UDC 518:62-50

MATHEMATICS

V. F. SHOLOKHOVICH

UNSTABLE EXTREMAL PROBLEMS

AND GEOMETRIC PROPERTIES OF BANACH SPACES

(Presented by Academician A. N. Tikhonov on 21 IV 1970)

1. We consider the extremal problem

[
\inf_{x\in M} F(x)=F_0,
\tag{1}
]

where (x) is an element of the real Banach space (X), and (M\subseteq X) is a convex closed set. Suppose that the functional (F(x)) attains the minimal value (F_0) on the set (\Omega_0\subseteq M), with (\Omega_0) nonempty. The optimization problem (1) is called stable (see ((^{1,2}))) if it is solvable and, for any sequence ({x_n}) minimizing (F(x)) on (M), the condition
[
\rho(x_n,\Omega_0)=\inf_{x\in\Omega_0}|x_n-x|\to 0
]
is satisfied. It is known, however, that there are whole classes of extremal problems which, although solvable, belong to the class of unstable ones ((^{1-5})).

The regularization method for constructing a strongly convergent minimizing sequence in such problems was first proposed by A. N. Tikhonov ((^{1,2})). A direct development of this method is given in the works ((^{3-5})). In the present note, regularization is carried out by constructing a sequence of sets, in each of which an element of least norm is sought. In terms of the geometry of the unit sphere of the (B)-space (X), conditions are found that are necessary and sufficient for the existence and convergence of a minimizing sequence constructed in this way for a very broad class of functionals (F(x)).

2. Everywhere below we use the following notation: (U) is the unit ball of the (B)-space (X), and its boundary is
[
S={x\in X:|x|=1}.
]
The functional (F(x)) is called convex if
[
F!\left[\frac{x_1+x_2}{2}\right]\le
\frac12 F(x_1)+\frac12 F(x_2)
]
for all (x_1,x_2); (F(x)) is called a quasiconvex functional if, for any (x_1,x_2),
[
F!\left[\frac{x_1+x_2}{2}\right]\le
\max{F(x_1),F(x_2)}.
]
Clearly, a convex functional is quasiconvex, whereas the converse is, in general, false. One says that the (B)-space (X) is strictly convex if (S) contains no line segments (of course, not degenerating to a point).

Following ((^{6})), we shall call a set (K\subseteq X) approximately compact if, for every (x\in X), any sequence ({y_n}\subseteq K) such that (\rho(x,y_n)\to\rho(x,K)) has a limit point (y\in K). Following ((^{7})), we shall say that the (B)-space (X) has the Efimov–Stechkin property if every sequentially weakly closed set in (X) is approximately compact. A Banach space (X) is called an (E)-space if it is strictly convex and has the Efimov–Stechkin property ((^{7})).

It is shown in ((^{6})) that uniformly convex (B)-spaces are (E)-spaces, for example (L_p), (p>1). On the other hand, examples are known of strictly convex reflexive spaces that do not possess the Efimov–Stechkin property and are therefore not (E)-spaces.

properties ((^7,\, ^8)). Note that (E)-spaces are the largest known class of Banach spaces for which the operator (P) of metric projection onto a closed convex set is correct in the sense of Hadamard (see ((^9))). Let us give one equivalent definition of an (E)-space ((^{10})): a space (X) is an (E)-space if it is reflexive, strictly convex, and from the conditions ({x_n}\subset X), (x\in X), (|x_n|=|x|=1), and (x_n\to x) (weakly) it follows that (|x_n-x|\to 0).

3. We now consider a regularization algorithm for problem (1). We shall assume that there is a method allowing one to solve the optimization problem (1) with arbitrarily high accuracy in the functional, i.e., that for any (\delta>0) an (R_\delta) is given, defined by the relation (|F_0-F_\delta|\leq \delta). To problem (1) we assign the following problem: find

[
\inf |x| \quad \text{under the condition } x\in \Omega_\delta,
]

where

[
\Omega_\delta={x\in M:F(x)\leq F_\delta+\delta}.
\tag{2}
]

Theorem 1. In a Banach space (X) the following assertions are equivalent:

a) problem (2) has a unique solution (x_\delta) for any (\delta>0), (F_\delta), convex closed set (M), and quasiconvex lower semicontinuous functional (F(x));

b) the space (X) is strictly convex and reflexive.

Theorem 2. In a strictly convex reflexive (B)-space (X) the following assertions are equivalent:

a) the sequence ({x_\delta}) of solutions of problem (2) converges strongly as (\delta\to 0) to (x_0), the element of the set (\Omega_0) with least norm, for any (F_\delta), convex closed set (M), and quasiconvex lower semicontinuous functional (F(x));

b) the space (X) is an (E)-space.

We shall call problem (2) stable if, for any fixed (\delta>0), (F_\delta), it has a unique solution (x_\delta), and every sequence ({x_n}\subset \Omega_\delta) such that (|x_n|\to \rho(0,\Omega_\delta)) converges strongly to (x_\delta). This stability property is very important, since it makes it possible to solve problem (2) of finding (x_\delta) approximately by means of any method minimizing (|x|) on the set (\Omega_\delta) (one of such methods we consider in Sec. 4).

Theorem 3. In a strictly convex (B)-space (X) the following assertions are equivalent:

a) problem (2) is stable, whatever the convex closed set (M) and quasiconvex lower semicontinuous functional (F(x)) may be;

b) the space (X) is an (E)-space.

By definition, we shall call a set (R\subseteq M) a class of stabilization for the extremal problem (1) if, for every sequence ({x_n}\subset R) minimizing (F(x)) on (M), the condition (\rho(x_n,\Omega_0)\to 0) is satisfied. From the results of A. N. Tikhonov’s work ((^2)) it follows that a stabilization class is, for example, any compact (R\subseteq M) containing a unique point from (\Omega_0). Besides topological ones, in Banach spaces one can indicate geometric characteristics of stabilization classes for extremal problems. We formulate, in terms of the geometry of the unit sphere of the (B)-space (X), conditions necessary and sufficient for stabilization classes to be the sets (R_0=\rho_0 U\cap M), where (\rho_0=\rho(0,\Omega_0)).

Theorem 4. In a strictly convex (B)-space (X) the following assertions are equivalent:

a) the set (R_0) is nonempty and is a stabilization class for the extremal problem (1), whatever the convex closed set (M) and quasiconvex lower semicontinuous functional (F(x)) may be;

b) the space (X) is an (E)-space.

1. One of the methods for the numerical determination of elements (x_\delta) solving problem (2) is passage to a certain finite-dimensional analogue and the finding of approximations to (x_\delta) that are elements of finite-dimensional sets. Suppose that there is an increasing chain of convex closed finite-dimensional sets

[
M_n \subset M_{n+1} \subset \cdots \subset M, \qquad n=1,2,\ldots,
]

such that

[
\overline{\bigcup_{n=1}^{\infty} M_n}=M.
]

In computing the functional (F(x)), one usually uses its approximate values, i.e., finds (F^{(k)}(x)), such that

[
F^{(k)}(x)\to F(x)\qquad (k\to\infty).
\tag{3}
]

We associate with problem (2) the following finite-dimensional variational problem: find

[
\inf |x| \quad \text{under the condition } x\in \Omega_{\delta,n}^{k},
]

where

[
\Omega_{\delta,n}^{k}={x\in M_n:\ F^{(k)}(x)\leq F_\delta+\delta}.
\tag{4}
]

Theorem 5. Suppose (X) is an (E)-space, the functional (F(x)) is convex and continuous on (M), the functionals (F^{(k)}(x)) are lower semicontinuous on the sets (M_n), and the convergence (3) takes place uniformly, for fixed (n), on every bounded subset of (M_n) ((n=1,2,\ldots)). Then there exist (n_0) and (k_0) such that for (n>n_0) and (k>k_0) problem (4) has a solution (x_{\delta,n}^{k}). Moreover,

[
\lim_{n\to\infty}\lim_{k\to\infty} x_{\delta,n}^{k}=x_\delta .
]

Ural State University
Sverdlovsk

Received
30 III 1970

CITED LITERATURE

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