ON EXISTENCE OF SOLUTIONS TO DEGENERATE NONLINEAR OPTIMIZATION PROBLEMS

ON EXISTENCE OF SOLUTIONS TO DEGENERATE NONLINEAR OPTIMIZATION PROBLEMS

Agnieszka Prusi´ nska

and Alexey Tret’yakov

1

Institute of Mathematics and Physics

University of Podlasie, 3–go Maja 54, 08–110 Siedlce, Poland

2

System Research Institute

Polish Academy of Sciences, Newelska 6, 01–447 Warsaw, Poland

Abstract

We investigate the existence of the solution to the following problem min ϕ(x) subject to G(x) = 0,

where ϕ : X → R, G : X → Y and X, Y are Banach spaces. The question of existence is considered in a neighborhood of such point x

that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x

to the initial point x

.

Keywords: Lagrange function, necessary condition of optimality, p- regularity, contracting mapping, p-factor operator.

2000 Mathematics Subject Classification: Primary: 90C30, Secondary: 49M05, 47H10, 47A50, 47J05.

1. Introduction

As it is well known, one of the most important questions of applied mathe- matics, for example from the numerical point of view (but not only), is the problem of existence of solutions.

We investigate the conditions of existence of solutions for the following nonlinear optimization problem

(1) min ϕ(x) subject to G(x) = 0

in some neighborhood of an initial point x

, where ϕ : X → R and G : X → Y are sufficiently smooth and X, Y are Banach spaces.

Based on the classical approach, the answer to this question is given by one of the modifications of the implicit function theorem. Nevertheless, there are many cases of nonlinear type problems, where this theorem cannot be applied, at least not directly.

The purpose of our paper is to investigate the above mentioned type of problems, where the Hessian of the Lagrange function is degenerate at the initial point.

Let us consider the Lagrange function of the following form L(x, λ) = ϕ(x) + hG(x), λi, λ ∈ Y

and define the mapping F F (x, λ) =

 ϕ

(x) + G

(x)

· λ G(x)

 ,

where F : X × Y

→ X

× Y. As it is known, in accordance with necessary conditions, in an optimal point x

there exists λ

such that the following equality

(2) F (x

, λ

) = 0

holds (obviously under some regularity condition for G(x) at the point x

). If operator F

(x

, λ

) is non-degenerate in some initial point z

= (x

, λ

), then we can guarantee the existence of solutions for (2) in some neighborhood of z

(see for instance [2]). If F

(x

, λ

) is degenerate, then the question about the existence of solutions to (2) is open.

In this paper, we investigate a degenerate situation and obtain condi- tions for which there exists a solution to the problem (2) in some neighbor- hood of the initial point z

.

Our approach is based on the notion of p-regularity introduced in 1982 by the second author [9]. Among many applications the structure of p-factor operator can be used in the construction of numerical methods for solving degenerate nonlinear operator equations and optimization problems.

We would like to present how to apply the p-regularity theory, also

known as factor-analysis of nonlinear mappings to develop methods for find-

ing solutions to related singular problems, in particular, we would like to

show how these ideas can be applied in some specific situations, namely in optimization problems.

2. Elements of p-regularity theory We start with some important notions of the p-regularity theory.

The problem (2) is called regular at the initial point z

= (x

, λ

) if F

(x

, λ

) is surjection. Otherwise, the problem (2) is called nonregular, (irregular, singular, degenerate) at the point z

.

Let p be a natural number and let Z, W be Banach spaces. We consider a mapping F : Z → W which is continuously p-times Fr´echet differentiable on Z. Furthermore, we shall use the following notation

Ker

F

(z) = n

h ∈ Z : F

(z)[h]

= 0 o

for the k-kernel of the mapping F

(z) (the zero locus of F

(z)), where k = 1, . . . , p.

The set

M (z

) = {z ∈ U : F (z) = F (z

)}

is called the solution set for the mapping F in neighborhood U .

We call h a tangent vector to a set M ⊆ Z at z

∈ M if there exist ε > 0 and a mapping r : [0, ε] → Z with the property that for t ∈ [0, ε] we have z

+ th + r(t) ∈ M and kr(t)k = o(t).

The collection of all tangent vectors at z

is called the tangent cone to M at z

and it is denoted by T

M (z

). (For more detail we refer for instance to [3]).

For a linear operator Λ : Z → W, we denote by Λ

its right inverse, that is Λ

: W → 2

(Λ

is said to be a multimapping or a multivalued mapping from W into 2

) whose any element w ∈ W maps on its complete inverse image under the mapping Λ :

Λ

w = {z ∈ Z : Λz = w} . Furthermore, we will use the ”norm”

(3)

Λ

= sup

kwk=1

inf {kzk : Λz = w, z ∈ Z} .

Note that when Λ is one-to-one, kΛ

k can be considered as the usual norm

of the element Λ

in the space L(Z, W ).

Let U be a neighborhood of the point z

∈ Z. Consider a sufficiently smooth nonlinear mapping F : U → W, such that ImF

(z

) 6= W. We construct the p-factor operator under the assumption that the space W is decomposed into the direct sum:

(4) W = W

⊕ . . . ⊕ W

,

where W

= cl(ImF

(z

)) (the closure of the image of the first derivative of F evaluated at z

), and the remaining spaces are defined as follows. Let V

= W, V

be closed complementary subspace to W

(we are assuming that such a closed complement exists), and let P

: W → V

be the projection operator onto V

along W

. Let W

be the closed linear span of the image of the quadratic map P

F

(z

)[·]

. More generally, define inductively,

W

= cl(span ImP

F

(z

)[·]

) ⊆ V

, i = 2, . . . , p − 1,

where V

is a choice of a closed complementary subspace for (W

⊕. . .⊕W

) with respect to W, i = 2, . . . , p and P

: W → V

is the projection operator onto V

along (W

⊕ . . . ⊕ W

) with respect to W, i = 2, . . . , p. Finally, W

= V

. The order p is chosen as the minimum number for which (4) holds.

Now, define the following mappings (see [4]),

f

: U → W

, f

(z) = P

F (z), i = 1, . . . , p,

where P

: W → W

is the projection operator onto W

along (W

⊕ . . . ⊕ W

⊕ W

⊕ . . . ⊕ W

) with respect to W, i = 1, . . . , p.

Definition 1. The linear operator Ψ

(h) ∈ L(Z, W

⊕ . . . ⊕ W

) is defined for h ∈ Z by

Ψ

(h)z = f

(z

)z + f

(z

)[h]z + . . . + f

(z

)[h]

z, for z ∈ Z and is called the p-factor operator.

Sometimes it is convenient to use the following equivalent definition of the p-factor operator e Ψ

(h) ∈ L(Z, W

× . . . × W

) for h ∈ Z,

Ψ e

(h)z = (f

(z

)z, f

(z

)[h]z, . . . , f

(z

)[h]

z), for z ∈ Z.

Note that in the completely degenerate case, i.e., in the case that

F

(z

) = 0, r = 1, . . . , p−1, the p-factor operator is simply F

(z

)[h]

.

Roughly speaking, we construct a decomposition of a ”non-regular part”

of the mapping F on partiall mappings f

, in such a way that all of those mappings are completely degenerate up to the order i − 1, where i = 2, ..., p.

For our consideration we need the following generalization of the notion of the regular mapping.

Definition 2. We say that the mapping F is p-regular at z

along h if ImΨ

(h) = W.

Let us introduce the following auxiliary set H

(z

) = {z ∈ Z : Ψ

(z)z = 0}.

Definition 3. The mapping F is p-regular at the point z

if either it is p-regular along any h from the set H

(z

) \ {0} or H

(z

) = {0}.

The following theorem gives a description of a solution set in the degenerate case:

Theorem (Generalization of the Lyusternik theorem) [11]. Let Z and W be Banach spaces, and U be a neighborhood of z

∈ Z. Assume that Φ : Z→W, Φ ∈ C

(U ) is p-regular at z

. Then T

M (z

) = H

(z

).

3. Regular case

For solving many nonlinear problems in the regular case (that is when F

is non-degenerate at the initial point z

) classical results can be used, such as Lyusternik Theorem, Implicit Function Theorem, Lagrange-Euler’s op- timality conditions. In this case, the tangent cone to a solution set of the equation (2) is equal to the zero locus of the first derivative of the mapping F, i.e., T

M (z

) = KerF

(z

).

Besides description of the solution set and formulation of optimality conditions, a very important problem is to give a guarantee of the existence of a solution in some neighborhood of an initial point.

We quote one modification of the theorem about the existence of solu- tions of equation (2) in the regular case (see e.g. [2, 8]).

Consider a mapping F : Z → W and the existence of such point z

that

F (z

) = 0. Throughout this section we assume that F (z

) is regular, i.e.,

F

(z

)Z = W and that [F

(z

)]

is a multivalued mapping. Moreover, let

U

(z

) = {z ∈ Z : kz − z

k < ε} where 0 < ε < 1.

Theorem 1. Let F ∈ C

(U

(z

)), kF (z

)k = η, k[F

(z

)]

k = δ and sup

z∈Uε(z0)

kF

(z)k = c < ∞. In addition, if the following inequalities 1. δ · c · ε ≤

2. δ · η ≤

hold, then the equation F (z) = 0 has a solution z

∈ U

(z

).

4. Degenerate case

There are many interesting optimization problems which are of degenerate form. We shall prove a generalization of Theorem 1 in the degenerate case.

For this purpose we need three auxiliary lemmas. The first of these lemmas is a ”multivalued” generalization of the contraction mapping principle. By σ(A

, A

) we denote the deviation of the set A

from the set A

and by h(A

, A

) we mean the Hausdorff distance between sets A

and A

.

Lemma 1 (Contraction multimapping principle) [3]. Let Z be a complete metric space with distance ρ. Assume that we are given a multimapping

Φ : U

(z

) → 2

,

on a ball U

(z

) = {z : ρ(z, z

) < } ( > 0) where the sets Φ(z) are non- empty and closed for any z ∈ U

(z

). Further, assume that there exists a number θ, 0 < θ < 1, such that

1. h(Φ(z

), Φ(z

)) ≤ θρ(z

, z

) for any z

, z

∈ U

(z

) 2. ρ(z

, Φ(z

)) < (1 − θ), where

ρ(z

, Φ(z

)) = inf

u∈Φ(z⁰)

ρ(z

, u).

Then, for every number 

which satisfies the inequality ρ(z

, Φ(z

)) < 

< (1 − θ),

there exists an element z ∈ B = {ω : ρ(ω, z

) ≤ 

/(1 − θ)} such that

(5) z ∈ Φ(z).

Further, by the distance ρ we mean just a norm, that is ρ(z

, z

) = kz

−z

k.

Lemma 2 [3]. Let Z be a Banach space, and let M

and M

be linear manifolds in Z which are translations of a single subspace L. Then

h(M

, M

) = σ(M

, M

) = σ(M

, M

) = inf {kz

− z

k : z

∈ M

, z

∈ M

} . Lemma 3 [3]. Let Z and W be Banach spaces, and let Λ ∈ L(Z, W ). We set

C(Λ) = sup

kwk=1

inf {kzk : z ∈ Z, Λz = w} . If ImΛ = W, then C(Λ) < ∞.

Let us mention one of the consequences of the Mean Value Theorem, which is important for our further investigations. For the proof we refer the reader to [3, 7].

Lemma 4. Let f : U → W, where [a, b] ⊂ U ⊆ Z. Then kf (b) − f (a) − Λ(b − a)k ≤ sup

ξ∈[a,b]

kf

(ξ) − Λk · ka − bk,

for any Λ ∈ L(Z, W ).

Lemma 5. Let Λ : Z → W be a nonlinear operator of the form Λ[z]

= Λ

[z], Λ

[z]

, . . . , Λ

[z]

= (w

, . . . , w

) = w, where kwk 6= 0 and Λ

[z]

is r-form for r = 1, 2, . . . , p. If

(6) sup

Λ

w kwk

= inf



kzk : Λ[z]

= w kwk



= c < ∞, then

(7)

Λ

w ≤ cp 

kw

k + kw

k

+ . . . + kw

k

1 p

 .

P roof. For any w = (w

, . . . , w

) ∈ W there exists α > 0 such that

(8) kw

k

α + . . . + kw

k α

= 1.

Without loss of generality, we can assume that kw

k 6= 0, i = 1, . . . , p. Let us

denote

= e w

, . . . ,

= e w

. Then from (8) we have k e w

k + . . . + k e w

k = 1.

Taking into account (6) we obtain k

k ≤ c. Hence kzk ≤ cα. Suppose that α > p 

kw

k + kw

k

+ . . . + kw

k

1 p

 . Then

kw¹k

α

+ . . . +

<

p



kw¹k+...+kwpk

1 p



+ . . . +

p^p



kw¹k+...+kwpk

1 p

p

≤

1k

+ . . . +

pk

=

+ . . . +

= 1.

This contradicts our assumption (8). Hence α ≤ p 

kw

k + . . . + kw

k

1 p

 and finally (7).

Let us introduce the following additional notations and assumptions U

(z

) = {z ∈ Z : kz − z

k < ε} where 0 < ε < 1,

(9)

δ = kF (z

)k 6= 0, (10)

Λ(h) ∈ L(Z, W

× . . . × W

) where Λ(h) =



f

(z

), f

(z

)[h], . . . , 1

(p − 1)! f

(z

)[h]

 , (11)

Λ[h]

=



f

(z

)[h], f

(z

)[h]

, . . . , 1

(p − 1)! f

(z

)[h]

 , (12)

Λ

w = {h ∈ Z : w = Λ[h]

} , (13)

h = ˆ h

khk , where h ∈ Λ

[−F (z

)], h 6= 0, (14)

Λ

(h) = 1

(i − 1)! f

(z

)[h]

, i = 1, . . . , p, (15)

n

Λ 

ˆ h o

≤ c

, (16)

sup

z∈U^ε(z0)

f

(z)

≤ c

< ∞, i = 1, . . . , p, (17)

Λ

= sup

kwk=1

inf {kzk : w = Λ[z]

, z ∈ Z} = c

.

(18)

Remark. From the condition of p-regularity follows c

< +∞ (see [9]). Let the assumptions (9)–(18) be satisfied. Then the following generalization of Theorem 1 holds

Theorem 2. Let F : Z → W, F ∈ C

(U

(z

)) be a p-regular mapping at z

along some ˆ h . Moreover, assume the following inequalities to be satisfied

1. c

p

δ

1 p

≤

ε, 2. δ < 1,

3.

(4

− 1)c

c

ε ≤

.

Then the equation F (z) = 0 has a solution z

∈ U

(z

).

P roof. Consider a multivalued mapping Φ

: U

(z

) → 2

, such that Φ

(z) = z − {Λ(h)}

(f

(z

+ h + z), . . . , f

(z

+ h + z)), z ∈ U

(z

).

Similarly as in the regular case, the assumptions of the contraction mul- timapping principle hold for Φ

. Indeed, the sets Φ

(z) are non-empty be- cause Λ(h) is a surjection for any z ∈ U

(z

).

Moreover, for any w ∈ W

× . . . × W

the sets {Λ(h)}

w are linear manifolds parallel to KerΛ(h), and hence the sets Φ

(z) are closed for any z ∈ U

(z

). We will now prove that

(19) h (Φ

(u

), Φ

(u

)) ≤ 1

2 ku

− u

k, for u

, u

∈ U

2

(z

) such that ku

k ≤

, j = 1, 2, where R = max



R

: R

= max

 1, 2

ε · c

· 1 (i − 1)!

f

(z

)



, i = 1, . . . , p

 .

Let s

= z

+ h + u

, s

= z

+ h + u

. Then

h (Φ

(u

), Φ

(u

)) = inf {kz

− z

k : z

∈ Φ

(u

), j = 1, 2}

= inf {kz

− z

k : Λ(h)z

= Λ(h)u

− (f

(s

), . . . , f

(s

)) , j = 1, 2}

≤ inf {kzk : Λ(h)z = Λ(h)(u

− u

)− (f

(s

) − f

(s

), . . . , f

(s

) −f

(s

))}

= inf

 kzk : Λ

 h khk



z = (Λ

(h)(u

− u

) − f

(s

) + f

(s

),

. . . , 1

khk

(Λ

(h)(u

− u

) − f

(s

) + f

(s

))



≤ inf



kzk : z = n

Λ(ˆ h) o

(Λ

(h)(u

− u

) − f

(s

) + f

(s

),

. . . , 1

khk

(Λ

(h)(u

− u

) − f

(s

) + f

(s

)))



≤ c

· X

i=1

1

khk

kf

(s

) − f

(s

) − Λ

(h)(s

− s

)k . Taking into account Lemma 4, we have

(20)

kf

(s

) − f

(s

) − Λ

(h)(s

− s

)k

≤ sup

θ∈[0,1]

f

(s

+ θ(s

− s

)) − Λ

(h)

· ku

− u

k .

By a complete degeneration of f

up to the order i we obtain the following Taylor expansion

f

(s

+ θ(s

− s

)) = f

(z

) + . . . + f

(z

)

(i − 1)! [s

− z

+ θ(s

− s

)]

+ ω(h, u

, u

, θ)

= f

(z

)

(i − 1)! [s

− z

+ θ(s

− s

)]

+ ω(h, u

, u

, θ), (21)

where

kω(h, u

, u

, θ)k ≤ sup

z∈Uε(z0)

f

(z)[h + u

+ θ(s

− s

)]

.

On account of R and ku

k, j = 1, 2, we have kh + u

+ θ(s

− s

)k ≤ 4khk.

Let F (z

) = (w

, . . . , w

), where w

∈ W

, i = 1, . . . , p. Then from the

assumption and the definition of norm in Z we have kw

k + . . . + kw

k ≤ δ.

Taking into account Lemma 5 and assumptions 3 and 2 we have khk ≤ (1 + ∆)kΛ

(−F (z

))k ≤ (1 + ∆)c

p 

kw

k+kw

k

+ . . . + kw

k

1 p



≤ (1 + ∆)c

p

δ

≤

,

where 0 < ∆ <

. Hence, from previous formulas we get (22) kω(h, u

, u

, θ)k ≤ c

kh + u

+ θ(s

− s

)k

≤ 4

c

ε

2 khk

. Moreover,

(23)

f

(z

)[h + u

+ θ(s

− s

)]

=

i−1

X

k=0 i−1

k

f

(z

)[h]

[u

+ θ(s

− s

)]

= f

(z

)[h]

+

i−1

X

k=1 i−1

k

f

(z

)[h]

[u

+ θ(s

− s

)]

,

where

(24) ku

+ θ(s

− s

)k ≤ 3khk/R ≤ 3khk/R

. Taking into account the choice of R

,

(25)

i−1

X

k=1 i−1

k

f

(z

)[h]

[u

+ θ(s

− s

)]

≤

f

(z

) ·

i−1

X

k=1 i−1

k

khk

(3khk)

/R

≤

f

(z

)

· khk

· 4

/R

≤ 4

(i − 1)! ε

2 c

khk

. Now, inserting (21)–(25) into (20) we obtain

kf

(s

) − f

(s

) − Λ

(h)(s

− s

)k ≤ 4

εc

khk

· ku

− u

k.

Hence

h(Φ

(u

), Φ

(u

)) ≤ c

· X

1

khk

4

c

εkhk

ku

− u

k

= 4

3 (4

− 1) c

c

εku

− u

k ≤ 1

2 ku

− u

k so this proves (19).

Let us take z

, and an arbitrary element of Φ

(z

), such that z

∈ z

− Λ

(F (z

)) and kz

− z

k ≤ (1 + ∆)kΛ

(−F (z

)k, where 0 < ∆ <

. Thus we have kΦ

(z

) − z

k ≤ kz

− z

k ≤ (1 + ∆)c

p

δ

1

p

≤

ε. From the above and from (19) we obtain that for the mapping Φ

all the assumptions of the contraction multimapping principle hold and hence there exists an element z

such that z

∈ Φ

(z

). It means that

0 ∈ [Λ(h)]

(f

(z

+ h + z

), . . . , f

(z

+ h + z

))

and hence (f

(z

+ h + z

), . . . , f

(z

+ h + z

)) = 0. Then for i = 1, . . . , p, f

(z

+ h + z

) = 0 which is equivalent to F (z

+ h + z

) = 0. It follows that z

+ h + z

is the solution of (1).

We conclude the discussion by the following simple examples, which serve to illustrate how to apply Theorem 2 to degenerate optimization problems.

Example 1. Consider the optimization problem

(26) min ϕ(x

, x

) subject to G(x

, x

) = 0, where ϕ : R

→ R, G : R

→ R and ϕ(x

, x

) = (x

−

3·10³

)

+ x

and G(x

, x

) = x

x

with the solution x

= (

2·10³

, 0). Consider also the La- grange function of the form

L(x

, x

, λ) = (x

−

3·10³

)

+ x

+ λx

x

and the gradient of this function

F (x

, x

, λ) = ∇L(x

, x

, λ) = 2(x

−

) + λx

, 4x

+ λx

, x

x

.

For the following initial point (x

, x

, λ

) = (0, 0, 0), we obtain

F (0, 0, 0) =





−

0 0



 , F

(0, 0, 0) =





2 0 0 0 0 0 0 0 0





and hence F is singular at (0, 0, 0). Moreover the mapping F is 2-regular at the given initial point along ˆ h = (1, 0, 0)

because the 2-factor operator Λ(ˆ h) =





2 0 0

0 0

2·10³

0

0



 is a surjection. Then for h = (

3·10³

, 0, 0)

we obtain Λ[h]

=

, 0, 0

and we calculate δ =

, c

= 1, c

= 24, c

=

. Applying Theorem 2 we obtain that in U

(0, 0, 0) where ε =

1

10³

, there exists a solution of the equation F (x

, x

, λ) = 0 and, as it is easy to see, there exists a solution to the optimization problem (26) under consideration, namely x

= (

, 0) ∈ U

(0, 0).

Example 2. Similarly, let us consider

(27) min(x

+ x

) subject to x

x

= 1 7200 , where the solution is x

= (

60√ 2

,

60√

2

). The Lagrange function for this problem is as follows

L(x

, x

, λ) = x

+ x

+ λx

x

− λ

, and its gradient has the form

F (x

, x

, λ) = ∇L(x

, x

, λ) =



2x

+ λx

, 2x

+ λx

, x

x

− 1 7200



.

For the initial point (x

, x

, λ

) = (0, 0, −2) we get

F (0, 0, −2) =



 0 0

−



 , F

(0, 0, −2) =





2 −2 0

−2 2 0

0 0 0





and hence F is a singular mapping at (0, 0, −2). As it is easy to see F is 2-regular at this point along ˆ h = (

2

,

2

, 0)

and the 2-factor operator

Λ(ˆ h) =



 

2 −2

−2 2

√2 2

√2

2

0



  is nonsingular. For h = (

,

, 0)

we get

Λ[h]

= 0, 0,

. On account of (9)–(18) we obtain c

=

2

, c

= 1, c

=

. Applying Theorem 2 we conclude that in U

(0, 0, −2), where ε =

there exists a solution of equation (3), and hence there exists also a solution to the optimization problem (27), namely x

= (

60√ 2

,

60√ 2

).

References

[1] V.M. Alexeev, V.M. Tihomirov and S.V. Fomin, Optimal Control, Consultants Bureau, New York, 1987. Translated from Russian by V.M. Volosov.

[2] B.P. Demidovitch and I.A. Maron, Basis of Computational Mathematics, Nauka, Moscow 1973. (in Russian)

[3] A.D. Ioffe and V.M. Tihomirov, Theory of extremal problems, North-Holland, Studies in Mathematics and its Applications, Amsterdam 1979.

[4] A.F. Izmailov and A.A. Tret‘yakov, Factor-Analysis of Non-Linear Mapping, Nauka, Moscow, Fizmatlit Publishing Company, 1994.

[5] L.V. Kantorovitch and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford 1982.

[6] M.A. Krasnosel’skii, G.M. Wainikko, P.P. Zabreiko, Yu.B. Rutitskii and V. Yu.

Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen (1972), 39.

[7] K. Maurin, Analysis, Part I, Elements, PWN, Warsow 1971. (in Polish) [8] A. Prusi´ nska and A.A. Tret’yakov, The theorem on existence of singular solu-

tions to nonlinear equations, Trudy PGU, seria Mathematica, 12 (2005).

[9] A.A. Tret’yakov, Necessary Conditions for Optimality of p-th Order, Control and Optimization, Moscow MSU (1983), 28–35 (in Russian).

[10] A.A. Tret’yakov, Necessary and Sufficient Conditions for Optimality of p-th Order, USSR Comput. Math. and Math Phys. 24 (1984), 123–127.

[11] A.A. Tret’yakov, The implicit function theorem in degenerate problems, Russ.

Math. Surv. 42 (1987), 179–180.

Received 24 February 2006