Vector optimization : Singularities, Regularizations
Gomez Boll, Walter
Humboldt-Universitat zu Berlin Inst. fur Angewandte Mathematik.
Unter den Linden 6 D-10099 Berlin
Germany March 28, 1996
Abstract
We discuss three scalarizations of the multiobjective optimization from the point of view of the parametric optimization. We analize three important aspects:
i) What kind of singularities may appear in the dierent parametriza- tions
ii) Regularizations in the sense of Jongen, Jonker and Twilt, and in the sense of Kojima and Hirabayashi.
iii) The Mangasarian-Fromovitz Constraint Qualication for the rst parametrization.
keywords: multiobjective optimization, parametric optimization, singularities, regularizations
This papper is a short version of the thesis of the author at the University of Havanna, Department of Mathematics, Havanna, Cuba.
1
We consider the following multiobjective optimization problem:
min f f1(x) ::: fL(x) jx2M g (1) where M is dened by
M =f x2IRn j gi(x) = 0 i2I gk(x)0 k2K g (2) and wherefj j = 1:::L gk k 2 IK are given functions , I = f1:::mg and K =f1:::pgnI pm .
We use the following well-known notions of "optimality" for a multiob- jective optimization problem.
De nition 1.1
(c.f. e.g. 3])A point x2IRn is called an ecient point if
( f(x) + D )\f(M) = (3)
where f = (f1 ::: fL) and D = ;IRL+nf0g.
The concepts of -ecient point and weakly ecient point are dened analogously writing
D =f y2IRLnf0g j dist(y D)jjyjj g and Dd =int(D) respectively instead ofD.
There are local versions of these concepts that have a trivial denition. We denote the set of all (loc) ecient points by Meff(Mloceff) and analogously the setsMeff(Mloceff) and Mw;eff (Mlocw;eff) .
It is posible to characterize these sets by means of the solution sets of certain parametric optimization problems ( c.f. e.g. 4]). We use here the following parametrizations of the problem (1),(2).
1st Parametrization: (c.f. e.g. 4]) P1() : min
( XL
k=10kfk(x) jx2M fk(x)k k2L
)
(4) where L =f1 ::: Lg k 2IRf+1g k 2L and 0 2;Dd.
2
We denote by 1() 1loc(), the set of global and local optimal points of the problemP1(). The following relation is known. (c.f. e.g. 5])
Meff =
2IRL1() Mloceff =
2IRL1loc():
2nd.Parametrization: (c.f. e.g. 3, 4]) s(f(x) ) = max
i 2 L 0i (fi(x);i) +XL
k=10k(fk(x);k):
where 2(0 1) suciently small is xed.
The problem
min s(x ) x2M (5)
has the following properties: (c.f. e.g. 5])
2IRL2()Meff
2IRL2loc()Mloceff
Meff
2f(Meff )
s;1(f(x) )\2()
2IRL2()
and Mloceff
2IRL2loc():
Since s is not dierentiable, we transform (5) into the equivalent problem (c.f. e.g. 3, 4])
P2() : min
(
XL
k=10k(fk(x);k) +v
x2M 8k2L 0k(fk(x);k)v
)
(6) 3rd Parametrization: (c.f. e.g. 3, 4])
P3() : minf v j x2M fk(x);v k k 2L g (7) If we consider 3() (3loc()) as the projection of the global (local) solution set of (7) into the x space. We obtain the relations: (c.f. e.g. 5])
Mw;eff =
2IRL4() Mlocw;eff =
2IRL4loc():
3
We use the reduction of the multiparametric optimization problems (4), (6) and (7) to a sequence of one-parametric optimization problems presented in 4].These sequence can be generated by a dialogue procedure with the decision maker (c.f. e.g. 4, 3]).
We obtain from the dialogue procedure some 0 1 2 IRL and we have to consider the following one-parametric optimization problems.
Pi(t) = Pi((t)) i = 1 ::: 3
(t) = (1;t)0+t1 t20 1] (8) We denote by Mi(t) i = 1 2 3: the corresponding feasible sets for the para- metrizations (8).
From the dialogue procedure (c.f. e.g. 4]) we know that1 expresses the wishes of the decision maker. He is mainly interested to know wherther his wish 1 was realistic or not. If the goal point 1 is a realistic one, then the decision maker wants to nd a point ~x2M such that
fj(~x)1j j = 1 ::: L We call a point 1 2IRL a realistic goal if
M1(1) =fx2Mjfj(x)1j j = 1 ::: Lg6= and a point x2M1(1) a goal realizer.
Our purpose in these papper is to analyse the singularities and regulari- zations of these 3 parametrizations from the point of view of the parametric optimization theory.
2 Theoretical Background
We consider the following general one-parametric optimization problem:
P(t) : min f f(x t)j x2M(t) g where
M(t) =f x2IRn jgi(x t) = 0 i2I gk(x t)0 k 2K g
I and K are index sets dened as in (2). We rst recall the classF of Jongen, Jonker and Twilt 8].
4
De nition 2.1
(c.f. e.g. 3, 8])gc is the set of all (x t) such that x2M(t) and the vectors Dxf(x t) Dxgk(x t) k 2IK0(x t)
are linearly dependent, where K0(x t) =f j 2K j gj(x t) = 0 g
HereDxf denotes for the row vectors of rst partial derivatives with respect to x. If (x t)2gc there exits numbersuj j 2IKf0g, such that:
u0Dxf(x t) + X
j2IKujDxgj(x t) = 0 (9) where uj = 0 for j2 KnK0.
De nition 2.2
(c.f. e.g. 3])LICQ
The linear independence constraint qualication is satised at x 2 M(t) if the vectorsDxgj(x t) j 2IK0(x t) are linearly independent.
MFCQ
The Mangasarian Fromovitz constraint qualication is satised at x2M(t) if:1. Dxgi(x t) i2I are l.i.
2. There exits a vector 2IRn with:
Dxgi(x t) = 0 i2I
Dxgj(x t) < 0 j2K0(x t)
If x is a local minimizer for P(t) and LICQ or MFCQ is satised at x, then (x t)2gcand we can nd u in (9), such that u0 = 1 and uj 0,j 2K0(x t)
De nition 2.3
(c.f. 8])1. We call every (x t)2gc a generalized critical point.
2. If (x t) 2 gc and in (9) we can nd u, such that u0 = 1 and uj
0 j 2K0(x t), then we call (x t) an stationary point.
5
Dene:
stat =n (x t)2IRn+1 j x is a stationary poit of P(t)o
In 8, 9] the local structure of gcis completelydescribed in the neighborhood of generalized critical poits of the following 5 types. (c.f. 3], too)
Type 1:
(x t)2gc is of Type 1 if the following conditions are fullled:
A1)
LICQ is satised at x2M(t).Then there exits only one uj j 2IK0(x t) such that (9) is satised with u0 = 1
A2)
DxL(z) = 0, where z = (x t).A3)
uj 6= 0 j 2K0(z).A4)
D2xLjTzM is nonsingular.where L(z) denotes the Lagragian :
L(z) = f(z) + X
k2IK0( z)ukgk(z) and TzM the tangent subspace:
TzM =f 2IRn j Dxgk(z) = 0 k 2IK0(z) g
Dx2LjTxM represents VTDx2LV , where V is a matrix whose columns form a basis for TzM.
The 2-5 types represent 4 basic degeneracies of Type 1.
Type 2.-
The violation of A3.Type 3.-
The violation of A4.Type 4.-
The violation of A1, butjIj+jK0(z)j< n + 1.Type 5.-
The violation of A1, butjIj+jK0(z)j=n + 1.6
stat
gc
MFCQ holds(k) z
MFCQ violated(l) z
MFCQ violated(m) z
Type 5
J0(z)(g)6= z
J0(z)(h)6= z
J0(z) =(i) z
J0(z) =(j) z
Type 4
(e) z
(f) z
Type 3
(a) z
(b) z
(c) z
(d) z Type 2
Type 1 z
t
Figure 1: The 5 types of generalized critical points.
7
F =
(
(f g)2C3(IRn+1 IRp+1)
each point of gc
belongs to one of the 5 types
)
According to the investigations 8, 9] we have the following posibilities for the local structure of gc (see g. 1)
The setF is open and dense with to the strong (or Whitney)Cs3-topology.
In 8] is proved the following result:
Theorem 2.1
If (f g) 2 F then, the set KT is a 1-dimensional manifold with boundary and
z 2KT is a boundary point , K0(z)6= and MFCQ fails to hold.
where KT (c.f. 8]) is by denition the clousure of the set of all stationary point of Type 1.
With the results presented (c.f. 8]) we obtained a condition (F) for f and g , such that gc is suitable for application of continuation methods (c.f.
e.g. 3]). The continuation methods for the singularities are explained in 3]
, it is posible to jump from one connected component in gc to another one, if there is no continuation in gc, but not in any cases. (c.f. 3, 6]). It is not posible to jump in many cases of types 4 and 5.
Furthermore we use another regularity condition for (f g).
Let z = (x u t)2IRn+p+1, and u+ = max(u 0) , u; = min(u 0). Dene the Kojima map: (c.f. e.g.11])
H(z) =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
Dxf(x t) + Pi2IuiDxgi(x t) + Pj2Ku+j Dxgj(x t)
;g1(x t)
... ... ...
;gm(x t)
u;m+1 ; gm+1(x t)
... ... ...
u;p ; gp(x t)
3
7
7
7
7
7
7
7
7
7
7
7
7
5
(10)
Lemma 2.1
(c.f. 10])If02IRn+p is a regular value ofH and MFCQ is satised at all (x t)2stat
then, stat is a one-dimensional manifold and each connected component of H;1(0) is homeomorc with a circle (loop) or with IR (path).
8
Here isH a PC1mapping as a generalization ofC1mappings and analogously regular value.
De nition 2.4
Let T IR . We dene:
1. (f g)(or the problemP(t)) is JJT-regular with respect to T ,denoted by(f g)2FjT (orP(t)2 FjT), if and only if each point ofgc\IRnT belongs to one of the 5 types.
2. P(t) is KH-regular on T if 0 is a regular value of the Kojima map (10) of P(t) restricted to IRnIRpT.
3 Singularities
In these section we analize the singularities for the parametrizations (8). In the general case the 5 types of singularities may appear when we follow curves in gc.
If MFCQ is fullled in M ( from the original problem (1, 2) ),it is known that M2(t) and M3(t) satisfy MFCQ for all t20 1] (c.f. e.g. 3, 4]). Using Lemma 4.1 of 8] we obtain that, following KT inPi(t) i = 2 3 ,points of Type 4 may not appear and points of Type 5 only in the "good" case (with continuation) (c.f. e.g. 3]). (see the gures of section 2).
We have examples from all the posible singularities for the 3 parametriza- tions, . We recall that ,when we follow gc, the previous reduction of Type 4 is not posible, as shown in 3, 4].
We give a completion to the tables of singularities presented in 3, 4] and examples that show the dierences.
Sing. KT gc
P1(t) 1,2,3,4,5 1,2,3,4,5 P2(t) 1,2,3,5 1,2,3,4,5 P3(t) 1,2,3,5 1,2,3,4,5
9
Example 3.1
( Parametrization 2) :01 =02 = 1 f1(x1 x2) =;x1 f2(x1 x2) =x2 g1(x1 x2) =x31;3x1;x2
0 = (;1:5 0) 1 = (3 ;3) For this example the following point is of Type 4.
t=0.345155, x1 = 0:816558 x2 =;1:90522 v =;0:869755 u1 !+1 u2 !;1 u3!+1
Where theu represent the Lagrange multipliers.
Example 3.2
( Parametrization 3) :01 =02 = 1 f1(x1) =x21 f2(x1) =x1 0 = (;10 12) 1 = (10 ;12) For this example the following point is of Type 4.
t=0.494318345155, x1 = 0:5 v = 0:363586 u1 !+1 u2!;1
In the two points of Typ 4 presented is posible to jump , because they are not endpoints of a curve of local minimizers. (see 6]). It is not dicult to make an examplewith one point of Typ 4 without jump in the rst parametrization.
We know now that could be imposible nd a goal realizer (t = 1) using the rst parametrization. We need a numerical description of all connected components in gc in the worst case.
4 Regularizations
We begin these section with the introduction of the Kojima regularization in details.(c.f. e.g. 11])
We call a partition of IRn a countable familyQ of polyedrons such that:
1. SS2QS = IRn
2. The intersection of any two elements of Q is empty or is a common face
3. Q is locally nite
10
Given a partition Q of IRn in polyedrons, we call a function H : IRn ! IRq PCr dierenciable (H 2PCr(IRn IRq)) with respect to Q if:
8S 2Q there is a neighborhood US of S and a function HS such that 1. HS 2Cr(US IRq)
2. HSjS =HjS
b 2 IRq is a regular value of H 2 PCr(IRn IRq) on V (open subset of IRn) with respect toQ (partition of IRn), if8S2Q the following conditions hold:
1. 9(US HS) such that b is a regular value of HS on US\V . 2. 8x2H;1 \\V , where is any face of S,
range
"
DHS(x) B
#
=q + n;q (11)
whereB is a matrix of size (n;q) n with range (n;q) and d 2IRn; q such that span() =fx 2IRn j B x = d g
The equation (11) must hold for any representation (B d) of span(). (c.f.
e.g. 11])
For H as in (10) we have that H 2 PC1(IRn+p+1 IRn+p) with respect to the following partition of IRn.
Polyedrons:
P(S) =f z 2IRn+p+1 j uj 0 j 2S uk 0 k 2KnS g where S K
Faces :
(A B C) =fz 2IRn+p+1juj 0 j 2A ur= 0 r 2B uk 0 k 2Cg where (A B C) is a partition of K with B6=.
The faces of P(S) have of the form (A B K n(AB)) with A S S (AB). For a polyedron P(S) we obtain the corresponding reduction of H:
HP(S) =
2
6
4
Dxf(x t) + Pi2ISuiDxgi(x t)
;gi(x t) i2I S
uj ; gj(x t) j 2KnS
3
7
5 (12) We need the following important result.
11
Theorem 4.1 (Sard)
Let V an open subset of IRn
1. Let 2 Cr(V IRq) and r > (n;q)+. Then the set of singular values of has Lebesgue measure 0.(c.f. e.g. 12])
2. (Parametrized): Let 2Cr(IRn+p IRq) and r > (n;q)+. If 02IRq is a regular value of over V IRp, then for almost all 2 IRp 0 is a regular value of the function :V !IRq dened as (x) = (x ). (c.f. e.g. 1])
The parametrized version remainds true if we consider dened on V W, where V 2 IRn and W 2 IRp are open subsets. We will often refer to this last case of Sard's theorem .
For P1(t) we introduce the following perturbed problem
~P1(t) : min F(x) + (1;t)cTx
restricted to Gi(x) + (1;t)bi = 0 i2I Gj(x) + (1;t)bj 0 j 2K
fk(x) ; k(t) 0 k2L (13) Where F(x) =PLk=1 0kfk(x) and G = (g1 ::: gp).
Theorem 4.2
Letf G 0 such that f Dxf GandDxGare twice continously dierentiable, then for almost all (c b 0 1)2IRn+p+2L ~P1(t) is KH-regular on IRnf1g
Proof:
Let us introduce the following notation:
~L = p + 1 ::: p + L
~K = K ~L
gk+p(x t) = fk(x);k(t) k2L
We call ~H the Kojima function (10) of the problem (13). For S ~K the following relation is fullled
~HS =HS+ (1;t)
2
6
4
cT
;bT 0
3
7
5
12
Computing the derivative with respect to the variables c b 0 and 1 we obtain that 0 is a regular value of ~HS restricted to the set IRn+p+1(t 6= 1) where
IRn+p+1(t 6= 1) =fz 2IRn+p+1j t 6= 1g (14) From Theorem 4.1 we obtain that for almost all (c b 0 1) ~HS restricted to IRn+p+1(t 6= 1) has 0 as a regular value.
Now, for the second condition of (11). Let a face of P(S) for S K, then there is an index set B with B K such that.
span() =f z 2IRn+p+1 j MB z = 0 g
WhereMBis an (n+p+1;dim(span()))(n+p+1) matrix whose columns having index in B form an identity square matrix and all other elements are 0. It is easy to verify that over the set IRn+p+1(t6= 1)IRn+p+2L the matrix
"
DHMB
#
(15) has full range.We note that if we change the identity of MB for a regular matrix the matrix (15) has also full range.
We obtain using theorem 4.1 that for almost all (c b 0 1) the condition (11) for is fullled.
The numbers of faces ofP(S) is nite and so is the number of polyedrons, we obtain then that for almost all (c b 0 1) 0 is a regular value of ~H restricted to IRn+p+1(t 6= 1). Also ~P(t) is regular in the sense of Kojima and
Hirabayashi. 2
For the other parametrizations we obtain, similarly the following result
Theorem 4.3
Let f G 0 such that f Dxf GandDxG are twice continously dierenciable, then for almost all (c b 0 1)2IRn+p+2L ~P2(t) and ~P3(t) is KH-regular on IRf1g in the sense of Kojima, except for the points wheret = 1.
~P2(t) and ~P3(t) are dened anagously , with an aditional parameter cn+1. For the regularization in the sense of Jongen, Jonker and Twilt we con- sider the following perturbed problem:
P1(t) : min F(x) + (1;t) (12 xTAx + cTx)
restrictedto : x2 M(t B x0 0 1) (16) 13
where
M(t B x0 0 1) = f x2IRn jgi(x t) = 0 i2I gj(x t)0 j 2 ~K g gi(x t) = t gi(x t) + (1;t) bTi (x;x0) i2I
gj(x t) = t gj(x t) + (1;t) (bTi x + dj;m) j 2K gk(x t) = t fk;p(x t) + (1;t) bTix;k;p(t) k2 ~K
Here A 2 IR05n(n+1) represents a symetric matrix, B 2 IRn(p+L)+p;m with B = (b1 ::: bp+L d) where bj 2IRn, j 2I ~K d2IRp;m and x0 2IRn.
Theorem 4.4
If the 3rd partial derivatives of fk k 2L and gj j 2I K are continuosly dierentiable with respect tox, then for almost all (A c B x0 0 1) follows
P1(t) 2FjIRnf1g
We use "almost all" in the following sense: each measurable subset of the set
n(A c B x0 0 1)2IR0:5n(n+1)+n(p+L+2)+p;m+2Lj P1(t) 62 FjIRnf1g
o
has Lebesgue measure zero.
We use the following notation for a given (B x0 0 1) K0(x t) = fj 2 ~K jgj = 0 g
MF(B x0 0 1) = f(x t)2IRn+1j x2 M(t B x0 0 1) LICQ failsg
Lemma 4.1
For almost all (B x0 0 1) we have that
1. MF(B x0 0 1) \ f (x t) 2 IRn+1 j t 6= 1 g is a 0-dimensional manifold.
2. If (x t)2 MF(B x0 0 1)\f(x t) 2IRn+1j t 6= 1g, then the vectors
fDgj(x t), j 2 I K0(x t)g are linearly independent and if j, j 2 I K0(x t) are solution of
X
j2IK0( xt)jDxgj(x t) = 0 then j 6= 0 for j 2 K0(x t)
14
Proof of Lemma 4.1:
Using the Theorem of Fubini (c.f. e.g. 7]) and an inductive argument we obtain that the complement of the open set
B0=n (B x0 0 1)2IRn(p+L)+p;m j bi i2I are linearl. indep. o (17) has Lebesgue measure 0.
Let S ~K, q 2IS and ~S S be xed. We consider the functions
;(x t B x0 0 1) = X
j2(IS)nfqgjgj(x t) + gq(x t) 1(x t B x0 0 1) =
"
DTx;(x t B x0 0 1) gj(x t) j 2IS
#
2(x t B x0 0 1) =
"
1(x t B x0 0 1) j j 2 ~S
#
Where is formed by the j j 2S
Computing the derivatives of 1 with respect to bq dj j 2S\K x0 and 0k p +k 2S and 2 with respect to the same variables and with respect to j j 2 ~S too we obtain that 1 and 2 restricted over the set
IRn+jSj+1(t 6= 1)B0
whereIRn+jSj+1(t6= 1) is dened anagously to (14), have 0 as a regular value.
Using the theorem 4.1 and that B0 has complement with Lebesgue mea- sure 0 we obtain ,that for almost all (B x0 0 1), 1 and 2 have 0 as a regular value over the set IRn+jSj+1(t6= 1).
Since the ways to select S and ~S are nite we obtain that for almost all (B x0 0 1), 1and 2have 0 as a regular value over the setIRn+s+1(t 6= 1) for each selection of S, ~S and q.
The Assertions of the Lemma can be easily obtained using the Jacobi
matrix of 1 and 2. 2
Proof of Theorem 4.4:
We make the proof in two steps:
Step 1:
We prove here that for almost all (A c B x0 0 1) ,each generalized critical point with LICQ and t6= 1 is of Type 1, 2 or 3.
15
Let S ~K xed. Supose that S = fm + 1 ::: qg with q m. Let T f1 ::: n + qg and T S.
LetMn+q(T)IR05(n+q)(n+q+1) the set of all symmetric matrix with size (n + q)(n + q) and range jTj, such that the columns with index in T are linearly independent.
Mn+q(T) is a manifold of codimension 12(n+q;jTj)(n+q;jTj+1): We denote by the equations that dene these manifold locally as
We use H andHS as the Kojima matrix (10) and the corresponding re- duction (12) for a problem such that the following relation is fullled:
HS =HS+
2
6
6
6
6
4
(1;t) Ax + c +Pi2ISui bi ]
;(1;t) bTi (x;x0) i2I
;(1;t) bTj x + dj ] j 2K
;(1;t) bTk x k2 ~L
3
7
7
7
7
5
(18) where HS the corresponding reduction (12) of the Kojima matrix (10) of the problem (16).
If we derivate the equation (18) with respect to x and ui i 2 I S the rows with index l such that l = 1 ::: n or (l;n) 2 I S, we obtain (by denition)
MS =MS+
"
(1;t)A (1;t)bj j 2IS
;(1;t)bTj j 2I S 0
#
The size of MS and MS are (n + q)(n + q) , but they are not symmetric.
We consider also the matrix
MS+ = MS
"
;In 0 0 Iq
#
which is symmetric and has the same rank as MS.
We will consider all the symmetric matrices of size ll as elements of the space IR05(l+1)l.
We dene the function dened over
IR2(n+p+L)+1+05(n+q)(n+q+1)+05n(n+1)+2n+n(p+L)+p;m+2L and with values on
IR2(n+p+L)+05(n+q)(n+q+1)+05(n+q;jTj)(n+q;jTj+1)+jT)j 16
We consider the variable of ,that we calls, subdidived as s = (z z ~z A c B x0 0 1)
Where z = (x u t)2IRn+p+L+1), z 2IRn+p+L and ~z 2IR05(n+q)(n+q+1). We denote by ej the canonical vectors .
(s) :=;zj+ej HS(z A c B 0 1) for j = 1 ::: n + p + L
n+p+L+j(s) :=;z~j+ej MS+(z A c B 0 1) for j = 1 ::: 0 5(n + q)(n + q + 1)
n+p+L+05(n+q)(n+q+1)+j(s) := zj
for j = 1 ::: n + p + L
2(n+p+L)+05(n+q)(n+q+1)+j(s) := j(~z) for j = 1 ::: 0 5(n + q;jTj)(n + q;jTj+ 1)
2(n+p+L)+05(n+q)(n+q+1)+05(n+q;jTj)(n+q;jTj+1)+j(s) := uj
for j = 1 ::: jTj.
It can be proved that if fn + 1 ::: n + qgT then 0 is a regular value of dened over the open set
f(z z ~z A c B x0 0 1) j t6= 1 (B x0 0 1)2B0g
where B0 is dened by (17).( see 2] ,for details) With the same idea used in the lema 4.1 can be obtained that for almost all (A c B x0 0 1) and for each selection of S T T such that
fn + 1 ::: n + qg T
, that means LICQ, has 0 as regular value over the open set Z(t6= 1) =f(z z ~z)jt6= 1g
17
Supose that we have a "good" selection of (A c B x0 0 1), and we supose that these selection is 0 , we obtain H = H.
For xed T and T, is dened over
IR2(n+p+L)+1+05(n+q)(n+q+1) and with values on
IR2(n+p+L)+05(n+q)(n+q+1)+05(n+q;jTj)(n+q;jTj+1)+jT)j Since 0 is a regular value of , we have then only 3 posibilities:
1. 0 5(n+q;jTj)(n+q;jTj+1)+jT)j> 1, and then ;1(0)\Z(t6= 1) is empty.
2. 0 5(n+q;jTj)(n+q;jTj+1)+jT)j= 1, and then ;1(0)\Z(t6= 1) is a 0-dimensional manifold.
3. 0 5(n+q;jTj)(n+q;jTj+1)+jT)j= 0, and then ;1(0)\Z(t6= 1) is a 1-dimensional manifold.
If (z z ~z)2 ;1(0)\Z(t6= 1) then there is only 3 posibilities:
Case 1
jTj=n + q and T =Case 2
jTj=n + q;1 and T =Case 3
jTj=n + q and jTj= 1For each point z = (x t)2gc where LICQ is satised we obtain that there exitsS T and T such that:
(z HS(z) MS+(z))2 ;1(0) Choosing
S = ~K0 (x t)
T =f j 2 ~K0(x t) juj = 0 g
and T an index set such that jTj = range(MS(z)) , the columns of MS(z) with index in T linearly independent and
f n + 1 ::: n + m + j~K0(x t)j g T
The 3 cases (Case 1, 2 and 3) represents the Typ 1, 3 and 2. (c.f. 11]) Step 2:
Taking into acount the two following facts:
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1. Let (x t) xed with (t6= 1), then for almost all (A c)2IR05n(n+1)+n it holds that:
tDxf(x t) + (1;t)A x + c]62 spanfDxgi(x t) i2ISg for each S ~K with fDxgi(x t) i2ISg < n.
2. IfIR is a matrix with range greater than 0 and (t6= 1) then for almost all (A c)2IR05n(n+1)+n it holds that:
tDxf(x t)+(1;t)A x + c]T IR tDxf(x t)+(1;t)A x + c] 6= 0 and using the same argument as in 11] it can be proved that if (B x0 0 1)2 B0,B0dened by (17) then for almost all (A c)2IR05n(n+1)+n each generali- zed criticalpoint without LICQ is of Typ 4 or 5.
We note that we can get the parameters so small as needed, and such that, the perturbed problem (16) belongs to the set F(t 6= 1) and x0 is a feasible point for P1(0). The same property is not posible to hold if we want that x0 be a generalized critical point too.
For P2(t) and P3(t) we obtained similar regularizations results (see 2]).
We are now interested in the MFCQ condition for the set M1(t).
Theorem 4.5
Let L L not empty. If x is a locally ecient points, a -locally ecient point or a weakly locally ecient point of the problem:
min f fj(x) j 2 L j x2M g (19) then MFCQ is not fullled globally over the set
~M =f x2IRn j x2M fk(x)fk(x) k 2 L g Proof:
Supose L = f1 ::: lg with l L. If MFCQ fails to hold in x for M then fails to hold for ~M too. We can then supose that MFCQ is fullled for these case.
If MFCQ is fullled at x for M, then there exists a function x(t) such that:
x(0) = x
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x(t)2 ~M M 8 t2(0 t)
We obtain using a continuity argument of fk k 2 L that there exists a positive ~ttsuch that:
fk(x(t)) < fk(x) 8 t2(0 ~t) 8 k 2 L
That is a contradiction with the asumption that x is a locally ecient point, a -locally ecient point or a weakly locally ecient point of the problem
(19) 2
We note that if x is a point of interest , then there exists a subset of IRL which can't be intersected by the segment(t) t20 1], if we are interested in MFCQ for M1(t) t20 1].
The next theorem give us a measure of these "bad" set in IR2L.
Theorem 4.6
Let k 2L, such that the set
M(k) = f x2IRn j x2M fk(x) k g
doesn't fulll MFCQ for some k 2IR. Then the set
f (0 1)2IR2L j 9t20 1] where MFCQ fails to hold for M1(t) g (20) has a subset of measure +1.
Proof:
Supose that k = 1, and at x2M(1) MFCQ fails to hold. We consider the following subset ofIRL
~IRL=f 2IRL j 1 = 1 j fj(x) j = 2 ::: L g and using ~IRL we dene
~IR2L =f (0 1)2IR2L j 9 t20 1] with (1;t)0+t1 2 ~IRL g We will prove that the set ~IR2L, a subset of (20) , has L-measure +1.
It's easy to prove that ~IR2L is closed and then measurable.
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We consider the variable (0 1)2 IR2L subdivided in the following way (~0 ~1)2IR2L where:
~0 = (0 11)2IRL+1 and ~1 = (12 ::: 1L)2IRL;1 If ~0 62IR0 IRL+1 then the set IR1(~0)IRL;1 is empty, where
IR0 =f ~0 2IRL+1 j 01 1 11 or 01 1 11 g and IR1(~0) =f ~1 2IRL;1 j(~0 ~1)62 ~IR2L g If ~0 2IR0 is such that 01 6= 1 then , dening t as
t= 1 ;01 11 ;01 we obtain the representation
IR1(~0) =f ~1 2IRL;1 j 1j 1
t fj(x) + (t;1)0j] j = 2 ::: Lg This set has measure +1 in IRL;1. Using that the set IR0 has L-measure +1in IRL+1 and the theorem of fubini we obtain the result. 2
Remark:
If the original set M of (2) has a connected compact component, then the set (20) has a subset of measure +1.
References
1] Abraham R., J. Robbin : Transversal mappings and !ows. Ben- jamin, New York, 1967.
2] G"omez W. : Optimizaci"on vectorial: singularidades, regulari zacio- nes. Trabajo de Diploma. Fac. Mat.-Cib. Universidad de la Habana.
1994.
3] Guddat J., Guerra F., Jongen H.T. : Parametric Optimization:
singularities, pathfollowing and jumps. John Wiley, Chichester and Teubner, Stuttgart 1990.
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