Reports of the Academy of Sciences of the USSR

1964, Volume 158, No. 4

MATHEMATICS

B. S. VERKHOVSKII

ON THE EXISTENCE OF A SOLUTION OF A MULTI-INDEX LINEAR PROGRAMMING PROBLEM

(Presented by Academician A. A. Dorodnitsyn on 25 IV 1964)

All unexplained notation is to be found in \((^{1})\).
Consider the problem of minimizing the functional

\[ L=\sum_{k\in M} p_{i_1\ldots i_s}x_{i_1\ldots i_s} \tag{1} \]

subject to the conditions

\[ \sum_{k\in M_j} a^{(j)}_{i_1\ldots i_s}x_{i_1\ldots i_s} \begin{cases} \leqslant b_{M_j(i_1\ldots i_s)},\\ = b_{M_j(i_1\ldots i_s)}; \end{cases} \tag{2} \]

\[ x_{i_1\ldots i_s}\geqslant 0;\qquad j=1,\ldots,t;\qquad i_k=1,\ldots,n_k;\qquad k=1,\ldots,s. \tag{3} \]

All \(a^{(j)}_{i_1\ldots i_s}\) and \(p_{i_1\ldots i_s}\) are prescribed real numbers. In the particular case when all \(a^{(j)}_{i_1\ldots i_s}=1\), we obtain a multi-index linear programming problem of transportation type. In addition, all \(b_{M_j(i_1\ldots i_s)}\geqslant 0\) are given.

We shall call the problem (1)—(3) an \(X\)-problem. The totality of all constraints for a fixed \(j\) will be called the \(j\)-block of constraints.

Let there exist \(\widetilde M\ne\varnothing\) such that, if \(\widetilde M\cap M_j\ne \widetilde M\), then all constraints entering the \(j\)-block of constraints are given in the form of inequalities. For definiteness, let \(\widetilde M=\{r+1,\ldots,s\}\).

Consider

\[ \prod_{k=r+1}^{s} n_k \]

independent problems:

\[ \sum_{k\in M_j\setminus \widetilde M} a^{(j)}_{i_1\ldots i_s}y_{i_1\ldots i_r} \begin{cases} \leqslant b_{M_j(i_1\ldots i_s)}\\ = b_{M_j(i_1\ldots i_s)} . \end{cases} \quad \text{for all } j \text{ such that } M_j\cap \widetilde M=\widetilde M; \tag{4} \]

\[ \sum_{k\in M_j\setminus \widetilde M\cap M_j} a^{(j)}_{i_1\ldots i_s}y_{i_1\ldots i_r} \leqslant b_{M_j(i_1\ldots i_s)} \quad \text{for all } j \text{ such that } M_j\cap \widetilde M\ne \widetilde M; \tag{5} \]

\[ y_{i_1\ldots i_r}\geqslant 0. \tag{6} \]

In (4) and (5) all indices \(i_{r+1},\ldots,i_s\) are fixed.

Theorem 1. Suppose that for at least one set of values of the indices \(i_{r+1}=\omega_{r+1},\ldots,i_s=\omega_s\) there exists a feasible solution of the problem (4)—(6). All \(1\leqslant \omega_k\leqslant n_k,\ k=r+1,\ldots,s\). Then the \(X\)-problem also has a feasible solution.

Proof. By direct substitution it is easy to verify that

\[ x_{i_1\ldots i_s}= \begin{cases} y_{i_1\ldots i_r}, & \text{if } \displaystyle\bigcap_{k\in \widetilde M} p(i_k=\omega_k)=1,\\[6pt] 0, & \text{if } \displaystyle\bigcap_{k\in \widetilde M} p(i_k=\omega_k)=0, \end{cases} \]

is a feasible solution of the \(X\)-problem. Here \(p(i_k=\omega_k)\) is a logical condition,

\[ p(i_k=\omega_k)= \begin{cases} 1, & \text{if } i_k=\omega_k,\\ 0, & \text{if } i_k\ne \omega_k. \end{cases} \]

We shall say that the \(X\)-problem is cut on the set of indices \(\widetilde M\) with respect to the tuple \((\omega_{r+1},\ldots,\omega_s)\).

Consider the \(X(t)\)-problem, which differs from the \(X\)-problem in that it does not include the \(t\)-block constraints.

Theorem 2. Suppose:

1) the \(X(t)\)-problem is cut on the set of indices

\[ \widetilde M \equiv \bigcup_{j=1}^{t-1} M_j \setminus M_t \]

for all tuples \((\omega_{r+1},\ldots,\omega_s)\);

2) there exists at least one \(j_* \ne t\) such that all \(a^{(j_*)}_{i_1\ldots i_s}=a^{(t)}_{i_1\ldots i_s}\);

3) the \(j_*\)-block constraints and the \(t\)-block constraints are given in the form of equalities;

4) the compatibility conditions are satisfied,

\[ \sum_{k\in M_{j_*}\cup M_t\setminus M_{j_*}} b_{M_{j_*}(i_1\ldots i_s)} = \sum_{k\in M_{j_*}\cup M_t\setminus M_t} b_{M_t(i_1\ldots i_s)} . \]

Then the \(X\)-problem has a feasible solution.

Proof. Every solution of the \(X(t)\)-problem can be written in the form

\[ x^{(\omega_{r+1}\ldots\omega_s)}_{i_1\ldots i_s} = \begin{cases} y_{i_1\ldots i_r}, & \text{if } \displaystyle\bigcap_{k\in \widetilde M} p(i_k=\omega_k)=1,\\ 0, & \text{if } \displaystyle\bigcap_{k\in \widetilde M} p(i_k=\omega_k)=0. \end{cases} \]

We shall show that there exist such

\[ \lambda_{\omega_{r+1}\ldots\omega_s}\ge 0,\qquad \sum_{\omega_{r+1}=1}^{n_{r+1}}\cdots \sum_{\omega_s=1}^{n_s} \lambda_{\omega_{r+1}\ldots\omega_s}=1, \tag{7} \]

for which the convex combination

\[ \sum_{\omega_{r+1}=1}^{n_{r+1}}\cdots \sum_{\omega_s=1}^{n_s} x^{(\omega_{r+1}\ldots\omega_s)}_{i_1\ldots i_s} \lambda_{\omega_{r+1}\ldots\omega_s} = \bar x_{i_1\ldots i_s} \]

satisfies the \(t\)-block constraints

\[ \sum_{k\in M_t} a^{(t)}_{i_1\ldots i_s}\bar x_{i_1\ldots i_s} = \sum_{k\in M_t} a^{(t)}_{i_1\ldots i_s} \left( \sum_{\omega_{r+1}=1}^{n_{r+1}}\cdots \sum_{\omega_s=1}^{n_s} x^{(\omega_{r+1}\ldots\omega_s)}_{i_1\ldots i_s} \lambda_{\omega_{r+1}\ldots\omega_s} \right) = \]

\[ = \sum_{k\in M_t} a^{(t)}_{i_1\ldots i_s} x^{(i_{r+1}\ldots i_s)}_{i_1\ldots i_s} \lambda_{i_{r+1}\ldots i_s} = \sum_{k\in M_t} a^{(t)}_{i_1\ldots i_s} y_{i_1\ldots i_r}\lambda_{i_{r+1}\ldots i_s} = b_{M_t(i_1\ldots i_s)} . \]

Since, by definition, \(\widetilde M\cap M_t=\varnothing\), it follows that

\[ \lambda_{i_{r+1}\ldots i_s} = \frac{b_{M_t(i_1\ldots i_s)}} {\displaystyle\sum_{k\in M_t} a^{(t)}_{i_1\ldots i_s}y_{i_1\ldots i_r}} . \]

It remains to show that condition (7) is satisfied. From the compatibility conditions,

\[ \sum_{k\in M_{j_*}\cup M_t\setminus M_t} b_{M_t(i_1\ldots i_s)} = \sum_{k\in M_{j_*}\cup M_t} \left( \sum_{\omega_{r+1}=1}^{n_{r+1}}\cdots \sum_{\omega_s=1}^{n_s} a^{(t)}_{i_1\ldots i_s} x^{(\omega_{r+1}\ldots\omega_s)}_{i_1\ldots i_s} \lambda_{\omega_{r+1}\ldots\omega_s} \right) = <!-- source-page: 003 --> \[ = \left( \sum_{\omega_{r+1}=1}^{n_{r+1}} \cdots \sum_{\omega_s=1}^{n_s} \lambda_{\omega_{r+1}\ldots\omega_s} \right) \left( \sum_{k \in M_{j_*}\cup M_t} u_{i_1\ldots i_s}^{(j_*)} x_{i_1\ldots i_s}^{(\omega_{r+1}\ldots\omega_s)} \right) = \]

\[ = \left( \sum_{\omega_{r+1}=1}^{n_{r+1}} \cdots \sum_{\omega_s=1}^{n_s} \lambda_{\omega_{r+1}\ldots\omega_s} \right) \left( \sum_{k \in M_{j_*}\cup M_t \setminus M_{j_*}} b_{M_{j_*}(i_1\ldots i_s)} \right) \]

it follows that

\[ \sum_{\omega_{r+1}=1}^{n_{r+1}} \cdots \sum_{\omega_s=1}^{n_s} \lambda_{\omega_{r+1}\ldots\omega_s} = 1. \]

The theorem is proved.

Consider a multi-index linear programming problem of transportation type, in which all constraints are given in the form of equalities, i.e., conditions (2) are written as follows:

\[ \sum_{k \in M_j} x_{i_1\ldots i_s} = b_{M_j(i_1\ldots i_s)}. \tag{2′} \]

Denote \(J_0 \equiv \{1,\ldots,t\}\), \(M_j^{(0)} \equiv M_j\). For each \(r=1,\ldots,t-2\), consider

\[ M_j^{(r)} \equiv M_j^{(r-1)} \setminus \bigcap_{j \in J_{r-1}} M_j^{(r-1)} \quad \text{for all } j \in J_{r-1}, \]

\[ M^{(r)} \equiv \bigcup_{j \in J_{r-1}} M_j^{(r-1)}. \]

Theorem 3. If for each \(r=1,\ldots,t-2\) there exists, in the last resort, one \(j_r \in J_{r-1}\) such that

\[ M^{(r)} \setminus M_{j_r}^{(r)} \subset \bigcap_{j \in J_r} M_j^{(r)}, \quad \text{where } J_r = J_{r-1}\setminus j_r, \]

then the multi-index problem of transportation type has a feasible solution if and only if the consistency conditions are satisfied.

Sufficiency. For \(t=2\), the consistency conditions are sufficient for any \(M_1\) and \(M_2\). In this case the feasible solution is written as follows:

\[ x_{i_1\ldots i_s} = \frac{ b_{M_1(i_1\ldots i_s)}\, b_{M_2(i_1\ldots i_s)} }{ \sum\limits_{k \in M_2^{(1)}} b_{M_1(i_1\ldots i_s)} }. \]

Consider the case \(t=3\). There exists \(j_1\) such that

\[ M^{(1)} \setminus M_{j_1}^{(1)} \subset \bigcap_{j \in J_1} M_j^{(1)}. \tag{8} \]

It is not hard to see that \(J_r\) contains \(t-r\) elements, i.e., for \(t=3\), \(J_1\) contains two elements.

Since for the \(X(j_1)\)-problem all the conditions of Theorem 2 are satisfied, for \(t=3\) the problem (1), (2′), (3) has a feasible solution.

Suppose that the theorem is true for \(t=\tau\). We shall show that it is true for \(t=\tau+1\). In this case condition (8) is also satisfied, and \(J_1\) contains \(\tau\) elements. Since the \(X(j_1)\)-problem has a solution, the \(X\)-problem also has a solution by Theorem 2. Necessary and sufficient conditions for the existence of a feasible solution to problem (2′), (3) were obtained in [2] by another method, in considering mixed strategies in coalition games.

Consider the case when all \(a_{i_1\ldots i_s}^{(j)} \ge 0\) and upper and lower bounds are imposed on all \(x_{i_1\ldots i_s}\), i.e.,

\[ 0 \le m_{i_1\ldots i_s}^{(1)} \le x_{i_1\ldots i_s} \le m_{i_1\ldots i_s}^{(0)} \le \infty. \tag{3′} \]

Denote

\[ \sum_{k\in M_j} m^{(l)}_{i_1\ldots i_s}x_{i_1\ldots i_s}\equiv A^{(l,l)}_{M_j(i_1\ldots i_s)} . \]

We shall compute

\[ m^{(l)}_{i_1\ldots i_s} = (-1)^l \min_{1\le j\le t} \left[ (-1)^l \left( \frac{b_{M_j(i_1\ldots i_s)}-A^{(j,l-1)}_{M_j(i_1\ldots i_s)}}{a^{(j)}_{i_1\ldots i_s}} + m^{(l-1)}_{i_1\ldots i_s} \right), \, (-1)^l m^{(l-2)}_{i_1\ldots i_s} \right]. \]

Theorem 4. In order that problem (2), (3′) have a solution, it is necessary that the condition

\[ (-1)^l\left[m^{(l)}_{i_1\ldots i_s}-m^{(l-1)}_{i_1\ldots i_s}\right]\ge 0 \]

be satisfied for all \(l=2,3,\ldots;\ i_k=1,\ldots,n_k;\ k=1,\ldots,s\). Moreover, after a finite number of steps \(l\), either all

\[ m^{(l)}_{i_1\ldots i_s}=m^{(l-2)}_{i_1\ldots i_s}, \]

and in this case all lower and upper bounds are balanced, or at least in one cell the condition

\[ (-1)^l\left[m^{(l)}_{i_1\ldots i_s}-m^{(l-1)}_{i_1\ldots i_s}\right]\ge 0 \]

is violated, i.e. problem (2), (3′) has no feasible solution.

Consider the multi-index LP problem of transportation type.

Theorem 5. If \(\mu_{i_1\ldots i_s}\) and \(M_{i_1\ldots i_s}\) are, respectively, the values of the balanced lower and upper bounds, then the inequalities

\[ \sum_{k\in M_j} M_{i_1\ldots i_s} + \left(\prod_{l\in M_j} n_l-1\right) \sum_{k\in M_j}\mu_{i_1\ldots i_s} \le \left(\prod_{l\in M_j} n_l\right)b_{M_j(i_1\ldots i_s)} \le \]

\[ \le \sum_{k\in M_j}\mu_{i_1\ldots i_s} + \left(\prod_{l\in M_j} n_l-1\right) \sum_{k\in M_j} M_{i_1\ldots i_s} \qquad \text{for all } j=1,\ldots,t. \]

Special cases of Theorems 4 and 5 were considered by the author in work \((^3)\).

Central Economic-Mathematical Institute
Academy of Sciences of the USSR

Received
23 IV 1964

CITED LITERATURE

1. G. S. Verkhovskii, DAN, 151, No. 3 (1963).
2. N. N. Vorob’ev, Theory of Probability and Its Applications, 7, 2 (1962).
3. G. S. Verkhovskii, On an Algorithm for Solving a Multi-Index Transportation Problem on an Electronic Computer, Novosibirsk, 1962.