On a generic class of regular one-parametric variational inequalities

On a generic class of regular one-parametric variational inequalities.

Gomez Boll, Walter

Institut fur Mathematik, Humboldt Universitat zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

Abstract

In this paper the regularity of one-parametric optimization prob- lems in the sense of Jongen, Jonker and Twilt is extended to one- parametric variational inequalities. The ve singularities are dened in this context and suitable indices are described around them. Many of the local properties of the singularities are also proved for this case.

The generic property of the dened class of regular one-parametric variational inequalities is proved and a corresponding linear quadratic perturbation result is stated.

Keywords: one-parametric variational inequalities, singularities, genericity

1 Introduction.

Let F : M ^!IRⁿ be a mapping and M IRⁿ be a closed set dened as M =^fx²IR^njH(x) = 0 G(x)0^g

where (H G) : IR^{n !} IR^m+p. The variational inequality problem is dened as follows: Find some point x²M such that

F(x)(y^;x)0 ⁸y²M: (1)

1

This problem has found wide application as an adequate mathematical framework for a number of economic, game-theoretic and equilibrium prob- lems. For extensive surveys we cite e.g. 5] and 13].

Special cases of variational inequalities are for example equality systems (M = IRⁿ), nonlinear complementarity problems (M = IRⁿ⁺), and convex minimization problems (F =^rf).

IfM is convex ( ^;gj are convex and hi are ane linear) the solutions of the variational inequalities can be characterized, under constraint qualica- tion, by a suitable version of the well-know KKT-system. This convex case have been intensively studied and various reformulations of the variational inequality problem are given, for example as a system of equations (e.g. the normal map), as a generalized equation (via the normal cone) and as an optimization problem (constrained and unconstrained). Many of the solu- tion approaches proposed are very similar and base on the use of generalized Newton methods for nonsmooth equations (see e.g. 13], 5], 6], 3], 2], 8]).

A further approach was given recently (see e.g. 15], 18], 19] 17]), which base on the use of continuation methods. In this paper we are also interested in the application of such continuation techniques for the solution of varia- tional inequalities. Our main purpose is to extend the well-known regularity of one-parametric optimization problems in the sense of Jongen, Jonker and Twilt (see 11]) to the case of one-parametric variational inequalities. This regularity deals with the local structure of the solution sets of one-parametric optimization problems and is an important theoretical issue, for instance by the so called path-following methods with jumps (see e.g. 4]).

The paper is organized as follows. In the second section we redene the singularities, which characterize the regularity in the sense of Jongen, Jonker and Twilt, in the case of one-parametric varational inequalities and state the most important local properties. In the third section other properties are stated, for instance the generic character of the regular problems.

2 One-parametric variational inequalities.

Let us consider variational inequalities (F(x t) H(x t) G(x t)) depending of a one-dimensional real parametert. Such problems are dened by its data (F H G) : IR^{n+1 !} IR^n+m+p and will also be denoted by V I(F H G). The corresponding feasible set M IRⁿ⁺¹ will also be denoted by M(H G).

Let us denotez = (x t)²IRⁿ⁺¹,I =^f1 ::: m^gand J =^f1 ::: p^g. For 2

a given parameter valuet the notation V I(F H G)(t) and M(H G)(t)IRⁿ (or simply M(t)) state for the corresponding variational inequality and its feasible set.

Many well-known concepts from optimization theory concerning feasible sets apply also to variational inequalities, for instance constraint qualica- tions as LICQ or MFCQ, active index set, etc.

A feasible point z ² M(H G) of the V I(F H G) is a critical point if there exist ( )²IR^m+jJ0(z)j satisfying the following relation

L(z ) = 0 (2)

where ^L(z ) = F(z)^;P_mi⁼¹iDxhi(z)^;Pj²J⁰⁽z⁾jDxgj(z). If j 0,

8j²J⁰(z) then z is called astationary point.

If the vectors ^fF(z) Dxhi(z) Dxgj(z) i²I j ²J⁰(z)^g are linearly de- pendent, z is dened as a generalized critical point, shortly g.c. point (see 11]).

We consider in this paper mainly the sets

gc = ⁿz ²IR^n+1jz is a g:c: point of V I(F H G)^o

stat = ⁿz ²IR^n+1jz is a stationary point of V I(F H G)^o If necessary, the notation (F H G) will be used for recalling the data.

De nition 1

A g.c. point z of V I(F H G) is called non-degenerated if the LICQ holds and the following conditions are fullled:

VU-ND1

: The uniquely determined solution ( ) ² IR^m+jJ0(z)j of (2) satises j ⁶= 0 ⁸j ²J⁰(z).

VU-ND2

: The matrix Dx^L(z )^jTxM⁽t⁾ is nonsingular.

Here TxM(t) is the tangential space to the set M(t) at the point x and Dx^L(z )^jTxM⁽t⁾ is a matrix of the form V^TDx^L(z )V , where the columns from V build a linear basis for TxM(t). In general, Dx^L(z ) is not symmetric. The sign^hdet(V^TDx^L(z )V )ⁱ is independent of the possible selection of V . Let us dene for K J the following mapping

HK(z ) =

0

B

@

F(z)^;P_mi⁼¹iDxhi(z)^;Pj²KjDxgj(z) GH(z)K(z)

1

C

A: 3

Remark 1

Under the LICQ the Condition

VU-ND2

is equivalent to the regularity from D⁽x ^)HJ⁰(z ). The set of critical points is given locally under

VU-ND1

by the projection onto the z-components of the zeros of ^HJ⁰ and, therefore, due to the non-degeneracy it builts a smooth one-dimensional manifold that is parametrizable in t^.

At a non-degenerated g.c. point theLI and LCI are uniquely determined (number of negative and positive entries of ). sign^hdet(Dx^L(z )^jTxM⁽t⁾)ⁱ is also uniquely determined. Around the dierent singularities we are then going to describe the changes of the triple

(LI LCI sign^hdet(Dx^L(z )^jTxM⁽t⁾)ⁱ) (3) Taking into account the relation

sign^hdet(Dx^L(z )^jTxM⁽t⁾)ⁱ=sign^hdet(D⁽x ^)HJ⁰(z ))ⁱ

we dene sign(det(Dx^L(z )^jTxM⁽t⁾)) = +1 for the case of TxM(t) =^f0^g.

2.1 Singulariti es and local properties.

A g.c. point z is of Type 1 if it is non-degenerated as in Denition 1.

Remark 2

In the neighbourhood of a g.c. point zof Type 1 the points of gc are non- degenerated and the triple (3) remain constant.

De nition 2

(see also 11])

A g.c. point zof V I(F H G) is of Type 2 if the following conditions hold:

Type2: 1-VU

: The LICQ is fullled

Type2: 2-VU

^{: Let} ( )² IR^m+jJ0(z)j solve (2), then there exists exactly one index l ²J⁰(z), such that l = 0, and k ⁶= 0, ⁸k²J⁰(z)^nf l^g.

Type2: 3-VU

: Dx^L(z )^jTxM⁽t⁾ is nonsingular.

Type2: 4-VU

: Dx^L(z )^j_Tx⁺M⁽t⁾ is nonsingular, where Tx⁺M(t) =^{n 2}IR^njDxH(z) = 0 DxG_J0⁺

(z⁾(z) = 0^o and J⁰⁺(z) =^fj ²J⁰(z)^jl ⁶= 0^g=J⁰(z)^nfl^g.

4

Type2: 5-VU

: Let us denote

⁺ = (H G_J0⁺

(z⁾)^T, and

W is a matrix columns of which form a basis of Tx⁺M. It holds that ⁶= 0, where

= Dxgl(z)( + ) + Dtgl(z) = ^;((Dx⁺(z))^y)^TDt⁺(z),

= ^;W(W^T Dx^LW)^;1W^T Dx^L + Dt^L] and B^y = (B^TB)^;1B^T represent the Moore-Penrose Inverse of B^. Denote by (x^J0+(t) t ^J0+(t) ^J0+(t)) (resp. (x^J0(t) t ^J0(t) ^J0(t))) a t- parametrization for the zeros of ^H_J0⁺ (resp. ^HJ⁰).

Remark 3

It is easy to see that, around z, the set gc is given by the curve (x^J0(t) t) and the feasible part of (x^J0+(t) t).

Type2: 4-VU

implies that the tangent lines of the two curves are not parallel.

For the description of the change from the triple (3) we use the charac- teristic numbers sign() and sign(), where

= det(Dx^L(z )^jTxM⁽t⁾)det(Dx^L(z )^j_Tx⁺_M⁽_t⁾):

De nition 3

Let A be a matrix divided as follows A =

B C

D E

!

where B and E are quadratic. Let B be regular. The matrix S(A^jB) = E^;DB^;1C is called the Schur-complement of B in A.

Lemma 1

(see 14])

Let A be as in Denition 3. Then it holds:

1. det(A) = det(E)det(S(A^jB)):

2. If A is symmetric, it follows: In(A) = In(B) + In(S(A^jB)). 5

3. If A is regular, its inverse is given by:

A^;1=

B^;1+B^;1CS(A^jB)^;1DB^{;1 ;}B^;1CS(A^jB)^;1

;S(A^jB)^;1DB^;1 S(A^jB)^;1

!

: Here In(P) = (p(P) n(P) z(P)) represents the so-called innertia triple of a symmetricquadratic matrixP, where p(P) (resp. n(P) and z(P)) means the number of positive (resp. negative or zero) eigenvalues of P.

Proposition 1

Let z be a g.c. point of Type 2 with vanishing multiplier l. It holds then that sign()sign() =^;sign(_^J_l⁰(t)) :

Proof :

Let the columns of the matrixV¹build a basis forTxM(t) and the columns of V⁰ =V¹...v¹ one for Tx⁺M(t). It holds:

Dx^LjT_x⁺M⁽t⁾ =

V^1TDx^LV¹ V^1TDx^Lv¹ v^1TDx^LV¹ v^T1Dx^Lv¹

!

:

Type2: 3-VU

implies the regularity of V^1TDx^LV¹. Lemma 1 (1.-) gives that det(Dx^LjT_x⁺M⁽t⁾) = det(Dx^LjTxM⁽t⁾)det(S(Dx^LjT_x⁺M⁽t^)jDx^LjTxM⁽t⁾)) where

S(Dx^LjT_x⁺M⁽t^)jDx^LjTxM⁽t⁾) = v^T1Dx^Lv^1;

v^T1Dx^LV¹V^1TDx^LV^1;1V^1TDx^Lv¹: Consequently, it holds:

sign() = signS(Dx^LjT_x⁺M⁽t^)jDx^LjTxM⁽t⁾): (4) Since the derivatives of the components (H G_J0⁺) in^H_J0⁺ and^HJ⁰ coincide, it holds that _x^J0+(t)^;x_^J0(t) ²Tx⁺M(t). Then there exists w⁰ with _x^J0+(t)^; x_^J0(t) = V⁰w⁰. Multiplying the Jacobi-matrix of ^L from the left-hand side by V⁰ we obtain thatV^0TDx^LV⁰w⁰+ _^Jl⁰(t)V^0TDxgl(z) = 0: Consequently,

x_^J0+(t)^;x_^J0(t) =^;_^Jl⁰(t)V⁰V^0TDx^LV^0;1V^0TDTxgl(z): (5) 6

x x

t t

(a b c⁾

(a⁺¹b^;1c⁾

(a b^;1c⁾

sign() = 1 sign() = 1 t t

(a b c⁾

(a^;1b⁺¹c⁾

(a^;1b^;c⁾

sign() = 1 sign() =^;1 x

x

t t

(a b c⁾

(a^;1b⁺¹c⁾

(a^;1b c⁾

sign() =^;1 sign() = 1 t t

(a b c⁾

(a⁺¹b^;1c⁾

(a b^;1;c⁾

sign() =^;1 sign() =^;1 Figure 1: Type 2

Since Dxgl(z)x_^J0(t) + Dtgl(z) = 0, it follows that

= Dxgl(z)x_^J0+(t) + Dtgl(z) = Dxgl(z)(_x^J0+(t)^;x_^J0(t)):

Now (5) provides = ^;_^J_l⁰(t)Dxgl(z)V⁰V^0TDx^LV^0;1V^0TDTxgl(z): Since V⁰ =V¹...v¹, it follows that

=^;_^Jl⁰(t) 0...Dxgl(z)v^1T

V^1TDx^LV¹ V^1TDx^Lv¹ v^T1Dx^LV¹ v^T1Dx^Lv¹

!

;1

0

v^1TDTxgl(z)

!

: Lemma 1 and (4) together give:

=^;_^Jl⁰(t)Dxgl(z)v^T12Dx^LjT_x⁺M⁽t^)jDx^LjTxM⁽t⁾

;1: Substituting (4) in the above relation concludes the proof.

Using Proposition 1 we obtain the changes of the triple (3) as shown in Figure 1.

De nition 4

(see also 11])

A g.c. point zof V I(F H G) is of Type 3 if the following conditions hold:

7

Type3: 1-VU

: The LICQ is fullled.

Type3: 2-VU

^:

VU-ND1

is fullled.

Type3: 3-VU

^: Dx^L(z )^jTxM⁽t⁾ has rank n^;m^;jJ⁰(z)^j;1.

Type3: 4-VU

: Let us denote

= (H GJ⁰⁽z⁾),

W is a matrix whose columns form a linear basis of TxM(t),

w^{1 6}= 0 is a vector, with W^TDx^L(z )W w¹= 0, v¹ =W w¹, and

w^{2 6}= 0 is a vector, with W^TDTx^L(z )W w² = 0, v² =Ww^2. It holds that ^{12 6}= 0, where

¹ = v^1T(D²x^Lv²)v^1;2v^T1DTx^L((DTx)^y)^T (v^1TD²xv²)

;v^2TDx^L((D_Tx)^y)^T(v^T1D²xv¹), ² = Dt^Lv^2;DTt(DTx)^yDTx^Lv² and the following notation is used:

v^1T(Dx^2Lv²)v¹ =v^1ThDx(DTx^Lv²)ⁱv^1T v^T1Dx²v² =v^1TD²xhiv² i²I v^1TDx²gjv² j ²J^0T:

Remark 4

(see also 16])

Under the LICQ and

Type3: 3-VU

the condition

Type3: 4-VU

is equiv- alent to the non-degeneracy of the vector (z ) as a critical point of the optimization problem:

PV U min (z ) = t

(z )^2XV U

XV U =ⁿ(z ) ²IR^{n+m+jJ0(z)j+1jH}J⁰(z ) = 0^o:

Remark 5

gc is locally described by the projection of^XV U onto thez-components. Con- sequently, gc can be parametrized around zby one of the x-components, and the local structure is as shown in Figure 2.

8

x x

t t

(a b c⁾

(a b^;c⁾

sign(¹²) = 1 t t

(a b c⁾

(a b^;c⁾

sign(¹²) =^;1 Figure 2: Type 3

In z the curve gc has a turning point in the variablet, since (z ) is a locally maximizer or minimizer of ^PV U. The position of the turning point z (t tas locally minimizer, the other t tas locally maximizer) depends directly on sign(¹²) (see Figure 2). It is easy to note that sign(¹) and sign(¹²) are independent of the selected vectors v¹ and v² in

Type3: 4- VU

.

The linear indices remain locally constant along gc. The local changes of sign^hdet(Dx^L(z )^jTxM⁽t⁾)ⁱare explained in Theorem 1 and in Remark 17.

De nition 5

(see also 11])

A g.c. point zof V I(F H G) is of Type 4 if the following conditions hold:

Type4: 1-VU

: 0< m +^jJ⁰(z)^j< n + 1.

Type4: 2-VU

: The matrix Dx

"

GJH(z)⁰⁽z⁾(z)

#

has rank m +^jJ⁰(z)^j;1.

Type4: 3-VU

^{: Let} ( ) satisfying ( )Dx

"

GH(z)J⁰⁽z⁾(z)

#

= 0 be xed.

Then it holds that j ⁶= 0, ⁸j²J⁰(z).

Type4: 4-VU

^: Dt

L

(z) ⁶= 0 and the matrix A = Dt

L

(z)W^TDx²

L

(z)W ^is regular, where

L

(z) = ^Pi²I ihi(z) +^Pj²J⁰⁽z⁾jgj(z) and the columns of W form a basis for the (n^;m^;J⁰(z)+1)-dimensional linear space

T =^{f 2}IR^njDhi(z) = 0 Dgj(z) = 0 i ²I j ²J⁰(z)^g 9

Type4: 5-VU

: F^T(z)WA^;1W^TF(z) ⁶= 0.

Let us xl ²J⁰(z) and denote = (⁰ i j i²I j ²J⁰(z)^nfl^g).

Remark 6

(see also 16])

If

Type4: 1-VU

,

Type4: 2-VU

and

Type4: 3-VU

are fullled, the conditions

Type4: 4-VU

and

Type4: 5-VU

together are equivalent to the following non-degeneracy:

The vector (z ), where = (0 j j ²J⁰(z)ⁿl), is a non-degenerated critical point of the problem ^PeV U:

e

PV U min ~(z ) = t (z )^2XfV U

f

XV U =ⁿ(z )²IR^{n+m+jJ0(z)j+1j};V U(z ) = 0^o Here ;V U(z ) is dened by:

;V U(z ) =

0

B

@

⁰F⁽z^);P_i2IiDxhi⁽z^);P_j2J⁰⁽z^)nfl^gjDxgj⁽z^);_lDxg_l⁽z⁾ H⁽z⁾

G_J⁰⁽_z⁾⁽z⁾

1

C

A:

Remark 7

Since W^TF(z)⁶= 0, the vectors

fF(z) Dxhi(z) Dxgj(z) i²I j ²J⁰(z)^nfk^gg (6) are linearly independent for arbitrary k ² J⁰(z). Therefore, the projection of ^XfVU onto the z-components describes the set gc around z^{. Xf}V U can be parametrized with a smooth mapping (z() ()) dened around zero satis- fying (z(0) (0)) = (z ). From the LICQ it follows that _t(0) = 0 ⁶= _⁰(0). Consequently, LICQ is fullled around z at each point of gc dierent from z. The linear indices change their values for < 0 and > 0, since the Lagrange-multipliers are given by 0¹

(⁾(i() j() i²I j ²J⁰(z)^nfl^g l).

Lemma 2

(see 12])

Let A be a symmetric(nn)-matrix and B an(nm)- matrix with rankm. If C =

A B

B^T 0

!

it follows In(C) = In(C^jKernB^T) +In(m m m^;m) . 10

HereKernB^T =^{n 2}IR^njB^T = 0^o and C^jKernB^T denotes a matrix of the form ~B^TC ~B where the columns of ~B form a linear basis for KernB^T.

Proposition 2

Let (z() ()) parametrize the curve ^XfV U around a g.c. point z of Type 4 with (z(0) (0)) = (z )). Then, for ⁶= 0, z() is a non-degenerated g.c.

point. sign^hdet(Dx^L(z())^jT_x^{( )}M⁽t⁽⁾⁾)ⁱ is the same (or dierent) for > 0 and < 0 if and only if n^;m^;J⁰(z) is even (resp. odd).

Proof:

Letl ²J⁰(z) be xed. We use the following notations:

B⁰(z) = Dx

0

B

@

GJ⁰H(z)⁽z^)nfl^g(z) F^T(z)

1

C

A B¹(z) = Dx

H(z)

GJ⁰⁽z^)nfl^g(z)

!

and

L(z ) = ⁰F(z)^;Pi²IiDxhi(z)^;Pj²J⁰⁽z^)nfl^gjDxgj(z)^;lDxgl(z):

Let W⁰(z) be dened in a neighbourhood Uz of z such that its columns form a linear basis for KernB⁰(z). W⁰(z) can be choosen depending from z as smooth as B⁰(z). Let the columns of a matrix W¹ = W⁰(z)...b⁰ form a basis for KernB¹(z).

Consider near zero with z() ² Uz. The columns of W⁰(z()) form then a basis for Tx⁽⁾M(t()) and, since LICQ and

VU-ND1

are fullled by Remark 7, the non-degeneracy of z() holds if W^0T(z())Dx^L(z())W⁰(z()) is regular, where Dx^L(z()) = ⁰⁽¹⁾Dx^L(z() ()): Consequently, it is su- cient to show the regularity of

W^0T(z(0))Dx^L(z(0) (0))W⁰(z(0)) =^;W^0T(z)D²x

L

(z)W⁰(z) (7) where

L

(z) is intended as in Denition 5. The equation

det(^{W0T(z())DxL(z())W0(z())}) =^{01( )}

(n^;m^;J⁰⁽z⁾⁾det(^{W0T(z())DxL(z()())W0(z())}) explains the changes of sign^hdet(Dx^L(z())^jT_x^{( )}M⁽t⁽⁾⁾)ⁱ. (8)

Using Lemma 2 we obtain that the matrix (7) is regular if and only if

0

B

@

Dx²

L

(z) B¹(z) F(z) B^1T(z) 0 0 F^T(z) 0 0

1

C

A (9)

11

is so, too, and that the submatrix

D²_x

L

(z) B¹(z) B^1T(z) 0

!

is regular, since A (as in Denition 5) is nonsingular. From Lemma 1 the regularity of (9) is obtained if

F^T(z) 0

D_x²

L

(z) B¹(z) B^1T(z) 0

!

;1

F(z) 0

!

6= 0:

However the above number is equal to F^T(z)W^1hW^1TDx²

L

(z)W^1i;1W^1TF(z) and the proof is concluded by using

Type4: 5-VU

.

Remark 8

x(0)_ ⁶= 0, since the vectors (6) are linearly independent and _t(0) = 0. gc is then parametrizable aroundzwith respect to an x-component (see Figure 3).

Around z gc has a turning point in t, since (z ) is a non-degenerated stationary point of ^PeV U. The geometric position of the turning point is determined by the number

D²⁽z ⁾L(z )^j_T(z^)XeV U = ^;1

Dt

L

(z)F^T(z)W ^hW^TD²x

L

(z)W^i;1W^TF(z) where L is the Lagrange-function corresponding to ^PeV U and W is intended as in Denition 5.

Sincesign(D²⁽_z ⁾L(z )^j_T(z^)XeV U) =^;sign(F^T(z)WA^;1W^TF(z)), we can use = sign(F^T(z)WA^;1W^TF(z)) for the characterization of the geometric position of the curve gc (see Figure 3).

Finally sign() = (^;1)^n;m;J0(z) determines according to Proposition 2 the changes of sign^hdet(Dx^L(z())^jT_x^{( )}M⁽t⁽⁾⁾)ⁱ.

Remark 9

Let us consider a point zof Type 4 in the case that the sets M(t)are convex (hi linear ane and then J⁰(z)⁶=). Let(z() ())be a parametrization as in the above Proposition 2, and such that z()is a stationary point for < 0^. It follows that every j j ² J⁰(z), in Denition 5 has the same sign (i.e., the MFCQ is not fullled). Let us x then j > 0 ⁸j ²J⁰(z). It follows that ⁰() > 0 for < 0.

The LICQ holds for < 0 and the convexity of M(t()) implies that D²xgj(z()) j ²J⁰(z), are negative semidenite on Tx⁽⁾M(t()). Therefore,

12

x x

t t

(a b c⁾

(b a c⁾

sign() = 1 sign() = 1 t t

(a b c⁾

(b a c⁾

sign() = ^;1 sign() = 1 x

x

t t

(a b c⁾

(b a^;c⁾

sign() = 1 sign() =^;1 t t

(a b c⁾

(b a^;c⁾

sign() =^;1 sign() =^;1 Figure 3: Type 4

W^0TD²_x

L

(z)W⁰ is negative semidenite, where W⁰ is intended as in the proof of the above Proposition 2 and

L

as in Denition 5. Since W^0TD²_x

L

(z)W⁰ is regular, it is then negative denite, and sign^hdet(^;W^0TD²x

L

(z)W⁰)ⁱ = +1. The relations (7) and (8) imply then that sign^hdet(Dx^L(z())^jT_x^{( )}M⁽t⁽⁾⁾)ⁱ= +1 for < 0.

On the other hand, W^TD²x

L

(z)W, where W is intended as in Denition 5, has at most one positive eigenvalue. Applying the results from Theorem 5.3 (cases 2. and 3.) in 10] about the homotopic changes of parametric feasible sets it follows that W^TDx²

L

(z)W must be negative denite and that M(t), by passing the parameter value t, vanishes or is created.

De nition 6

(see also 11])

A g.c. point zof V I(F H G) is of Type 5 if the following conditions hold:

Type5: 1-VU

: m +^jJ⁰(z)^j=n + 1.

Type5: 2-VU

: The matrix D

"

GH(z)J⁰⁽z⁾(z)

#

has rank n + 1.

Type5: 3-VU

: It holds

Type4: 3-VU (

( ) are xed

)

. 13

Type5: 4-VU

: Let ( )²IR^m IR^jJ0(z)j solve the system

F(z) 0

!

; X

i²IiD^Thi(z)^{; X}

j²J⁰⁽z⁾ jD^Tgj(z) = 0:

Then jk ⁶= 0, ⁸j k ²J⁰(z)^, j ⁶=k^{, where} jk = j ^; k j k.

Remark 10

From

Type5: 1-VU

,

Type5: 2-VU

and

Type5: 3-VU

it follows that each g.c. point in a neighbourhood ofzsolves (with corresponding multipliers) one of the systems

HJ⁰⁽z^)nfk^g(z ) = 0 (10) where k²J⁰(z)^.

For each k ² J⁰(z) there exists a parametrization (x^k(t) t ^k(t) ^k(t)) around (z ^{k k}) of the solution set from (10), with (x^k(t) t ^k(t) ^k(t)) = (z ^{k k}).

Remark 11

gc is described aroundzby the union of the feasible parts of the n+1 ^curves (x^k(t) t). It also holds that _dt^dgk(x^k(t) t) = D_Txgk(z)_x^k(t) + Dtgk(z) ⁶= 0 for k ²J⁰(z). Since DTxgk¹(z)_x^k2(t)+Dtgk¹(z) = 0, it follows thatx_^k1(t)⁶= _x^k2(t) for k^{1 6}=k² (see Figure 4).

Remark 12

^{(see 11])}

If the MFCQ fails to hold in z, then every k has the same sign. Then each curve belonging to gc runs in the samet-direction. Therefore, zis a turning point from gc. Analogously as for optimization problems it holds that stat

around zconsists of only one curve (x^k(t) t)and zis a border point ofstat. If the MFCQ is fullled andzis a stationary point, then there exist exactly two curves (x^k1(t) t)^and (x^k2(t) t) consisting of stationary points and with opposite t-directions. Hence, stat has a continuation.

The local structure of the sets gc and stat around a point of Type 5 is shown in gure 4.

Remark 13

If the M(t) are convex (hi ane and J⁰(z) ⁶=, since ^jI^j<= n^{) Theorems} 4.1 (Type 3 and 4) and 5.3 (cases 2. and 3.) in 10] imply that the feasible set M(t)in the parameter value tvanishes or appears when the MFCQ is not fullled.

14

z z MFCQ is

violated at z

z holds at MFCQz

t x

t

x gcⁿstat

stat

Figure 4: Typ 5

3 Some properties of regular one-parametric variational inequalities.

3.1 The points of Type 3.

In this section we prove a basic result similar to Theorem 10.2.2 in 9]. Let q n be xed and consider mappings with the structure

T(x u t) =

T¹(x u t) T²(x t)

!

(11) where x ² IRⁿ, u ² IR^q, T : IR^{n+q+1 !} IR^n+q, T¹ : IR^{n+q+1 !} IRⁿ and therefore T² :IR^{n+1 !}IR^q. Let us consider two matrices,BL =

C¹ C² 0 Iq

!

and BR =

C³ 0 C⁴ Iq

!

, where C¹ and C² are quadratic regular matrices with size n, and Iq states for the identity matrix of size q. BL and BR are then quadratic and regular.

By T⁽BLBR⁾(y v t) = BLT(BR(y v) t) =

T⁽¹_BLBR⁾(y v t) T⁽²_BLBR⁾(y t)

!

let us denote another mapping. From the structure of BL and BR it follows that

15

T⁽BLBR⁾ has the same dependence on its variables as T (pointed out by the notations T⁽¹_BLBR⁾ and T⁽²_BLBR⁾). It holds that T(x u t) = 0 if and only if T⁽B_LB_R⁾(B_R^;1(x u) t) = 0. If we dene (T) =^f(x u t)^jT(x u t) = 0^g it follows that

(T) =

BR 0 0 1

!

(T⁽B_LB_R⁾): (12) Let us consider the following assumption for a mappingT(x u t).

P¹(T) : 0 is a regular value of T

Remark 14

The relation

DT⁽B_LB_R⁾⁾(y v t) = BLDT(BR(y v) t)

BR 0 0 1

!

(13) implies that P¹(T)holds if and only if P¹(T⁽BLBR⁾) holds, too.

Let us dene ^s(T) =ⁿ(x u t)²(T)^jdet(D⁽x u⁾T(x u t)) = 0^o:

Proposition 3

Let (x u t) ^{2 s}(T) and (y v t) ^{2 s}(T⁽BLBR⁾) be xed with (x u) = BR

(y v). Let P¹(T) (and also P¹(T⁽B_LB_R⁾)) be fullled around (x u t) (resp.

(y v t)). Then, (x u t) is a non-degenerated critical point of t^j(T⁾ if and only if (y v t) is a non-degenerated critical point of t^j(T⁽_BLBR⁾⁾.

Heret^j(T⁾(analogously tot^j(T⁽_BLBR⁾⁾) represents the optimization prob- lem with the objective function F(x u t) = t and the feasible set (T).

Proposition 3 is proved easily writting down booth conditions.

Remark 15

Let P¹(T) be fullled and (y() v() t()) be a local parametrization of the curve(T⁽BLBR⁾)around(y v t)^2s(T⁽BLBR⁾), which is dened in a neigh- bourhood of zero and (y(0) v(0) t(0)) = (y v t). By (12) the mapping

0

B

@

x()u() t()

1

C

A=

BR 0 0 1

! 0

B

@

y()v() t()

1

C

A

is also a local parametrization of (T) around (x u t). 16