UDC 51:330.115

MATHEMATICS

A. M. RUBINOV

EFFECTIVE TRAJECTORIES OF A DYNAMIC MODEL OF PRODUCTION

(Presented by Academician L. V. Kantorovich on 19 VI 1968)

The note studies the properties of certain classes of trajectories of a positively homogeneous dynamic model of production. This model may be regarded, on the one hand, as a generalization of the discrete model with variable technology \((^{1,2})\) (and, consequently, of the models of Neumann \((^3)\) and Gale \((^4)\)); on the other hand, as a generalization of the problem of continuous convex programming \((^5)\) (and, consequently, of Bellman’s “bottleneck” problem \((^6)\), and of the problem of continuous linear programming \((^7)\)).

\(1^\circ.\) Below we use the following definitions.

1) Let \(X_i\) be normed spaces, partially ordered respectively by closed, convex, reproducing, salient cones \(K_i\) \((i=1,2)\). A mapping* \(a\) of the cone \(K_1\) into the collection of subsets of the cone \(K_2\) is called superlinear if

\[ a(0)=0;\qquad \bigcup_{x\in K_1} a(x)=K_2;\qquad a(x_1+x_2)\supset a(x_1)+a(x_2)\quad (x_1,x_2\in K_1); \]

\[ a(\lambda x)=\lambda a(x)=\lambda a(x)\quad (\lambda>0,\ x\in K_1); \]

and from the relations \(x_n\in K_1,\ x_n\to x,\ y_n\in a(x_n),\ y_n\to y\) it follows that \(y\in a(x)\).

2) The normal hull of a mapping \(a\) will mean the mapping \(na\) which assigns to each \(x\) in \(K_1\) the set \(na(x)\), which is the union of the conic intervals \(\langle 0,y\rangle\) over all \(y\in a(x)\).

3) If \(a\) is a superlinear mapping, then on the cone \(K_2\) one may define the mapping \(\hat a\), inverse to the mapping \(a\): for \(y\in K_2\),

\[ \hat a(y)=\{x\in K_1\mid y\in a(x)\}. \]

4) If \(a_i\) \((i=1,2)\) is a mapping of the set \(Y_i\) into the collection of subsets of the set \(Y_{i+1}\), then the product of the mappings \(a_1\) and \(a_2\) is understood to be the mapping \(a_2\circ a_1\), defined on \(Y_1\) by the formula

\[ a_2\circ a_1(x)=\bigcup_{y\in a_1(x)} a_2(y)\qquad (x\in Y_1). \]

5) Let \(\xi\) be a subset of the vector space \(X\); a point \(x\in X\) will be called a boundary-below point of the set \(\xi\) if \(x\in \xi\) and \(\lambda x\notin \xi\) for any \(\lambda<1\).

\(2^\circ.\) In the model defined here, the time parameter \(t\) ranges over a set \(E\) of nonnegative real numbers, more than one point in size, containing zero. To each \(t\in E\) we assign a finite-dimensional real vector space \(X_t\), partially ordered by a convex closed, reproducing, salient cone \(K_t\). In addition, to each pair of indices \(t,\tau\in E,\ \tau>t\), we assign a mapping \(a_{\tau,t}\) of the cone \(K_t\) into the collection of subsets of \(K_\tau\). If the mappings \(a_{\tau,t}\) are superlinear and, moreover,

\[ a_{t,t'}\circ a_{t',t''}=a_{t,t''}\qquad (t>t'>t'';\ t,t',t''\in E), \]

then

* Here and in what follows we assume that the mappings under consideration carry each point of their domain into a nonempty subset.

We shall call an object \(\mathfrak M=\{E,\ (X_t)_{t\in E},\ (K_t)_{t\in E},\ (a_{\tau,t})_{t,\tau\in E;\ \tau>t}\}\) a positively homogeneous dynamic model of production (abbreviated p.h.m.).

Lemma 1. If \(\mathfrak M\) is a p.h.m., \(\tau,t\in E\), \(\tau>t\), \(x\in K_t\), then the set \(a_{\tau,t}(x)\) is convex, closed, and bounded.

A trajectory of a p.h.m. \(\mathfrak M\) will mean a family \(\chi=(x_t)_{t\in E}\) such that \(x_t\in K_t\) \((t\in E)\); \(x_\tau\in a_{\tau,t}(x_t)\) \((\tau,t\in E;\ \tau>t)\).

Theorem 1. If \(\mathfrak M\) is a p.h.m., \(t',t''\in E\), \(t'>t''\), \(x'\in K_{t'}\), \(x''\in a_{t'',t'}(x')\), then there exists a trajectory \((x_t)_{t\in E}\) of the model \(\mathfrak M\) such that \(x_{t'}=x'\), \(x_{t''}=x''\).

We shall say that a trajectory \((x_t)_{t\in E}\) issues from the point \(x\) if \(x_0=x\). Theorem 1 guarantees that trajectories issue from any point of the cone \(K_0\).

\(3^\circ\). Consider a p.h.m. \(\mathfrak M\). Put \(T=\sup E\). We shall call the model \(\mathfrak M\) finite if \(T\in E\). If \(x\in K_0\), then by the symbol \(\Gamma_T^x\) we denote the face of the cone \(K_T\) generated by the set \(a_{T,0}(x)\) (\(\Gamma_T^x\) coincides with the conical hull of the set \(n a_{T,0}(x)\); if \(x\) is an interior point of \(K_0\), then \(\Gamma_T^x=K_T\)). A trajectory \(\bar\chi=(\bar x_t)_{t\in E}\) of a finite p.h.m. will be called efficient if there exists a functional \(f\ne0\) from the space \((\Gamma_T^x-\Gamma_T^x)^*\), positive on \(\Gamma_T^x\), and such that
\[ f(\bar x_T)=\max_{y\in a_{T,0}(x)} f(y)\quad (\text{here }x=\bar x_0). \]

Lemma 2. Let \(\mathfrak M\) be a finite p.h.m. A trajectory \(\bar\chi=(\bar x_t)_{t\in E}\) of the model \(\mathfrak M\) is efficient if and only if \(\bar x_0\) is a lower boundary point of the set \(n a_{T,0}(\bar x_T)\) (here \(T=\sup E\)).

Let us now consider an arbitrary (not necessarily finite) p.h.m. \(\mathfrak M\).

For each trajectory \(\chi=(x_t)_{t\in E}\) of the model \(\mathfrak M\) put
\[ \hat a(\chi)=\bigcup_{t\in E}\widehat{n a}_{t,0}(x_t). \]
(If \(\mathfrak M\) is finite, then \(\hat a(\chi)=\widehat{n a}_{T,0}(x_T)\).) A trajectory \(\bar\chi=(\bar x_t)_{t\in E}\) of the model \(\mathfrak M\) will be called weakly efficient if * there exists \(f\ne0\) from \(K_0^*\) such that
\[ f(\bar x_0)=\min_{x\in \hat a(\bar\chi)} f(x); \]
efficient if \(\bar x_0\) is a lower boundary point of the set \(\hat a(\bar\chi)\); strongly efficient if there exists \(f\in K_0^*\) such that
\[ f(\bar x_0)=\min_{x\in \hat a(\bar\chi)} f(x)>0. \]

It is clear that a strongly efficient trajectory is efficient; in turn, an efficient trajectory is weakly efficient. At the same time, a weakly efficient trajectory issuing from an interior point of the cone \(K_0\) will be strongly efficient.

Theorem 2. If \(\mathfrak M\) is a p.h.m., then from any point \(x\) of the cone \(K_0\) such that \(a_{t,0}(x)\ne0\) \((t\in E)\), there issues an efficient trajectory.

Theorem 3. In order that a trajectory \(\bar\chi=(\bar x_t)_{t\in E}\) of a p.h.m. \(\mathfrak M\) be strongly efficient, it is necessary and sufficient that there exist a family \((f_t)_{t\in E}\) \((f_t\in K_t^*)\) possessing the following properties: 1) for any trajectory \(\chi=(x_t)_{t\in E}\) of the model \(\mathfrak M\), the function \(h\), defined on \(E\) by the formula \(h(t)=f_t(x_t)\), does not increase; 2) \(h(t)\equiv f_t(\bar x_t)=1\) for any \(t\in E\). If the family \((f_t)_{t\in E}\) satisfies conditions 1) and 2), then for any \(t\) and \(\tau>t\) \((\tau,t\in E)\)
\[ \max_{y\in a_{t,0}(\bar x_0)} f_t(y)=f_t(\bar x_t)=\min_{z\in \hat a_{\tau,t}(\bar x_\tau)} f_t(z). \tag{1} \]

We shall give a characterization of weakly efficient trajectories. First we introduce the following definition. Let \(\mathfrak M\) be a p.h.m., \(t,\tau\in E\), \(\tau>t\). For \(f\in K_t^*\) put
\[ a'_{\tau,t}(f)=\{g\in K_t^*\mid f(x)\ge \max_{y\in a_{\tau,t}(x)} g(y)\ \text{for any }x\in K_t\}. \tag{2} \]

* By the symbol \(K^*\) we denote the cone conjugate to \(K\).

Theorem 4. Let \(\overline{\chi}=(\overline{x}_t)_{t\in E}\) be a trajectory of the finite p.o.m. \(\mathfrak{M}\). In order that there exist a family \((f_t)_{t\in E}\) \((f_t\in K_t^*,\, f_t\ne 0)\) such that: 1) for any trajectory \(\chi=(x_t)_{t\in E}\) the function \(h(t)=f_t(x_t)\) does not increase; 2) the function \(\overline{h}(t)=f_t(\overline{x}_t)\) is constant, it is necessary and sufficient that there exist a functional \(f\) from the cone \(K_0^*\) possessing the following properties:

\[ \alpha)\quad f(\overline{x}_0)=\min_{x\in a_{T,0}(x_T)} f(x);\qquad \beta)\quad a'_{T,0}(f)\ne\{0\}\quad (T=\sup E). \]

Corollary. Let the finite p.o.m. \(\mathfrak{M}\) be such that \(a'_{T,0}(f)\ne\{0\}\) for every \(f\ne 0\). Then for every weakly efficient trajectory \(\overline{\chi}=(\overline{x}_t)_{t\in E}\) of this model there exists a family \((f_t)_{t\in E}\) satisfying conditions 1) and 2) of Theorem 4.

4°. In this section an infinite-dimensional analogue of a p.o.m. is considered.* Let \(Q\) be a metric compactum. By the symbol \(\Phi_{KR}(Q)\) we shall denote the space of all completely additive functions defined on the \(\sigma\)-algebra of Borel subsets of the compactum \(Q\), endowed with the Kantorovich–Rubinstein norm \((^8)\). By the symbol \(\operatorname{Lip}(Q)\) we shall denote the space of all functions defined on \(Q\) and satisfying there the Lipschitz condition \((\operatorname{Lip}(Q)\) is the space conjugate to \(\Phi_{KR}(Q)\)). By the symbol \(\operatorname{Lip}^+(Q)\) we shall denote the cone of nonnegative functions from \(\operatorname{Lip}(Q)\). Consider the model

\[ \mathfrak{M}=\{E,\ (Q_t)_{t\in E},\ (\operatorname{Lip}(Q_t))_{t\in E},\ (\operatorname{Lip}^+(Q_t))_{t\in E},\ (a_{\tau,t})_{\tau,t\in E;\ \tau>t}\}. \tag{3} \]

Here, as above, \(E\) is a more than one-point set of nonnegative numbers containing zero; \(Q_t\) is a metric compactum \((t\in E)\); \(a_{\tau,t}\) is a superlinear mapping of the cone \(\operatorname{Lip}^+(Q_t)\) into the collection of subsets of the cone \(\operatorname{Lip}^+(Q_\tau)\) such that: 1) \(a_{\tau,t}(u)\) is a compactum \((u\in\operatorname{Lip}^+(Q_t))\); 2) if \(\varphi\in\Phi_{KR}(Q_\tau)\), \(\varphi\geq 0\), then the functional \(q_\varphi\):

\[ q_\varphi(u)=\max_{v\in a_{\tau,t}(u)} \int_{Q_t} v\,d\varphi \qquad (u\in\operatorname{Lip}^+(Q_t)) \]

is upper semicontinuous in \(\sigma(\operatorname{Lip}(Q_t),\Phi_{KR}(Q_t))\). Moreover,

\[ a_{t,t'}\circ a_{t',t''}=a_{t,t''}\qquad (t>t'>t'';\ t,t',t''\in E). \]

A trajectory of model (3) will mean a family \(\chi=(u_t)_{t\in E}\) such that: 1) \(u_t\in\operatorname{Lip}^+(Q_t)\) \((t\in E)\); 2) if \(\tau,t\in E,\ \tau>t\), then \(u_\tau\in a_{\tau,t}(u_t)\). Theorem 1 (on the existence of trajectories) is also valid for the model under consideration. A trajectory \(\overline{\chi}=(\overline{u}_t)_{t\in E}\) will be called efficient if \(\overline{u}_0(s)>0\) \((s\in Q_0)\) and, in addition, for every \(t\in E\) there exists \(\varphi_t\in\Phi_{KR}(Q_t)\) \((\varphi_t\geq 0,\ \varphi_t\ne 0)\) such that

\[ \int_{Q_t} \overline{u}_t\,d\varphi_t = \max_{v\in a_{\tau,t}(\overline{u}_t)} \int_{Q_t} v\,d\varphi_t . \]

(This definition, as is not difficult to verify, is equivalent to the definition of an efficient trajectory proceeding from an interior point in a p.o.m.)

Theorem 5. Let \(u\in\operatorname{Lip}(Q_0)\), \(u(s)>0\) \((s\in Q_0)\). Then there exists an efficient trajectory of model (3) proceeding from \(u\).

Theorem 6. Let \(u\in\operatorname{Lip}(Q_0)\), \(u(s)>0\) \((s\in Q_0)\). In order that the trajectory \(\overline{\chi}=(\overline{u}_t)_{t\in E}\), proceeding from \(u\), be efficient, it is necessary and sufficient that there exist a family \((\varphi_t)_{t\in E}\) \((\varphi_t\in\Phi_{KR}(Q_t),\ \varphi_t\geq 0)\), possessing the following properties: 1) for any trajectory \(\chi=(u_t)_{t\in E}\) of the model \(\mathfrak{M}\) the function

\[ h(t)=\int_{Q_t} u_t\,d\varphi_t \]

does not increase; 2)

\[ \overline{h}(t)=\int_{Q_t} \overline{u}_t\,d\varphi_t=1 \]

for

\[ \text{* The importance of this analogue was pointed out to the author by G. Sh. Rubinstein.} \]

any \(t \in E\). If the family \(\left((\varphi_t)_{t \in E}\right)\) satisfies conditions 1) and 2), then for any \(t\) and \(\tau > t\) \((\tau, t \in E)\)

\[ \max_{v \in a_{T,0}(\bar u_t)} \int_{\dot Q_t} v\,d\varphi_t = \int_{\dot Q_t} \bar u_t\,d\varphi_t = \min_{v \in \hat a_{\tau,t}(\bar u_t)} \int_{\dot Q_t} v\,d\varphi_t . \]

5°. The results presented above were obtained with the aid of the duality apparatus developed in (2). The following plays an important role here.

Theorem 7. Let \(\mathfrak M=\{E,\ (X_t)_{t\in E},\ (K_t)_{t\in E}(a_{\tau,t})_{\tau,t\in E;\ \tau>t}\}\) be a p.o.m. Then the object \(\mathfrak M'=\{E,\ (X_t^*)_{t\in E},\ (K_t^*)_{t\in E},\ (a_{\tau,t})_{\tau,t\in E;\ \tau>t}\}\) is also a p.o.m. (here \(a_{\tau,t}\) is the mapping defined by formula (2)).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
12 VI 1968

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