UDC 519.35

MATHEMATICS

A. I. SOTSKOV

NECESSARY CONDITIONS FOR A MINIMUM FOR ONE TYPE OF NONSMOOTH PROBLEMS

(Presented by Academician L. V. Kantorovich, 28 III 1969)

The note considers a minimization problem with a nonsmooth equality-type constraint. To investigate the problem, one of the schemes of the method of feasible directions is applied: the derivative on the cone of feasible directions is nonnegative (one of the schemes was developed in \((^1)\)). The main difficulty of the investigation consists in describing the cone of feasible directions or the directional derivative. In particular, some minimax problems, problems of minimizing a function on the set of extrema of another function, are included in our problem, and for them necessary conditions for a minimum are written out.

Problem (1)—(4). Let real functions \(\varphi(y,x)\), \(f_\alpha(y,x)\), \(\alpha\in A\), \(y\in Y\), \(x\in X\), and an operator \(H:Y\times X\to Z\) be given, where \(A\) is some set of indices; \(Y,X,Z\) are complete normed spaces. For every \(y\in Y\) for which the set

\[ \mathfrak{M}(y)=\{x\mid H(y,x)=0,\ \max_{\alpha\in A} f_\alpha(y,x)\leq 0\}\ne\varnothing \]

is given, define uniquely the operator

\[ P:y\to P(y)\in\mathfrak{M}(y). \]

It is required to find

\[ \min_{(y,x)}\varphi(y,x) \tag{1} \]

on the set of pairs \((y,x)\) satisfying the constraints:

\[ H(y,x)=0, \tag{2} \]

\[ \max_{\alpha\in A} f_\alpha(y,x)\leq 0, \tag{3} \]

\[ x=P(y). \tag{4} \]

We shall agree to denote by \((y^*,x^*)\) a minimum point in problem (1)—(4). With respect to the operator \(H\) and the functions \(f_\alpha\), everywhere in what follows we assume the following: the operator \(H(y,x)\) is continuously differentiable with respect to \(y,x\), and \(H'_x\) maps \(X\) onto all of \(Z\) at each point \((y,x)\), \(x\in\mathfrak{M}(y)\); the functions \(f_\alpha(y,x)\) are convex jointly in \(y,x\) and satisfy a Lipschitz condition with a uniformly bounded Lipschitz constant on every bounded domain in \(Y\times X\); there exists a vector \((\bar y',\bar x')\in Y\times X\) such that

\[ H'_y(y^*,x^*)+H'_x(y^*,x^*)\bar x'=0,\qquad \max_{\alpha\in A} f_\alpha(y^*+\bar y',x^*+\bar x')<0. \]

Introduce the notation:

\[ \mathcal{K}=\{(\bar y,\bar x)\mid \exists\lambda'>0,\ \max_{\alpha\in A} f_\alpha(y^*+\lambda'\bar y,\ x^*+\lambda'\bar x)<0\}, \]

\[ L=\{(\bar y,\bar x)\mid H'_y(y^*,x^*)\bar y+H'_x(y^*,x^*)\bar x=0\}, \]

\[ K=L\cap\mathcal{K},\qquad K(\bar y)=\{\bar x\mid(\bar y,\bar x)\in K\}, \]

\[ \overline K(\bar y)=\{\bar x\mid(\bar y,\bar x)\in \overline K\}. \]

\(\operatorname{Pr}_{Y}K\) is the projection of the cone \(K\) onto the space \(Y\).

Assertion 1. Suppose that for any sequence \(y_\varepsilon\) of the form
\(y_\varepsilon = y^* + \varepsilon \bar y + o(\varepsilon \bar y)\), \(\bar y \in \operatorname{Pr}_{Y}K\), \(\varepsilon \ge 0\), for which \(\mathfrak M(y_\varepsilon) \ne \varnothing\), there exists a finite or infinite limit

\[ \lim_{\varepsilon\to +0} \frac{\varphi(y_\varepsilon, P(y_\varepsilon))-\varphi(y^*,x^*)}{\varepsilon} = \frac{\partial \varphi(y^*,P(y^*))}{\partial \bar y}, \]

called the directional derivative in the direction \(\bar y\) of the function \(\varphi(y,P(y))\) at the point \((y^*,x^*)\).

In order that the point \((y^*,x^*)\) furnish a minimum in problem (1)—(4), it is necessary that

\[ \inf_{\bar y\in \operatorname{Pr}_{Y}K} \frac{\partial \varphi(y^*,P(y^*))}{\partial \bar y} \ge 0. \tag{5} \]

The principal difficulty lies in establishing the existence and finding an explicit form of
\(\partial \varphi(y^*,P(y^*))/\partial \bar y\). Here these questions are considered for the operator
\(P: y \to P(y)\in \mathfrak M(y)\), where \(P(y)\) is the solution of a certain maximization problem:

\[ \psi(y,P(y))=\max_{x\in \mathfrak M(y)} \psi(y,x). \]

The minimax problem considered here is included in problem (1)—(4) when
\(\varphi(y,x)\equiv \psi(y,x)\). It is an essential complication and generalization of the usual minimax problem of the form

\[ \min_{y\in Q_y}\max_{x\in Q_x}\psi(y,x),\qquad Q_y\subset Y,\quad Q_x\subset X, \]

since both variables \(y,x\) are connected by relations (2), (3).

Problem (6). Find a point \((y^*,x^*)\) furnishing a minimax of \(\psi(y,x)\), i.e.

\[ \psi(y^*,x^*)=\max_{x\in \mathfrak M(y^*)}\psi(y^*,x^*) = \min_{y:\mathfrak M(y)\ne \varnothing}\max_{x\in \mathfrak M(y)}\psi(y,x). \tag{6} \]

A necessary condition for a minimax of \(\psi(y,x)\) at the point \((y^*,x^*)\) is given by the following

Theorem 1. Let the operator \(H(y,x)\) be linear in \(x\) for each \(y\); if
\(K(0)=\varnothing\), then \(\overline K(0)=\{0\}\), and the Hausdorff distance
\(\rho(\overline K(\bar y),\overline K(0))\to 0\) as \(\|\bar y\|\to 0\),
\(\bar y\in \operatorname{Pr}_{Y}K\). The function \(\psi(y,x)\) is continuously differentiable in \(y,x\), concave in \(x\), attains its unique maximum on \(\mathfrak M(y)\), and the maximization problem for \(\psi(y,x)\) on \(\mathfrak M(y)\) is stable with respect to the solution (i.e. from \(y\to y^*\) it follows that \(P(y)\to P(y^*)\), if \(\mathfrak M(y)\ne \varnothing\)).

Then, in order that the point \((y^*,x^*)\) furnish a minimax of the function \(\psi(y,x)\), it is necessary that for any linear functionals
\(w_1=(y_1^*,x_1^*)\in \mathcal K^*\), \(w_2=(y_2^*,x_2^*)\in L^*\), for which

\[ \psi'_x(y^*,x^*)+x_1^*+x_2^*=0, \tag{7} \]

the relation

\[ (\psi'_y(y^*,x^*)+y_1^*+y_2^*,0)\in \mathcal K^*+L^* \tag{8} \]

hold; moreover, if \(K(0)\ne \varnothing\), then (8) is equivalent to

\[ \psi'_y(y^*,x^*)+y_1^*+y_2^*=0. \]

The set of linear functionals \(w_1\in \mathcal K^*\), \(w_2\in L^*\), satisfying (7), is nonempty.

The conditions of the theorem ensure the differentiability of the function

\[ \psi(y,P(y))=\max_{x\in \mathfrak M(y)}\psi(y,x) \]

in each direction \(\bar y\in \operatorname{Pr}_{Y}K\) at the point

\((y^*, x^*)\) and \(\partial \psi(y^*, P(y^*)) / \partial \bar y = \sup\limits_{\bar x\in K} \psi'(y^*, x^*)\bar y\bar x\). Hence it is not difficult to obtain that

\[ -\inf_{\bar y\in \operatorname{Pr}_Y K}\ \sup_{\bar x\in K(\bar y)} \psi'(y^*,x^*)\bar y\bar x=0. \tag{9} \]

Condition (9) is equivalent to the assertion of the theorem (7), (8). In proving this fact the following fact is used.

Let in the normed space \(Y\) a convex cone \(\Omega\) with an interior point, with vertex at \(O\), be given, and let on it a linearly concave functional \(f(\bar y)\) be defined, i.e.

\[ \begin{gathered} f(\alpha\bar y)=\alpha f(\bar y),\quad \alpha>0,\quad \bar y\in\Omega,\\ f(\bar y_1+\bar y_2)\ge f(\bar y_1)+f(\bar y_2),\quad \bar y,\bar y_2\in\Omega,\\ |f(\bar y)|\le C\|\bar y\|,\quad \bar y\in\Omega; \end{gathered} \]

then for any \(\bar y^0\in\Omega^0\) there exists a supporting functional \(l\in Y^*\): \(l(\bar y)\ge f(\bar y)\), \(\bar y\in\Omega\), such that \(l(\bar y^0)=f(\bar y^0)\).

Remark 1. The condition of concavity of \(\psi(y,x)\) with respect to \(x\) may be omitted when \(K(0)=\{0\}\).

Let us note that the necessary condition formulated, as a first-order condition, is only a stationarity condition and does not separate a minimum–maximum from a maximum–maximum.

Remark 2. If there exists a vector \((\bar y,\bar x)\in K\), \(\psi'(y^*,x^*)\bar y\bar x>0\), then the necessary minimax condition may be written in the form

\[ -\psi_x'(y^*,x^*)\in \overline{K}(0)^*, \]

\[ \operatorname{Pr}_Y K\subseteq \operatorname{Pr}_Y\{(\bar y\bar x)\mid \psi'(y^*,x^*)\bar y\bar x\ge 0\}\cap K\}. \]

Theorem 2. Let the functions \(f_\alpha(y,x)\), \(\alpha\in A\), the operator \(H\), and the function \(\psi(y,x)\) be as in Theorem 1, and let \(\varphi(y,x)\) be continuously differentiable with respect to \(y,x\). Suppose, in addition, that the following conditions are satisfied:

a) \(|\varphi(y,x)-\varphi(y,x')|\le k_1|\psi(y,x)-\psi(y,x')|\), \(0<k_1<\infty\), for \(x,x'\in \mathfrak M(y)\);

b) the problem \(\sup\limits_{\bar x\in \overline K(\bar y)}\psi'(y^*,x^*)\bar y\bar x\) has a solution for every \(\bar y\in\operatorname{Pr}_Y K\).

Then, in order that the point \((y^*,x^*)\) deliver a minimum in problem (1)—(4), it is necessary that

\[ \inf_{(\bar y,\bar x)\in\Gamma}\varphi'(y^*,x^*)\bar y\bar x=0, \]

where

\[ \Gamma=\{(\bar y\bar x)\mid \psi'(y^*,x^*)\bar y\bar x= \sup_{\bar x'\in \overline K(\bar y)} \psi'(y^*,x^*)\bar y\bar x',\ \bar y\in\operatorname{Pr}_Y K\}. \]

Example. Find \(\min \varphi(y,x)\) under the constraints \(y+a_1x\le 0\), \(-y-a_2x\le 0\), \(x=P(y)\), where \(P(y)\):

\[ \varphi(y,P(y))=\max_{x\in\mathfrak M(y)}\varphi(y,x),\quad \mathfrak M(y)=\{x\mid y+a_1x\le 0,\ -y-a_2x\le 0\}. \]

Assume that the point \((0,0)\) gives a minimum of \(\varphi(y,x)\), and write the necessary conditions for a minimum. \(K^*=\{\lambda_1(-1,-a_1)+\lambda_2(1,a_2),\ \lambda_1,\lambda_2\ge 0\}\).

According to Theorem 1, for any linear functional \(w_1\in K^*\) for which

\[ \varphi_x'(0,0)-\lambda_1a_1+\lambda_2a_2=0, \]

there exists \(w_2\in K^*\), \(w_2=\lambda_3(-1,-a_1)+\lambda_4(1,a_2)\), \(\lambda_1,\lambda_2\ge 0\), such that

\[ \varphi_y'(0,0)-\lambda_1+\lambda_2=-\lambda_3+\lambda_4,\quad -\lambda_3a_1+\lambda_4a_2=0. \]

Let \(a_2\ne 0\) and \(0<a_1/a_2\le 1\). Then \(\varphi_y'(0,0)-\lambda_1+\lambda_2=\lambda_3(-1+a_1/a_2)\le 0\) for all \(\lambda_2\ge 0\), \(\lambda_2=[-\varphi_x'(0,0)+\lambda_1a_1]/a_2\), where \(\lambda_1\ge 0\).

The latter is either solvable for any \(\lambda_1 \geqslant 0\), and then \(\varphi_y'(0,0)-\varphi_x'(0,0)/a_2 \leqslant 0\), or solvable for \(\lambda_2=0,\ \lambda_1 \geqslant 0\), and then \(\varphi_y'(0,0)-\varphi_x'(0,0)/a_1 \leqslant 0\).

If \(a_1a_2<0\), then \(\lambda_3=\lambda_4=0\), and, consequently,
\[ \varphi_y'(0,0)-\lambda_1+\lambda_2=0 \]
for all those \(\lambda_1 \geqslant 0,\ \lambda_2 \geqslant 0\) for which
\[ \varphi_x'(0,0)-\lambda_1a_1+\lambda_2a_2=0. \]
It follows easily from this that \(\varphi_y'(0,0)=\varphi_x'(0,0)=0\). If \(a_1=0,\ a_2\ne0\), we obtain that
\[ \operatorname{sgn}\varphi_x'(0,0)=-\operatorname{sgn}a_2,\qquad \varphi_y'(0,0)-\frac{1}{a_2}\varphi_x'(0,0)\leqslant0. \]

Thus, a necessary condition for a minimum of \(\varphi(y,x)\) at the point \((0,0)\) is
\[ \varphi_y'(0,0)-\frac{1}{a_2}\varphi_x'(0,0)\leqslant0 \]
or
\[ \varphi_y'(0,0)-\frac{1}{a_1}\varphi_x'(0,0)\leqslant0 \]
for \(0<a_1/a_2\leqslant1\), and equality when \(a_1=a_2\);
\[ \varphi_y'(0,0)-\frac{1}{a_2}\varphi_x'(0,0)\leqslant0 \]
and
\[ \operatorname{sgn}\varphi_x'(0,0)=-\operatorname{sgn}a_2 \]
for \(a_1=0,\ a_2\ne0\);
\[ \varphi_y'(0,0)=\varphi_x'(0,0)=0 \]
for \(a_1a_2<0\).

Central Economics and Mathematics Institute
Academy of Sciences of the USSR
Moscow

Received
27 II 1969

REFERENCES

1. A. Ya. Dubovitskii, A. A. Milyutin, Zhurn. vychisl. matem. i matem. fiz., 5, No. 3, 395 (1965).