on existence of solutions to degenerate nonlinear optimization problems

Discussiones Mathematicae Differential Inclusions, Control and Optimization 27 (2007 ) 151–164

ON EXISTENCE OF SOLUTIONS TO DEGENERATE NONLINEAR OPTIMIZATION PROBLEMS ´ ska1 and Alexey Tret’yakov1,2 Agnieszka Prusin 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 x0 that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x∗ to the initial point x0 . Keywords: Lagrange function, necessary condition of optimality, pregularity, 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 mathematics, 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

´ ska and A. Tret’yakov A. Prusin

152

in some neighborhood of an initial point x 0 , 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) =

ϕ0 (x) + G0 (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 0 (x0 , λ0 ) is non-degenerate in some initial point z 0 = (x0 , λ0 ), then we can guarantee the existence of solutions for (2) in some neighborhood of z0 (see for instance [2]). If F 0 (x0 , λ0 ) is degenerate, then the question about the existence of solutions to (2) is open. In this paper, we investigate a degenerate situation and obtain conditions for which there exists a solution to the problem (2) in some neighborhood of the initial point z0 . 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 finding solutions to related singular problems, in particular, we would like to

On existence of solutions to degenerate nonlinear ...

153

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 0 = (x0 , λ0 ) if 0 F(x,λ) (x0 , λ0 ) is surjection. Otherwise, the problem (2) is called nonregular, (irregular, singular, degenerate) at the point z 0 . 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 n o Kerk F (k) (z) = h ∈ Z : F (k) (z)[h]k = 0 for the k-kernel of the mapping F (k) (z) (the zero locus of F (k) (z)), where k = 1, . . . , p. The set M (z0 ) = {z ∈ U : F (z) = F (z0 )} 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 0 ∈ M if there exist ε > 0 and a mapping r : [0, ε] → Z with the property that for t ∈ [0, ε] we have z0 + th + r(t) ∈ M and kr(t)k = o(t). The collection of all tangent vectors at z 0 is called the tangent cone to M at z0 and it is denoted by T1 M (z0 ). (For more detail we refer for instance to [3]). For a linear operator Λ : Z → W, we denote by Λ −1 its right inverse, that is Λ−1 : W → 2Z (Λ−1 is said to be a multimapping or a multivalued mapping from W into 2Z ) whose any element w ∈ W maps on its complete inverse image under the mapping Λ : Λ−1 w = {z ∈ Z : Λz = w} . Furthermore, we will use the ”norm”

−1

Λ = sup inf {kzk : Λz = w, z ∈ Z} . (3) kwk=1

Note that when Λ is one-to-one, kΛ−1 k can be considered as the usual norm of the element Λ−1 in the space L(Z, W ).


154

Let U be a neighborhood of the point z 0 ∈ Z. Consider a sufficiently smooth nonlinear mapping F : U → W, such that ImF 0 (z0 ) 6= W. We construct the p-factor operator under the assumption that the space W is decomposed into the direct sum: (4)

W = W1 ⊕ . . . ⊕ Wp ,

where W1 = cl(ImF 0 (z0 )) (the closure of the image of the first derivative of F evaluated at z0 ), and the remaining spaces are defined as follows. Let V1 = W, V2 be closed complementary subspace to W 1 (we are assuming that such a closed complement exists), and let P V2 : W → V2 be the projection operator onto V2 along W1 . Let W2 be the closed linear span of the image of the quadratic map PV2 F (2) (z0 )[·]2 . More generally, define inductively, Wi = cl(span ImPVi F (i) (z0 )[·]i ) ⊆ Vi , i = 2, . . . , p − 1, where Vi is a choice of a closed complementary subspace for (W 1 ⊕. . .⊕Wi−1 ) with respect to W, i = 2, . . . , p and PVi : W → Vi is the projection operator onto Vi along (W1 ⊕ . . . ⊕ Wi−1 ) with respect to W, i = 2, . . . , p. Finally, Wp = Vp . The order p is chosen as the minimum number for which (4) holds. Now, define the following mappings (see [4]), fi : U → Wi , fi (z) = PWi F (z), i = 1, . . . , p, where PWi : W → Wi is the projection operator onto Wi along (W1 ⊕ . . . ⊕ Wi−1 ⊕ Wi+1 ⊕ . . . ⊕ Wp ) with respect to W, i = 1, . . . , p. Definition 1. The linear operator Ψp (h) ∈ L(Z, W1 ⊕ . . . ⊕ Wp ) is defined for h ∈ Z by Ψp (h)z = f10 (z0 )z + f200 (z0 )[h]z + . . . + fp(p) (z0 )[h]p−1 z, for z ∈ Z and is called the p-factor operator. Sometimes it is convenient to use the following equivalent definition of the e p (h) ∈ L(Z, W1 × . . . × Wp ) for h ∈ Z, p-factor operator Ψ e p (h)z = (f10 (z0 )z, f200 (z0 )[h]z, . . . , fp(p) (z0 )[h]p−1 z), for z ∈ Z. Ψ

Note that in the completely degenerate case, i.e., in the case that F (r) (z0 ) = 0, r = 1, . . . , p−1, the p-factor operator is simply F (p) (z0 )[h]p−1 .


155

Roughly speaking, we construct a decomposition of a ”non-regular part” of the mapping F on partiall mappings f i , 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 0 along h if ImΨp (h) = W. Let us introduce the following auxiliary set H p (z0 ) = {z ∈ Z : Ψp (z)z = 0}. Definition 3. The mapping F is p-regular at the point z 0 if either it is p-regular along any h from the set Hp (z0 ) \ {0} or Hp (z0 ) = {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 0 ∈ Z. Assume that Φ : Z→W, Φ ∈ C p (U ) is p-regular at z0 . Then T1 M (z0 ) = Hp (z0 ). 3.

Regular case

For solving many nonlinear problems in the regular case (that is when F 0 is non-degenerate at the initial point z 0 ) classical results can be used, such as Lyusternik Theorem, Implicit Function Theorem, Lagrange-Euler’s optimality 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., T1 M (z0 ) = KerF 0 (z0 ). 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 solutions 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 0 ) is regular, i.e., F 0 (z0 )Z = W and that [F 0 (z0 )]−1 is a multivalued mapping. Moreover, let Uε (z0 ) = {z ∈ Z : kz − z0 k < ε} where 0 < ε < 1.


156

Theorem 1. Let F ∈ C 2 (Uε (z0 )), kF (z0 )k = η, k[F 0 (z0 )]−1 k = δ and sup kF 00 (z)k = c < ∞. In addition, if the following inequalities z∈Uε (z0 )

1. δ · c · ε ≤ 2. δ · η ≤

1 6

ε 2

hold, then the equation F (z) = 0 has a solution z ∗ ∈ Uε (z0 ).

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(A1 , A2 ) we denote the deviation of the set A 1 from the set A2 and by h(A1 , A2 ) we mean the Hausdorff distance between sets A 1 and A2 . 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 (z0 ) → 2Z , on a ball U (z0 ) = {z : ρ(z, z0 ) < } ( > 0) where the sets Φ(z) are nonempty and closed for any z ∈ U (z0 ). Further, assume that there exists a number θ, 0 < θ < 1, such that 1. h(Φ(z1 ), Φ(z2 )) ≤ θρ(z1 , z2 ) for any z1 , z2 ∈ U (z0 ) 2. ρ(z0 , Φ(z0 )) < (1 − θ), where ρ(z0 , Φ(z0 )) =

inf

u∈Φ(z0 )

ρ(z0 , u).

Then, for every number 1 which satisfies the inequality ρ(z0 , Φ(z0 )) < 1 < (1 − θ), there exists an element z ∈ B = {ω : ρ(ω, z 0 ) ≤ 1 /(1 − θ)} such that (5)

z ∈ Φ(z).


157

Further, by the distance ρ we mean just a norm, that is ρ(z 1 , z2 ) = kz1 −z2 k. Lemma 2 [3]. Let Z be a Banach space, and let M 1 and M2 be linear manifolds in Z which are translations of a single subspace L. Then h(M1 , M2 ) = σ(M1 , M2 ) = σ(M2 , M1 ) = inf {kz1 − z2 k : z1 ∈ M1 , z2 ∈ M2 } . Lemma 3 [3]. Let Z and W be Banach spaces, and let Λ ∈ L(Z, W ). We set C(Λ) = sup inf {kzk : z ∈ Z, Λz = w} . kwk=1

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 kf 0 (ξ) − Λk · ka − bk, ξ∈[a,b]

for any Λ ∈ L(Z, W ). p Lemma 5. Let Λ : Z → W be a nonlinear operator of the form Λ[z] r = 2 p Λ1 [z], Λ2 [z] , . . . , Λp [z] = (w1 , . . . , wp ) = w, where kwk 6= 0 and Λr [z] is r-form for r = 1, 2, . . . , p. If

−1 w w p

(6) sup Λ = inf kzk : Λ[z] = = c < ∞, kwk kwk

then (7)

−1 1

Λ w ≤ cp kw1 k + kw2 k 12 + . . . + kwp k p .

P roof. For any w = (w1 , . . . , wp ) ∈ W there exists α > 0 such that (8)

kw1 k kwp k + ... + = 1. α αp

Without loss of generality, we can assume that kw i k 6= 0, i = 1, . . . , p. Let us w denote wα1 = w e1 , . . . , αpp = w ep . Then from (8) we have kw e1 k + . . . + kw ep k = 1.


158

Taking into account (6) we obtain kαz k ≤ c. Hence kzk ≤ cα. Suppose that 1 1 α > p kw1 k + kw2 k 2 + . . . + kwp k p . Then kw1 k α

+ ... +

kwp k αp

< ≤

kw1 k 1 p kw1 k+...+kwp k p kw1 k pkw1 k

+ ... +

+ ... +

kwp k pkwp k

pp

=

1 p

kwp k

1

kw1 k+...+kwp k p

+ ... +

1 p

p

= 1.

This contradicts our assumption (8). Hence α ≤ p kw1 k + . . . + kwp k and finally (7). Let us introduce the following additional notations and assumptions (9) (10)

(11)

(12) (13)

Uε (z0 ) = {z ∈ Z : kz − z0 k < ε} where 0 < ε < 1, δ = kF (z0 )k 6= 0, Λ(h) ∈ L(Z, W1 × . . . × Wp ) where 1 (p) p−1 0 00 Λ(h) = f1 (z0 ), f2 (z0 )[h], . . . , f (z0 )[h] , (p − 1)! p 1 p 0 00 2 (p) p Λ[h] = f1 (z0 )[h], f2 (z0 )[h] , . . . , f (z0 )[h] , (p − 1)! p Λ−1 w = {h ∈ Z : w = Λ[h]p } ,

(14)

ˆ = h

(15)

Λi (h) =

(16) (17)

h , where h ∈ Λ−1 [−F (z0 )], h 6= 0, khk

1 (i) fi (z0 )[h]i−1 , i = 1, . . . , p, (i − 1)!

n o −1

ˆ

Λ h

≤ c1 ,

(i+1) sup fi (z) ≤ c2 < ∞, i = 1, . . . , p,

z∈Uε (z0 )

(18)

−1

Λ = sup inf {kzk : w = Λ[z]p , z ∈ Z} = c3 . kwk=1

1 p


159

Remark. From the condition of p-regularity follows c 1 < +∞ (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 p+1 (Uε (z0 )) be a p-regular mapping ˆ . Moreover, assume the following inequalities to be at z0 along some h satisfied 1

1. c3 p2 δ p ≤ 31 ε, 2. δ < 1, 3.

4 p 3 (4

− 1)c1 c2 ε ≤ 12 .

Then the equation F (z) = 0 has a solution z ∗ ∈ Uε (z0 ). P roof. Consider a multivalued mapping Φ h : Uε (z0 ) → 2Z , such that Φh (z) = z − {Λ(h)}−1 (f1 (z0 + h + z), . . . , fp (z0 + h + z)),

z ∈ Uε (z0 ).

Similarly as in the regular case, the assumptions of the contraction multimapping principle hold for Φh . Indeed, the sets Φh (z) are non-empty because Λ(h) is a surjection for any z ∈ U ε (z0 ). Moreover, for any w ∈ W1 × . . . × Wp the sets {Λ(h)}−1 w are linear manifolds parallel to KerΛ(h), and hence the sets Φ h (z) are closed for any z ∈ Uε (z0 ). We will now prove that 1 h (Φh (u1 ), Φh (u2 )) ≤ ku1 − u2 k, 2

(19)

for u1 , u2 ∈ U 2ε (z0 ) such that kuj k ≤

khk R ,

j = 1, 2, where

2 1

(i)

R = max Ri : Ri = max 1, ·

f (z0 ) , i = 1, . . . , p . ε · c2 (i − 1)! i

Let s1 = z0 + h + u1 , s2 = z0 + h + u2 . Then

h (Φh (u1 ), Φh (u2 )) = inf {kz1 − z2 k : zi ∈ Φh (uj ), j = 1, 2} = inf {kz1 − z2 k : Λ(h)zj = Λ(h)uj − (f1 (sj ), . . . , fp (sj )) , j = 1, 2} ≤ inf {kzk : Λ(h)z = Λ(h)(u1− u2 )− (f1 (s1 ) − f1 (s2 ), . . . , fp (s1 ) −fp (s2 ))}


160

= inf kzk : Λ

h khk

z = (Λ1 (h)(u1 − u2 ) − f1 (s1 ) + f1 (s2 ),

1 (Λ (h)(u − u ) − f (s ) + f (s )) p 1 2 p 1 p 2 khkp−1 n o−1 ˆ ≤ inf kzk : z = Λ(h) (Λ1 (h)(u1 − u2 ) − f1 (s1 ) + f1 (s2 ), ...,

..., ≤ c1 ·

p X i=1

1 (Λp (h)(u1 − u2 ) − fp (s1 ) + fp (s2 ))) khkp−1

1 kfi (s1 ) − fi (s2 ) − Λi (h)(s1 − s2 )k . khki−1

Taking into account Lemma 4, we have

(20)

kfi (s1 ) − fi (s2 ) − Λi (h)(s1 − s2 )k

≤ sup fi0 (s2 + θ(s1 − s2 )) − Λi (h) · ku1 − u2 k . θ∈[0,1]

By a complete degeneration of fi up to the order i we obtain the following Taylor expansion (i)

fi0 (s2 + θ(s1 − s2 )) = fi0 (z0 ) + . . . +

fi (z0 ) [s2 − z0 + θ(s1 − s2 )]i−1 (i − 1)!

+ ω(h, u1 , u2 , θ) (i)

(21)

f (z0 ) = i [s2 − z0 + θ(s1 − s2 )]i−1 + ω(h, u1 , u2 , θ), (i − 1)!

where kω(h, u1 , u2 , θ)k ≤

(i+1)

sup fi (z)[h + u2 + θ(s1 − s2 )]i .

z∈Uε (z0 )

On account of R and kuj k, j = 1, 2, we have kh + u2 + θ(s1 − s2 )k ≤ 4khk. Let F (z0 ) = (w1 , . . . , wp ), where wi ∈ Wi , i = 1, . . . , p. Then from the assumption and the definition of norm in Z we have kw 1 k + . . . + kwp k ≤ δ.


161

Taking into account Lemma 5 and assumptions 3 and 2 we have 1 1 khk ≤ (1 + ∆)kΛ−1 (−F (z0 ))k ≤ (1 + ∆)c3 p kw1 k+kw2 k 2 + . . . + kwp k p 1

≤ (1 + ∆)c3 p2 δ p ≤ 2ε , where 0 < ∆ < 21 . Hence, from previous formulas we get (22)

ε kω(h, u1 , u2 , θ)k ≤ c2 kh + u2 + θ(s1 − s2 )ki ≤ 4i c2 khki−1 . 2

Moreover, (i)

fi (z0 )[h + u2 + θ(s1 − s2 )]i−1 (23)

=

i−1 X k=0

(i)

i−1 k

(i)

fi (z0 )[h]i−1−k [u2 + θ(s1 − s2 )]k

= fi (z0 )[h]i−1 +

i−1 X

i−1 k

k=1

(i)

fi (z0 )[h]i−1−k [u2 + θ(s1 − s2 )]k ,

where (24)

ku2 + θ(s1 − s2 )k ≤ 3khk/R ≤ 3khk/Ri .

Taking into account the choice of Ri ,

i−1

X

(i) i−1 i−1−k k f (z )[h] [u + θ(s − s )]

0 2 1 2 i k

k=1

(25)

i−1

X

(i)

≤ fi (z0 ) ·

k=1

i−1 k

khki−1−k (3khk)k /Rik

ε

(i)

≤ fi (z0 ) · khki−1 · 4i−1 /Ri ≤ 4i (i − 1)! c2 khki−1 . 2

Now, inserting (21)–(25) into (20) we obtain kfi (s1 ) − fi (s2 ) − Λi (h)(s1 − s2 )k ≤ 4i εc2 khki−1 · ku1 − u2 k.


162 Hence h(Φh (u1 ), Φh (u2 )) ≤ c1 ·

p X i=1

=

1 4i c2 εkhki−1 ku1 − u2 k khki−1

4 p 1 (4 − 1) c1 c2 εku1 − u2 k ≤ ku1 − u2 k 3 2

so this proves (19). Let us take z1 , and an arbitrary element of Φh (z0 ), such that z1 ∈ z0 − Λ−1 (F (z0 )) and kz1 − z0 k ≤ (1 + ∆)kΛ−1 (−F (z0 )k, where 0 < ∆ < 21 . 1

Thus we have kΦh (z0 ) − z0 k ≤ kz1 − z0 k ≤ (1 + ∆)c3 p2 δ p ≤ 12 ε. From the above and from (19) we obtain that for the mapping Φ h all the assumptions of the contraction multimapping principle hold and hence there exists an element z ∗ such that z ∗ ∈ Φh (z ∗ ). It means that 0 ∈ [Λ(h)]−1 (f1 (z0 + h + z ∗ ), . . . , fp (z0 + h + z ∗ )) and hence (f1 (z0 + h + z ∗ ), . . . , fp (z0 + h + z ∗ )) = 0. Then for i = 1, . . . , p, fi (z0 + h + z ∗ ) = 0 which is equivalent to F (z0 + h + z ∗ ) = 0. It follows that z0 + 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(x1 , x2 ) subject to G(x1 , x2 ) = 0,

1 2 4 where ϕ : R2 → R, G : R2 → R and ϕ(x1 , x2 ) = (x1 − 3·10 3 ) + x2 and 1 ∗ G(x1 , x2 ) = x1 x2 with the solution x = ( 2·103 , 0). Consider also the Lagrange function of the form

L(x1 , x2 , λ) = (x1 −

1 )2 3·103

+ x42 + λx1 x2

and the gradient of this function F (x1 , x2 , λ) = ∇L(x1 , x2 , λ) = 2(x1 −

1 ) 3·103

+ λx2 , 4x32 + λx1 , x1 x2

T

.


163

For the following initial point (x01 , x02 , λ0 ) = (0, 0, 0), we obtain     2 2 0 0 − 3·10 3  , F 0 (0, 0, 0) =  0 0 0  F (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 T ˆ the giveninitial point along  h = (1, 0, 0) because the 2-factor operator 2 0 0 1 ˆ   is a surjection. Then for h = ( 1 3 , 0, 0)T we Λ(h) = 0 0 2·103 3·10 1 0 0 2·103 T 2 2 obtain Λ[h]2 = 3·10 and we calculate δ = 3·10 3 , 0, 0 3 , c1 = 1, c2 = 24, √

15 c3 = 12·10 2 . Applying Theorem 2 we obtain that in U ε (0, 0, 0) where ε = 1 , there exists a solution of the equation F (x 1 , x2 , λ) = 0 and, as it is 3 10 easy to see, there exists a solution to the optimization problem (26) under 1 consideration, namely x∗ = ( 2·10 3 , 0) ∈ Uε (0, 0).

Example 2. Similarly, let us consider (27)

min(x21 + x22 ) subject to x1 x2 =

1 , 7200

where the solution is x∗ = ( 601√2 , 601√2 ). The Lagrange function for this problem is as follows 1 L(x1 , x2 , λ) = x21 + x22 + λx1 x2 − λ 7200 ,

and its gradient has the form F (x1 , x2 , λ) = ∇L(x1 , x2 , λ) =

1 2x1 + λx2 , 2x2 + λx1 , x1 x2 − 7200

T

.

For the initial point (x01 , x02 , λ0 ) = (0, 0, −2) we get     0 2 −2 0 F (0, 0, −2) =  0  , F 0 (0, 0, −2) =  −2 2 0  1 − 7200 0 0 0 and hence F is a singular mapping at (0, 0, −2). As it is easy to see F is ˆ = ( √1 , √1 , 0)T and the 2-factor operator 2-regular at this point along h 2 2


164 

2 −2  ˆ Λ(h) =  −2 2 √ √ 2 2

Λ[h]2

= 0, 0,

2 2 T 1 7200

√ 2 √2 2 2

0



 1 1 , 120 , 0)T we get  is nonsingular. For h = ( 120

. On account of (9)–(18) we obtain c1 =

√1 , 2

c2 = 1, √

c3 = 14 . Applying Theorem 2 we conclude that in U ε (0, 0, −2), where ε = 402 there exists a solution of equation (3), and hence there exists also a solution to the optimization problem (27), namely x ∗ = ( 601√2 , 601√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, WoltersNoordhoff 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 solutions 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