set (P) ((P\subseteq R)) and a function (Q(x)), defined on (R), such that (Q(x)\leq F(x)) for all (x\in P);

2) it is possible to construct an algorithm (\varphi) which, at the (k)-th step ((k=1,2,\ldots)), effectively finds an element (r_{j_k}\in R) having the property

[
Q(r_{j_k})=\min_{x\in R_k} Q(x),\qquad k=1,\ldots,m,
]

where (R_k=R_{k-1}\setminus r_{j_{k-1}}), and (R_1=R) *.

If, from the sequence (r_{j_1}, r_{j_2},\ldots) constructed with the aid of (\varphi), one removes the elements not belonging to the set (P), then for the remaining sequence (p_{i_1},p_{i_2},\ldots) the following holds.

Optimality criterion. If there exists a natural number (t) ((t\leq n)) such that

[
Q(p_{i_t})\geq \min F(p_{i_s})=F(p^*),
\tag{1}
]

then (p^) is an optimal element of problem A.*

Thus, the procedure (\psi) for finding an optimal element of problem A is as follows. At the (k)-th step ((k=1,2,\ldots)), with the aid of algorithm (\varphi), we find an element (r_{j_k}) and check whether it belongs to the set (P). If not, we pass to the next step of algorithm (\psi). If yes ((r_{j_k}=p_{i_t})), then we check whether condition (1) is satisfied. If yes, the process terminates and (p^*) is an optimal element of problem A. If not, we pass to the next step of algorithm (\psi).

* In other words, there must exist an effective method for constructing a sequence of elements of the set (R) in the order of nondecrease of the function (Q(x)) (cf. ((^{10}))).

Remarks. 1. If at the (k)-th step of algorithm (\psi) it is possible to refine the approximating function (Q(x)), i.e., to find a function (Q_k(x)) satisfying the condition

[
Q(x) \leq Q_{k-1}(x) \leq Q_k(x) \leq F(x)
]

for all (x \in R_k), then the procedure can be accelerated.

1. Obviously, if there is no number (t) satisfying condition (1), then algorithm (\psi) turns into complete enumeration. However, as was indicated in ((^{2,11})), at each step one can estimate the deviation of the best of the obtained elements of the set (P) from the optimal one.

2. If the function (Q(x)) coincides with (F(x)) on the set (P), then our procedure turns into a scheme close to the general scheme proposed in ((^{13})). In this case the optimal element of problem (A) is sought among the elements (r) of the set (R) satisfying the condition (Q(r) \leq F(p^*)) (7).

If, moreover, a lower bound (\bar F) of the values of the function (F(x)) on the set (P) is known, then this accelerates the procedure, since before checking whether the element (r_{j_k}) belongs to the set (P)*, we must make sure that (Q(r_j^k) \geq \bar F) (otherwise one must pass to the next step of algorithm (\psi)) (7).

Belorussian State University
named after V. I. Lenin
Minsk

Received
1 XII 1969

REFERENCES CITED

(^{1}) V. A. Emelichev, V. I. Komlik, Dokl. AN BSSR, 10, No. 10 (1966).
(^{2}) V. A. Emelichev, V. I. Komlik, Dokl. AN BSSR, 12, No. 10 (1968).
(^{3}) V. I. Komlik, Vesti AN BSSR, ser. fiz.-mat. navuk, No. 4 (1967).
(^{4}) V. A. Emelichev, P. Kilebanov, V. I. Komlik, Materials of the Interregional Seminar on the Location of Industry and Regional Development, Minsk, 1968.
(^{5}) V. A. Emelichev, V. I. Komlik et al., Proceedings of the Second Republican Conference of Mathematicians of Belorussia, Minsk, 1969.
(^{6}) A. M. Galperin, Technical and Economic Information, ser. Economics of the Chemical Industry, issue 8, Moscow, 1968.
(^{7}) V. I. Komlik, V. A. Emelichev, Dokl. AN BSSR, 11, No. 12 (1967).
(^{8}) V. I. Komlik, V. A. Emelichev, Vesti AN BSSR, ser. fiz.-mat. navuk, No. 4 (1968).
(^{9}) V. A. Emelichev, V. I. Komlik, Proceedings of the seminar Automated Enterprise Management Systems, issue 1, Kiev, 1968.
(^{10}) V. A. Emelichev, V. I. Komlik, DAN, 188, No. 2 (1969).
(^{11}) V. A. Emelichev, Vesti AN BSSR, ser. fiz.-mat. navuk, No. 6 (1969).
(^{12}) V. A. Emelichev, V. I. Komlik, Vesti AN BSSR, ser. fiz.-mat. navuk, No. 5 (1968).
(^{13}) E. Balas, Oper. Res., 16, No. 2 (1968).

* This check is often laborious; for example, in the case when (P) is the set of solutions of a system of inequalities ((^{6,7})).