E. Ya. Remez

ON FINDING BOUNDARY SOLUTIONS OF A SYSTEM OF LINEAR INEQUALITIES AND ON THE METHOD OF BALANCING DESCENTS

(Presented by Academician V. I. Smirnov on 3 IX 1964)

Despite the importance of systems of linear inequalities (s.l.i.) in mathematics itself, and especially in its modern applications (cf. certain programmatic statements in \((^1)\), as well as in connection with the extremal problems considered by us, \((^{2,3})\)), the state of development and introduction of computational methods for the effective solution of s.l.i. still lags behind the urgent needs.

In § 1 of the present paper we indicate a method, based on simple considerations and readily accessible to computational implementation, for finding a particularly important category—boundary solutions \((^{4,5})\) of s.l.i. of type (1) (i.e., solutions of s.l.i. (1) satisfying some maximal subsystem of linearly independent boundary equations)—by means of the standard Jordan transformation procedure (Jordan substitution, Jordan elimination \((^6)\)), applied iteratively either autonomously or as an elementary computational step of a one-phase \((^{13})\) simplex process of linear programming (l.p.)*.

In § 2 we indicate the application of the algorithm of § 1 as a decisive element in the standardization and increased efficiency of the method of balancing descents (see \((^{2,3})\), where the history of the question is also discussed) for constructing solutions of Chebyshev and generalized-Chebyshev minimax problems.

§ 1. Let an arbitrary finite s.l.i. of “closed type” be given

\[ \varphi_i(x)\equiv \sum_{j=1}^{n} a_{ij}x_j \leq c_i \quad (i=1,\ldots,m). \tag{1} \]

The following algorithm, while at the same time determining the rank \(r\) of the matrix \((a_{ij})\) and establishing the very fact of consistency or inconsistency of system (1), directly yields, in the case of consistency, \(\infty^{\,n-r}\) of some of its boundary solutions, constituting the general solution of one of its fundamental (i.e., of rank \(r\)) boundary subsystems of equations

\[ \sum_{j=1}^{n} a_{i_s j}x_j=c_{i_s}\quad (s=1,\ldots,r) \tag{2} \]

and at the same time satisfying, with the sign \(\leq\), the entire s.l.i. (1) as a whole.

Putting \(c_i-\varphi_i(x)=u_i\) \((i=1,\ldots,m)\), we represent the system \(m\)

* The fact is known (see \((^{7,8})\)) of the inverse reducibility (in principle, though not in a practically justified aspect) of a pair of mutually dual l.p. problems (or of the corresponding matrix game) to a certain special type of s.l.i. In D. Gale’s work \((^9)\) it was noted, on the other hand, that the reduction of finding a particular solution of an s.l.i. to a certain l.p. problem is possible. However, the algorithm formulated below in Gale’s book is based on complicated considerations, and the resulting l.p. problem thereby loses out because of its obviously almost complete degeneracy.

linear relations connecting \(\{u_i\}\) and \(\{x_j\}\) by the table

\[ \begin{array}{c|ccc|c} & -x_1 & \cdots & -x_n & 1\\ \hline u_1= & a_{11} & \cdots & a_{1n} & c_1\\ \vdots & \vdots & & \vdots & \vdots\\ u_m= & a_{m1} & \cdots & a_{mn} & c_m \end{array} \tag{3} \]

Performing successive Jordan substitutions \(( (6); (10),\ \text{p. }167)\), each time interchanging one of the \(\{u_i\}\) with some one of the \(\{x_j\}\) and carrying out the corresponding transformation of the \((m,n+1)\)-matrix* of the elements \(a_{ij}, c_i\), under the single mandatory condition that the pivot element \(a_{i_\nu j_\nu}\), located at the intersection of the row and column of the variables being interchanged (more precisely, \(a^{\nu}_{i_\nu j_\nu}\), where \(\nu\) is the current number of the table obtained after \(\nu\) Jordan transformations, with \(a^0_{ij}\equiv a_{ij}\)), be nonzero, after \(r\) successive substitutions (\(r\) is the rank of the matrix \((a_{ij})\)) we obtain (assuming, for definiteness and for greater expressiveness of notation, that \(r<n,\ r<m\)), with a possible change in the numbering of the variables \(u_i\) and \(x_j\), a table of the form:

\[ \left. \begin{array}{c} (4_1)\left\{ \begin{array}{r} u_{r+1}=\\ \vdots\\ u_m= \end{array}\right.\\[3.0em] (4_2)\left\{ \begin{array}{r} x_1=\\ \vdots\\ x_r= \end{array}\right. \end{array} \quad \begin{array}{c|ccc|ccc|c} & -u_1 & \cdots & -u_r & -x_{r+1} & \cdots & -x_n & 1\\ \hline & a^r_{11} & \cdots & a^r_{1r} & 0 & \cdots & 0 & c^r_1\\ & \vdots & & \vdots & \vdots & & \vdots & \vdots\\ & a^r_{m-r,1} & \cdots & a^r_{m-r,r} & 0 & \cdots & 0 & c^r_{m-r}\\ \hline & a^r_{m-r+1,1} & \cdots & a^r_{m-r+1,r} & a^r_{m-r+1,r+1} & \cdots & a^r_{m-r+1,n} & c^r_{m-r+1}\\ & \vdots & & \vdots & \vdots & & \vdots & \vdots\\ & a^r_{m1} & \cdots & a^r_{mr} & a^r_{m,r+1} & \cdots & a^r_{mn} & c^r_m \end{array} \right\}, \tag{4} \]

\[ \begin{array}{r|ccc|c} & -u_1 & \cdots & -u_r & 1\\ \hline u_{r+1}- & a^r_{11} & \cdots & a^r_{1r} & c^r_1\\ \vdots & \vdots & & \vdots & \vdots\\ u_m= & a^r_{m-r,1} & \cdots & a^r_{m-r,r} & c^r_{m-r} \end{array} \quad . \tag{\(\bar 4_1\)} \]

We shall refer to table (4) as the union of two tables \((4_1)\) and \((4_2)\), containing, respectively, the first \(m-r\) and the last \(r\) rows. In what follows we shall have to operate essentially with table \((4_1)\), which we reproduce here once more separately in the reduced form \((\bar 4_1)\). The table \((4_2)\), however, we reserve unchanged for use at the completion of the entire procedure.

If all the numbers \(c^r_1,\ldots,c^r_{m-r}\) in \((\bar 4_1)\) are nonnegative, then, putting \(u_1=0,\ldots,u_r=0,\ u_{r+1}=c^r_1\ge0,\ldots,u_m=c^r_{m-r}\ge0\) and substituting into \((4_2)\) the zero values \(u_1,\ldots,u_r\), with \(x_{r+1},\ldots,x_n\) left arbitrary, we directly obtain the definition of a family of \(\infty^{\,n-r}\) desired boundary solutions of the system of linear inequalities (1) (\(\infty^0=1\) for \(r=n\)), corresponding to the fundamental boundary subsystem \(\varphi_1(x)=c_1,\ldots,\varphi_r(x)=c_r\).**

If, however, among the numbers \(c^r_1,\ldots,c^r_{m-r}\) there are negative ones, then, treating the system of linear relations of table \((\bar 4_1)\) in the same way as, in linear programming, the system of restrictive relations reduced to explicit form is treated (in the search for an initial feasible basic solution), we introduce \(((8),\ \text{p. }67)\) an “artificial” nonnegative variable \(u_0\) (only one) and

* Each transformation requires \(m(n+1)\) operations of the second degree.
** For \(r=m\) we have not a subsystem, but one embracing all \(m\) boundary equations of the system; in this case (4) reduces to just one table \((4_2)\).

the auxiliary estimating function \(V\), equal to the expression \((-u_0)\) in terms of the nonbasic variables, and carry out an obviously finite one-phase \((^{13})\) simplex process for l.p. If \((-u_0)_{\max}<0\), then the s.l.i. (1) is inconsistent; if, however, \((-u_0)_{\max}=0\), then at our disposal there will be a tableau of type \((4_1)\), already with an obviously nonnegative right-hand column, only with a different partition of the system of the \(m\) variables \(\{u_i\}\) into two subsystems: at the top we shall have some headings \(-u_{i_1},\ldots,-u_{i_r}\), and on the left headings \(u_{i_{r+1}}=,\ldots,u_{i_m}=\). The fundamental boundary subsystem of equations will be \(\varphi_{i_1}(x)=c_{i_1},\ldots,\varphi_{i_r}(x)=c_{i_r}\). Into the unchanged tableau \((4_2)\) we now substitute those values of the variables \(u_1,\ldots,u_r\) which they have acquired in the optimal basic solution of the auxiliary l.p. problem, and with the remaining, again arbitrary, \(x_{r+1},\ldots,x_n\), we shall again obtain \(\infty^{\,n-r}\) boundary solutions of our s.l.i. (1).

§ 2. Turning to the application of the algorithm of § 1 indicated in the introduction, it will suffice for us to dwell on the most general discrete problem of conditional Chebyshev minimax (problem D in the notation of our paper \((^{11b})\); see also \((^3)\)):

\[ \max_{i=1,\ldots,N}\Phi_i(x)\equiv \max_{i=1,\ldots,N}[\varphi_i(x)+b_i]\equiv \max_{i=1,\ldots,N}\left(\sum_{j=1}^{n}a_{ij}x_j+b_i\right)\equiv L(x)=\min! \tag{5} \]

under the conditions (which determine the set of admissible sets \(\{x\}=\Xi\subset R_n\))

\[ \chi_\nu(x)\equiv \varphi_{N+\nu}(x)+l_\nu\equiv \sum_{j=1}^{n}a_{N+\nu,j}x_j+l_\nu\leqslant 0 \qquad (\nu=1,\ldots,N'). \tag{6} \]

For there to exist a nonempty set \(\mathfrak K_0=\{x^{(0)}\}\) of solutions of problem (5)—(6), it is necessary and sufficient \((^{3,12,11a})\) that, with \(\Xi\) nonempty (i.e. \(\Xi\ne\Lambda\)), there not exist a set \(x\in R_n\) for which \(\varphi_i(x)<0\) \((i=1,\ldots,N)\), \(\chi_\nu(x)\leqslant 0\) \((\nu=1,\ldots,N')\). The discovery of such a set (indicating that \(\inf L(x)=-\infty\)) will be denoted by us as the situation n.o. (a certificate of improper conditioning \((^{11a})\) of the given problem (5)—(6)). The iterative process formulated below, which is a development and computational implementation of earlier results \((^3)\) and \((^2)\), Chap. VI, aims at finding one of the solutions \(x^{(0)}\) or at detecting the fact that \(\mathfrak K_0=\Lambda\).

Suppose we already have some point \(x^*\in\Xi\). At the beginning of the process such a point can be obtained by direct application of the algorithm of § 1, unless it is thereby found that \(\Xi=\Lambda\), and hence \(\mathfrak K_0=\Lambda\). In general, by \(x^*\in\Xi\) we may mean either the mentioned initial point or any current one to which some number of already realized steps of the process has led us. We must formulate a mode of action ensuring that at the next step one of the following three possible results is obtained: either the determination of a correction \(\Delta x^*=\xi\) such that \(x^*+\xi\in\Xi\) and \(L(x^*+\xi)<L(x^*)\); or the ascertainment of the fact that the solution \(x^*\in\mathfrak K_0\) has already been attained; or, finally, the detection of the situation n.o.

Suppose that at the point \(x^*\), after a possible change in the numbering of the functions \(\Phi_i\) and \(\chi_\nu\), we have (with \(s+t=m\))

\[ L(x^*)=\Phi_1(x^*)=\ldots=\Phi_s(x^*)>\Phi_{i'}(x^*) \qquad (s\geqslant 1;\ i'=s+1,\ldots,N); \tag{7} \]

\[ \chi_1(x^*)=\ldots=\chi_t(x^*)=0;\qquad \chi_{\nu'}(x^*)<0. \qquad (t\geqslant 0;\ \nu'=t+1,\ldots,N'). \tag{8} \]

The key approach here consists in finding a solution of the system of equations (where \(z=(z_1,\ldots,z_n)\)):

\[ \varphi_i(z)=-1 \quad (i=1,\ldots,s), \qquad \varphi_{N+\nu}(z)=0 \quad (\nu=1,\ldots,t) \tag{9} \]

or, upon discovering that no such solution exists, in finding a boundary solution (in both cases at least one) of the s.l.i.:

\[ \varphi_i(z)\leq -1 \quad (i=1,\ldots,s); \qquad \varphi_{N+\nu}(z)\leq 0 \quad (\nu=1,\ldots,t). \tag{10} \]

Put

\[ u_i=-1-\varphi_i(z) \quad (i=1,\ldots,s); \qquad u_{s+\nu}=-\varphi_{N+\nu}(z) \quad (\nu=1,\ldots,t). \tag{11} \]

Representing the system of linear relations (11) in the form of a table of type (3) (with \(z\) instead of \(x\) and the other obvious replacements), and then transforming it to the form (4) \(\equiv (4_1)\cup(4_2)\) (with, if possible, again a change in the numbering of both systems of variables \(\{u_i\}_1^m\) and \(\{z_j\}_1^n\)), we shall distinguish three possible cases:

\(1^0\). The right-hand column \(c'=(c_1^r,\ldots,c_{m-r}^r)\) in \((4_1)\) consists of zeros alone (\(c'=0\)), or, for \(r=m\), the entire table (4) reduces to only \((4_2)\). Then
\[ \bar z=(c_{m-r+1}^r,c_{m-r+2}^r,\ldots,c_m^r,0_{r+1},\ldots,0_n), \]
or, more precisely, the set \(\tilde z\) obtained from \(\bar z\) when the original numbering \(z_1,\ldots,z_n\) is restored, gives a particular solution of the system of equations (9) (the point \(x^*\) is passable\({}^{(3)}\)). The possibility is not excluded that, for \(\tilde z\), one has \(\varphi_i(\tilde z)<0\) \((i=1,\ldots,N)\), \(\varphi_{N+\nu}(\tilde z)\leq 0\) \((\nu=1,\ldots,N')\), which would mean the n.o. situation and the immediate completion of the entire process with the conclusion \(\mathfrak K_0=\Lambda\). If, however, we do not encounter such a situation, then for the required correction \(\Delta x^*=\xi\) we put \(\xi=h\tilde z\) and take for \(h\) the smallest positive value for which, for some \(i'=i_0'\), one obtains
\[ \Phi_{i'}(x^*+h\tilde z)=L(x^*+h\tilde z), \]
or, for some \(\nu'=\nu_0'\),
\[ \chi_{\nu'}(x^*+h\tilde z)=0. \]

\(2^0\). \(c'\ne 0\), but \(c'\geq 0\) (the point \(x^*\) is nodal\({}^{(3)}\)). Then the same set \(\tilde z\) indicated above is one of the boundary solutions of the s.l.i. (10). Putting \(\Delta x^*=\xi=h\tilde z\), we choose \(h\), as in item \(1^0\), with the same reservation concerning the possible discovery of the n.o. situation.

\(3^0\). Among the numbers \(c_1^r,\ldots,c_{m-r}^r\) there are negative ones (the point \(x^*\) is again nodal). Having formed and solved the auxiliary l.p. problem as in § 1, if \((-u_0)_{\max}<0\) we have the immediate completion of the process—now with the conclusion that a solution has been attained (\(x^*\in\mathfrak K_0\)) for problem (5)—(6). If, however, \((-u_0)_{\max}=0\), then, on the basis of the final simplex table, the assessment of the situation and the course of action are as in item \(2^0\).

From the results of the general analysis of the question in \({}^{(3)}\) it follows that a finite number of transitions of the indicated type \((x^*,x^*+h\tilde z)\) will always lead to the finding of a solution \(x^{(0)}\in\mathfrak K_0\) of problem (5)—(6), or to the establishment of the fact \(\mathfrak K_0=\Lambda\).

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
1 IX 1964

REFERENCES

\({}^{1}\) L. V. Kantorovich, Siberian Mathematical Journal, 3, No. 5 (1962).
\({}^{2}\) E. Ya. Remez, General Computational Methods of Chebyshev Approximation, Publishing House of the Academy of Sciences of the Ukrainian SSR, 1957.
\({}^{3}\) E. Ya. Remez, A. S. Shteinberg, Reports of the Academy of Sciences of the Ukrainian SSR, No. 8 (1961).
\({}^{4}\) S. N. Chernikov, UMN, 8, No. 2 (1953); Proceedings of the 4th All-Union Mathematical Congress, 2, 1964.
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\({}^{10}\) D. B. Yudin, E. G. Golshtein, Problems and Methods of Linear Programming, 1961.
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\({}^{12}\) L. V. Kantorovich, Economic Calculation of the Best Use of Resources, Publishing House of the Academy of Sciences of the USSR, 1959.
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