UDC 519.8

MATHEMATICS

V. F. Dem’yanov

ON DIRECTIONAL DIFFERENTIATION OF A MAXIMIN FUNCTION

(Presented by Academician L. V. Kantorovich, 1 VI 1967)

In \((^{1,2})\), for the solution of minimax problems the maximum function was studied, and directional differentiability was proved for the function

\[ \varphi(X)=\min f(X,Y). \]

The use of directional differentiability of such functions makes it possible to develop effective methods for solving a number of problems (best-approximation problems, minimax problems, etc.).

For the study of some other problems (for example, pursuit problems in the theory of differential games) it may be useful to study a function of the form

\[ \varphi(Z)=\max_X \min_Y f(X,Y,Z). \]

Below we present some results connected with the question of the differentiability of the function \(\varphi(Z)\) in directions.

Although a finite-dimensional case is considered here, many of the results can be generalized to the infinite-dimensional case (just as the results obtained in \((^1)\) for the finite-dimensional case were extended in \((^2)\) to a more general case).

\(1^\circ\). Let \(f(X,Y,Z)\) be a function continuous in \(X\) and \(Y\) and continuously differentiable in \(Z\) on \(\Omega_X \times \Omega_Y \times \Omega_Z\), where \(\Omega_X \subset E_n,\ \Omega_Y \subset E_m;\ \Omega_Z \subset E_p\).

On \(\Omega_Z\) consider the function

\[ \varphi(Z)\equiv \max_{X\in\Omega_X}\ \min_{Y\in\Omega_Y} f(X,Y,Z). \tag{1} \]

The sets \(\Omega_X,\Omega_Y,\Omega_Z\) are bounded and closed.

Fix some \(Z\in\Omega_Z\). Let \(g\in E_p;\ g\ne 0\) be such that, for \(\alpha\in[0,\alpha_0]\) \((\alpha_0=\alpha_0(g)>0)\), the point \(Z_\alpha=Z+\alpha g\in\Omega_Z\). Such a \(g\) will be called an admissible direction.

It is required to find the derivative of \(\varphi(Z)\) in the direction \(g\)

\[ \varphi'_Z(g)\equiv \lim_{\alpha\to +0}\frac{\varphi(Z+\alpha g)-\varphi(Z)}{\alpha}. \tag{2} \]

Introduce into consideration the sets \(R(Z)\subset\Omega_X\) and \(Q(X,Z)\subset\Omega_Y\):

\[ R(Z)=\left\{X\mid X\in\Omega_X;\ \min_{Y\in\Omega_Y} f(X,Y,Z)=\max_{X\in\Omega_X} f(X,Y,Z)\right\}, \]

\[ Q(X,Z)=\left\{Y\mid Y\in\Omega_Y;\ f(X,Y,Z)=\min_{Y\in\Omega_Y} f(X,Y,Z)\right\}. \]

In the present section we shall assume that the sets \(Q(X,Z)\) satisfy the following rather stringent condition:

Condition A. For a given \(Z\), the set \(Q(X,Z)\) depends continuously on \(X\) on the set \(R(Z)\) \((Q(X',Z)\to Q(X,Z)\) as \(X'\to X;\ X'\in\Omega_X;\ X\in R(Z))\).

This means that

\[ \rho\bigl(Q(X',Z),Q(X,Z)\bigr)\equiv \]

\[ \equiv \sup_{Y\in Q(X',Z)}\inf_{V\in Q(X,Z)}\|V-Y\|+ \sup_{V\in Q(X,Z)}\inf_{Y\in Q(X',Z)}\|V-Y\| \longrightarrow 0,\qquad X'\to X. \tag{3} \]

We note that this condition is satisfied, for example, if \(Q(X,Z)\) consists, for every \(X\in R(Z)\), of a single point.

Theorem 1. If at the point \(Z \in \Omega_Z\) condition A is satisfied, then the function \(\varphi(Z)\) is differentiable in any admissible direction, and

\[ \varphi'_Z(g)=\max_{X\in R(Z)}\min_{Y\in Q(X,Z)}(\partial f(X,Y,Z)/\partial Z,g). \]

\(2^\circ\). Let the function \(f(X,Y,Z)\) be given and continuous on \(E_n \times \Omega_X \times \Omega_Z\), where \(\Omega_X \subset E_m\), \(\Omega_Z \subset E_p\); \(\Omega_Y,\Omega_Z\) are bounded closed sets. Fix a point \(Z \in E_p\). Consider the sets \(R(Z)\subset E_n\) and \(Q(X,Z)\); they were defined in \(1^\circ\), only here already

\[ R(Z)=\{X\mid X\in E_n;\ \min_{Y\in\Omega_Y} f(X,Y,Z)=\max_{X\in E_n}\min_{Y\in\Omega_Y} f(X,Y,Z)\}. \]

We shall assume that:

1. For the point \(Z \in E_p\) there does not exist a sequence of points \(\{X_j\}\), \(\|X_j\|\to\infty\), such that \(\Phi(X_i,Z)\to \sup_{X\in E_n}\Phi(X,Z)\), where
   \[ \Phi(X,Z)=\min_{Y\in\Omega_Y} f(X,Y,Z). \]

2. The function \(f(X,Y,Z)\) is twice continuously differentiable with respect to \(X\) and \(Z\) on \(E_n\times\Omega_Y\times\Omega_Z\) and is strictly concave with respect to \(X\) for any fixed \(Y\in\Omega_Y,\ Z\in\Omega_Z\), and the components of the vector functions \(\partial f(X,Y,Z)/\partial X\), \(\partial f(X,Y,Z)/\partial Y\) and the elements of the matrices \(\partial^2 f(X,Y,Z)/\partial X^2\), \(\partial^2 f(X,Y,Z)/\partial Z^2\), \(\partial^2 f(X,Y,Z)/\partial X\partial Z\) are bounded on \(E_n\times\Omega_Y\times\Omega_Z\).

Let \(g\in E_p\) be an admissible direction.

Theorem 2. If conditions 1 and 2 are satisfied at the point \(Z\), then the function

\[ \varphi(Z)=\max_{X\in E_n}\min_{Y\in\Omega_Y} f(X,Y,Z) \]

is differentiable in any admissible direction \(g\), and

\[ \varphi'_Z(g)\equiv \lim_{\alpha\to+0}\frac{1}{\alpha}[\varphi(Z+\alpha g)-\varphi(Z)] = \]

\[ = \max_{X\in R(Z)}\max_{V\in E_n}\min_{Y\in Q(X,Z)} \left[(\partial f(X,Y,Z)/\partial X,V)+(\partial f(X,Y,Z)/\partial Z,g)\right]. \tag{4} \]

\(3^\circ\). Finally, let the function \(f(X,Y,Z)\) be given and continuous on \(\Omega_X\times\Omega_Y\times\Omega_Z\), where \(\Omega_X\subset E_n\), \(\Omega_Y\subset E_m\), \(\Omega_Z\subset E_p\) are bounded closed sets, and let \(f(X,Y,Z)\) be twice continuously differentiable on \(\Omega_X\times\Omega_Y\times\Omega_Z\) and strictly convex with respect to \(X\) for any fixed \(Y\in\Omega_Y,\ Z\in\Omega_Z\). The sets \(R(Z)\) and \(Q(X,Z)\) are defined as above.

Theorem 3. The function

\[ \varphi(Z)\equiv \max_{X\in\Omega_X}\min_{Y\in\Omega_Y} f(X,Y,Z) \]

is differentiable in any admissible direction \(g\), and

\[ \varphi'_Z(g)= \]

\[ = \max_{X\in R(Z)}\max_{V\in M_X(\Omega_X)}\min_{Y\in Q(X,Z)} \left[(\partial f(X,Y,Z)/\partial X,V)+(\partial f(X,Y,Z)/\partial Z,g)\right], \tag{5} \]

where \(M_X(\Omega_X)\) is the cone of admissible directions in the broad sense of the word, constructed at the point \(X\) with respect to the set \(\Omega_X\) (see (3)).

Remark 1. Formulas (3), (4), (5) are also valid in the case when the sets \(\Omega_Y\) and \(\Omega_Z\) are not bounded. In this case an additional condition must be imposed on the function \(f(X,Y,Z)\): the sets \(Q(X,Z)\) must be bounded, i.e., for some \(\varepsilon>0\) the set
\[ \bigcup_{X\in R(Z),\,Z'\in S_\varepsilon(Z)} Q(X,Z'), \]
where \(S_\varepsilon(Z)\subset \Omega_Z\) is the sphere of radius \(\varepsilon\) with center at the point \(Z\), must be bounded.

Remark 2. In a similar way one can get rid of the boundedness of the set \(\Omega_X\) in \(1^\circ\).

Remark 3. In \(2^\circ\) and \(3^\circ\) the fulfillment of condition A was not required.

Leningrad State University
named after A. A. Zhdanov

Received
22 V 1967

REFERENCES

\(^{1}\) V. F. Demyanov, Cybernetics, 2, No. 6 (1966).
\(^{2}\) V. F. Demyanov, Vestnik Leningrad University, No. 7 (1966).
\(^{3}\) V. F. Demyanov, A. M. Rubinov, Economics and Mathematical Methods, No. 3 (1966).