On double one-parametric optimization problems and some applications

On double one-parametric optimization problems and some applications

Paulo Mbunga

Mathematisch-Naturwissenschaftliche Fakultt II Institut fr Mathematik Humboldt-Universitt zu Berlin

24th May 1999

Abstract

This paper deals with a special class of quadratic optimization prob- lems with quadratic constraints, which we call double parametric. We try to solve these problems using Pathfollowing Methods with a special case of the Standard Embedding. The singularity theory developed by Jongen, Jonker and Twilt plays a great role in our investigations. In cases where a jump from one connected component to another one in the set of local minimizers and in the set of generalized critical points is not possible, we prove that the turning points are in negative direction. We survey results with respect to the choice of the starting point and make some proposals in order to overcome cases where jumps are not possible. We show the role of the Mangasarian-Fromovitz constraints qualication in the existence of the jumps and the solution of considered problems. Some examples of ap- plications to global quadratic optimization and multicriteria optimization are given.

Keywords: parametric optimization, pathfollowing methods, jumps, gen- eralized critical point, turning point in the negative sense

1 Introduction 2

1 Introduction

In this paper, we consider the following double quadratic optimization problems min^f(x^;x^o)^TD(x^;x^o)^jx²Mi^g i = 1 2:

(^Pi) M¹ :=

(

x^2Rn g_j(x) = a_Tjx + b_j 0 j²J gs⁺i(x) = x^TAix + a_Ts⁺_ix + bs⁺i 0 i²I

)

M² :=

(

x^2Rn gj(x) = a_Tjx + bj 0 j²J gs⁺¹(x) = x^TA¹x + a_Ts⁺¹x + bs⁺¹ 0

)

J = ^f1 s^g I =^f1 2 3^g

where D is a positive-denite matrix, A¹ a negative semi-denite matrix, A² and A³positive semi-denite matrices, aja vector in^Rnand x^o2Rna specially chosen point. According to the properties of the matrices Ai,i = 1 ::: 3, the functions gs⁺i are the respective quadratic concave and convex constraints (for i = 2 3).

Let

P :=^fx^2Rnja_Tjx + bj0 (j = 1 s)^g

G¹:=x^2Rnjgs⁺¹(x) = x^TA¹x + a_Ts⁺¹x + bs⁺¹ 0 G²:=^fx^2Rnjgs⁺²(x) = x^TA²x + a_Ts⁺²x + bs⁺²0^g G³:=^fx^2Rnjgs⁺³(x) = x^TA²x + a_Ts⁺³x + bs⁺³0^g: The feasible sets M¹and M² can also be given by

M¹ = ^P\\3

i⁼¹Gi

M² = ^P\G¹:

We notice that the sets M¹ and M² are not necessarily convex (see Fig. 1).

Remark 1.

The kind of problems Pi,i = 1 2, was considered e.g. in 13].

Let us introduce some conditions, that we shall need later. First we assume:

(C1)

^P is a convex polyhedron

Then the sets M¹ and M² are compact. The sets

H¹ = ^fx^2RnjDxgs⁺³(x) = 2A³x + as⁺³= 0^g H² = ^fx^2RnjDxgs⁺¹(x) = 2A¹x + as⁺¹= 0^g represent the a ne subspaces with the dimension

n^;rank(A³) and n^;rank(A¹), respectively.

1 Introduction 3

H¹ x^o

M¹ gs⁺³ gs⁺²

P

gs⁺¹

(a)

H¹ x^o

M¹ M¹

P

gs⁺²

gs⁺¹

gs⁺³

H² (b)

x^o M²

P

gs⁺¹

(c)

H² x^o

M² M²

P

gs⁺¹

(d) Figure 1: possible structure of M¹ and M²

Starting situation

One of the important issues we have to deal with is the choice of the starting point x^o. We shall see later that solving the problems (Pi) i = 1 2 successfully will depend on the choice of this starting point. In this paper we suggest to choose the starting point in the following way (see Fig. 1):

Problem P¹:

C2(^a) x^o2int(^P\G^1\G²)ⁿ(H¹G³):

Problem P²:

C2(^b) x^{o 2} int^{P n}(G¹H²): Furthermore, we claim that the observed constraints must have full dimension(cf. 13]):

Mi = cl intMi i = 1 2:

(^C3)

With the condition (C3) we want to exclude cases where M¹@(^{P \\3}

i⁼¹Gi) M²@(^P\G¹):

Let us recall now the well-known concept of embedding (cf. e.g. Allgower 1], Guddat et al. 5], Dentcheva 3], Gfrerer et al. 10]) and propose the following embeddings in order to solve (Pi) i = 1 2:

Let x^o2P be arbitrarily chosen and dene

min^ff(x t)^jx²M¹(t)^g t²(^;1 1]

P

1(t) :

2 Theoretical Background 4

where

f(x t) := (x^;x^o)^TD(x^;x^o)

M¹(t) := ^fx^2Rnjgj(x t)0 (j = 1 s + 3)^g gj(x t) := gj(x) j = 1 s + 2

gs⁺³(x t) := gs⁺³(x) + (t^;1)gs⁺³(x^o) and

min^ff(x t)^jx²M²(t)^g t²(^;1 1]

P

2(t) : where

f(x t) := (x^;x^o)^TD(x^;x^o)

M²(t) := ^fx^2Rnjgj(x t)0 (j = 1 s + 1)^g gj(x t) := gj(x) j = 1 s

gs⁺¹(x t) := gs⁺¹(x) + (t^;1)gs⁺¹(x^o)

In Section 2 we present the necessary background for one-parametric problems, of which P_i(t) is a particular case. In Section 3 we study P_i(t) and in section 4 we analyze under which perturbations on P_i(t) we obtain a parametric problem P^D(t) belonging to the generic class ^F of Jongen, Jonker and Twilt 15]. In Section 5 we present some applications.

2 Theoretical Background

We consider the general one-parametric problem:

min^ff(x t)^jx²M(t)^g t^2R P(t) :

where

M(t) = ^fx^2Rnjhi(x t) = 0 i²I gj(x t)0 j²J^g I = ^f1 m^g m < n J =^f1 s^g

and f hi gj²C^q(^RnRR) q 2.

Furthermore, we introduce the following notations:

P

gc := ^f(x t)^2RnRj x is a g.c. point of P(t)^g

P

stat := ^f(x t)^2RnRj x is a stationary point of P(t)^g

Ploc := ^f(x t)^2RnRj x is a local minimizer of P(t)^g H := (h¹ h_m)^T G := (g¹ g_s)^T:

Note 1.

For the denition of a g.c. point we refer to 15], see also 5].

2 Theoretical Background 5

x^o M¹ gs⁺³

gs⁺²

P

gs⁺¹

gs⁺³(x 0) = 0

(a)

x^o gs⁺³(x 0) = 0

M¹ M¹

P

gs⁺²

gs⁺¹

gs⁺³

(b)

x^o

gs⁺³(x 0) = 0 M²

P

gs⁺¹

(c)

x^o

gs⁺³(x 0) = 0

M² M²

P

gs⁺¹

(d) Figure 2: possible structure of M¹(t) and M²(t)

The LinearIndependence Constraint Qualication (LICQ)is said to hold at x²M(t) if the vectors

Dxhi(x t) i²I Dgj(x t) j²Jo(x t) are linearly independent (Jo(x t) :=^fj²J^jgj(x t) = 0^g).

TheMangasarian-Fromovitz Constraint Qualication (MFCQ)is satised at x²M(t) if:

MF1 fDhi(x t) i²I^gare linearly independent,

MF2 there exists a vector ^2Rnsatisfying:

Dhi(x t) = 0 i²I Dgj(x t) < 0 j²Jo(x t):

Next we cite our short characterization of the class^F introduced by Jongen, Jonker and Twilt 15] from 2:5 in 5] . In 15] the local structure of^P_gc is completely described if (f H G) belongs to a C_s³-open and dense subset ^F of C³(^{Rn+1 R})^1+m+s, where C_s³denotes the strong (or Whitney-)C_s³-topology (see

2 Theoretical Background 6

also 5]). A typical baseneighbourhood of this topology with f ²C^k(^{Rn+1 R}) and ²C⁺(^{Rn+1 R}) is given by

B^k(f) :=

(

g²C^k(^{Rn+1 R})

j@g(x)^;@f(x)^j< (x)

8x^{2Rn+1 8} with^jjk

)

where

C⁺(^{Rn+1 R}) :=^f :^Rn+1;!Rj is continuous and (x) > 0 ⁸x^2Rn+1g: If (f H G)^2F , then^P_gccan be divided into 5 types.

Type

¹

:

A point z = (x t)^2P_gc is of Type 1 if the following conditions are satised:

There exists _{i j}^2R i²I j²J_o(z) satisfying

0

@Dxf +^X

i²I iDxhi+ ^X

j²J^o(z⁾ jDxgj

1

A

jz =

z

= 0 (1)

(^ND1) LICQ is satised at x²M(t),

therefore i j ^2Ri²I j²Jo(z) are uniquely dened, (^ND2) j⁶= 0 j²Jo(z),

(^ND3) D_x²L(x t) ^jT⁽z⁾ is nonsingular, where D²_xL is the Hessian of the Lagrangian

L(x t) = f(x t) +^X

i²I hi(x t) + ^X

j²J^o(z⁾ gj(x t) and the uniquely determined numbers _{i j} are taken from (1).

Furthermore,

T(z) =^f2RnjDxhi(z) = 0 Dxgj(z) = 0 j²Jo(z)^g

is the tangent space at z. D_x²L(z)^j_T⁽_z⁾represents V^TD²_xLV , where V is a matrix whose columns form a basis of T(z).

A point of Type 1 is a nondegenerate critical point. The set ^P_gc is the closure of the set of all points of Type 1, the points of the Types 2^;5 constitute a discrete subset of^P_gc.

The points of the Types 2^;5 represent basic degeneracies:

Type

²

:

- violation of (ND2)

Type

³

:

- violation of (ND3)

Type

⁴

:

- violation of (ND1) and^jI^j+^jJo(z)^j;1 < n

Type

⁵

:

- violation of (ND1) and^jI^j+^jJo(z)^j= n + 1

2 Theoretical Background 7

For each of these ve types Figure 3 illustrates the local structure of ^P_gc in the neighbourhood of stationary points. The full curves stand for the curve of stationary points z = (x t), and the dotted curve represents the curve of g.c.

points which are not stationary points. Let^P_gc ^{2 f}1 5^gbe the set of g.c. points of Type . The class^F is dened by

F:=

(

(f H G)²C³(^RnRR)^1+m+sjP_gc ⁵

⁼¹

Pgc

)

: Recall that^F is C_s³ open and dense in C³(^{RnR R})^1+m+s(cf. 15]).

The algorithm PATH III

This algorithm computes a numerical description of a compact connected com- ponent in^P_gc, i.e., in particular, it nds a discretization of the interval t_A t_B] t_A<

0 < tB(not necessarily tA tB] 0 1]) and corresponding g.c. points starting at (x^o 0)^2P_gc. The algorithm is based on the active index set strategy and is a so-called predictor-corrector scheme if the active index set is constant. A Newton corrector is used.

The main point of the approach consists in the computation of the new index sets for the possible continuations.

We note that we do not have any numerical di culties walking around turning points of the Types 3 or 4.

More precisely: if there exists a PC²-path connecting (x^o 0) and a point (x 1), then we obtain, in a nite number of predictor and corrector steps, a point lying in the radius of convergence of the Newton method for x with respect to the problem P(1). Since PATH III is not successful in nding a point (x 1)^2P_gc in general, we propose to jump from one connected component in cl^P_loc and

P

gc, respectively, to another one. For more information we refer the reader to Guddat et al. 5] and PAFO 11].

The algorithm JUMP I

This algorithm works in the set cl^P_loc. Starting at the known local minimizer x^oat to= 0, a connected component in cl^P_locfor increasing t will be numeric- ally described by using PATH III. Depending on the appearance of a singularity, a direction of descent will be computed. Using a feasible direction method, a local minimizer on another connected component in^P_locwill be calculated and PATH III starts again. We have to take into account that we have no proposals for jumps in any case. Jumps are possible if there occurs a turning point of Type 2 or a point of Type 3. Let t be near t t < t, and let xm(t) and xs(t) be the local minimizer and a point of^P_stat^nP_loc, respectively. Then, as t tends to t, the vector

u(t) := x^s(t)^;xm(t)

kxs(t)^;xm(t)^k

2 Theoretical Background 8

stat

gc

MFCQ holds (k)

z

MFCQ violated (l)

z

MFCQ violated (m) Type 5 z

J⁰(z)⁶= (g)

z

J⁰(z)⁶= (h) z

J⁰(z) = (i)

z

J⁰(z) = (j) Type 4 z

(e)

z

(f) Type 3 z

(a) z

(b) z

(c) z

(d) Type 2 z

Type 1 z

t

Figure 3: 5 Types of Jongen, Jonker and Twilt

3 Properties of the double parametric embedding 9

tends to a tangential vector, say u, which is a direction of (higher order) descent.

Hence, for t near t t < t, the vector xs(t)^;xm(t) provides an approximately tangential direction of descent (see Figure 4)

A g.c. point of Type 4 is a quadratic turning point and, when passing z along cl^P_loc, the local minimizer switches into a local maximizer. We have the following cases for t < tand t close to t:

Case

I

:

The value of f decreases.

Case

II

:

The value of f increases.

In Case I it is possible to jump to another brach of local minimizers. In fact, since the feasible set M(t) is compact, we compute a point on^P_gc beyond the turning point, say (xmax(t) t) with t < tclose to t. The point xmax(t) is a local maximizer for P(t) and we can start at xmax(t) with a descent method in order to nd a local minimizer. In Case II, there is no proposal for possible jumps (see Fig. 4).

If a point of Type 5 appears, we also do not know a jump in case the MFCQ is violated. Such a situation is characterized by the fact that the connected com- ponent in the feasible set shrinks to one point and becomes empty for increasing t.

The algorithm JUMP II

The algorith Jump II works in^P_gc and it is useful for the computation of as many connected components as possible using PATH III and jumps in ^P_gc. For more details, we refer the reader to Guddat et al. 5] for the pathfollowing algorithms in the Sections 4 and 5.

3 Properties of the double parametric embedding

The rst theorem includes basic properties of the embeddings Pi(t), i = 1 2.

Theorem 1.

It holds:

(E1) x^o is a global minimizer forPi(0). (E2) Pi(1)Pi,

(E3) Mi(t¹)Mi(t²) for t¹t²,t¹ t²²(^;1 1], (E4) Mi(t)⁶=and compact , for allt²(^;1 1],

(E5) i(t)⁶= ⁸t²(^;1 1] where i(t)denotes the set of all global minim- izers ofPi(t).

3 Properties of the double parametric embedding 10

Proof. Without loss of generality we consider the problem P¹(t). (E1) and (E2) are obvious.

(E3) Let t¹t², t¹ t²²(^;1 1]. If x²M¹(t²), then gs⁺³(x) + (t^2;1)gs⁺³(x^o)0.

We have gs⁺³(x) + (t^1;1)gs⁺³(x^o)gs⁺³(x) + (t^2;1)gs⁺³(x^o), since gs⁺³(x^o) > 0. So, x²M¹(t¹).

(E4) follows from (E3) and Mi⁶=. M¹(t) is compact, since ^P is compact and the set^T3_i⁼¹Giclosed.

(E5) follows from (E4).

For the problems Pi(t), i = 1 2 let us assume:

(^C4) (f g¹ gs⁺³)^2F and (f g¹ gs⁺¹)^2F respectively :

F is the class of Jongen, Jonker and Twilt.

Remark 2.

In this case the problemsP_i(t),i = 1 2are regular in the sense of Jongen, Jonker and Twilt (i.e., JJT-regular).

One important issue is to determine the kind of singularities which may appear. Theorem 2 gives us an answer.

Denition 1.

A point(x t)²Mi(t)^Ri = 1 2is a vertex if xis a vertex of the polyhedron ^P.

Theorem 2 (Singularity Theorem).

Assume that (C1), (C2), (C3), (C4) hold and let

I_ko(x t) :=^fji^2f1 s^gjgjⁱ(x t) = a_Tjⁱx + bjⁱ = 0 i = 1 k^g (2)

be the index-set of active linear inequalities of the problems Pi(t), i ^{2 f}1 2^g. Then the following assertions are true:

A) For^P(^t) :=^P1(^t)

For any z = (x t)²cl^P_loc^nP1_gcand t²(^;1 1], the following holds:

(i) z^2P2_gc^P3_gc^P4_gc^P5_gc.

(ii) (x^o 0)^2P2_gc^\P_loc and (x^o 0) is no turning point.

(iii) Each regular vertex zwithJ_o(z) =^fs + 3^gI_ko(z), is a point of Type 5and the MFCQ is fullled.

(iv) If zis no vertex of ^P and Jo(z) = ^fs + 3^gI_ko(z), or Jo(z) =^fs + 2^gfs + 3^gI_ko(x t), then

z^2P2_gc^P3_gc^P5_gc and the MFCQ is fullled.

(v) Ifs + 1²Jo(z), then it holds:

(a) If the MFCQ is not satised at z =⁾ z^2P4_gc^P5_gc, (b) If the MFCQ is fullled at z =⁾ z^2P2_gc^P3_gc^P5_gc.

3 Properties of the double parametric embedding 11

t x

x

t (~xmax(t) t)

Jump (xmin(t) t)

f decreases

x x

t (~xmax(t) t)

(xmin(t) t)

f increases

Figure 4: Jump in Points of Type 4

B) For^P(^t) :=^P2(^t)

For any z = (x t)²cl^P_loc^nP1_gcand t²(^;1 1], the following holds:

(i) z^2P2_gc^P3_gc^P4_gc^P5_gc.

(ii) (x^o 0)^2P2_gc^\P_loc and (x^o 0) is no turning point.

(iii) If(x t)is a regular vertex withJo(z) =^fs + 1^gI_ko(z), thenz^2P5_gc. (iv) If zis no vertex of withJo(z) =^fs + 1^gI_ko(x t), then it holds:

(a) If the MFCQ is violated =⁾ z^2P4_gc^P5_gc, (b) If the MFCQ is satised=⁾ z^2P2_gc^P3_gc^P5_gc.

C) If z²cl^P_loc^\P4_gc, then sign

0

@

;Dt^Xp j⁼¹ jgj(z)

1

A< 0:

Remark 3. C)

means that the jump is impossible.

Proof. ^A) ^P(^t) :=^P1(^t) (i) Follows from (ii)^;(v).

(ii) From the choice of x^oin (^V2) we have:

(a)

gs⁺³(x^o t)

8

>

<

>

:

< 0 if t < 0

= 0 if t = 0

> 0 if t > 0 (b) gj(x^o t) = gj(x^o)⁶= 0 j⁶= s + 3,

(c) it follows from (a) and (b) that Jo(x^o 0) =^fs + 3^g.

3 Properties of the double parametric embedding 12

x Dxgs⁺³(x t) Dxgs⁺¹(x t)

gs⁺³(x t) = 0 M(t) gs⁺¹(x t)

Figure 5: Violation of the MFCQ

Since x^o is a global minimizer of P¹(t), t 0 we also have that x^o is a g.c. point. Next, we verify that Dxgs⁺³(x^o) ⁶= 0, since x^{o 62} H¹, ^jJo(x^o 0)^j = 1, and D_x²L(x^o 0) = D. Consequently the LICQ is satised and the rst conclusion follows from (C4). The latter statement follows from the fact that D²_xL(x^o 0)=T(x^o 0) is positive- denite and the arguments described in (5:2:20) 5].

(iii) Since zis a regular vertex, we have^jJo(z)^j=^jI_ko(z)^j+ 1 = n + 1. Con- sequently, we obtain the rst statement. Using the convexity of gj j² Jo(x t) we have

Dxgj(x t)((x t)^;(x t))gj(x t):

With our choice of ^x²int M¹M¹(t) we obtain g_j(^x t) < 0. We set := (^x t)^;(x t): Here is an MFCQ-vector. For a given we have D_xg_j(x t) < 0: Hence the MFCQ is satised.

(iv) Readily the same as (iii). Of course, if^jJo(z)^j= n + 1

(^jI_ko(z)^j= n^;1) and Jo(z) =^fs + 2^gfs + 3^gjI_ko(z)^j, then z is a point of Type 5 and the MFCQ is satised.

(v) (a) follows from Theorem 2.5.4 in 5] and (b) is obvious.

B) P(t) := P²(t) For (i), (ii) and (iv) we proceed as in

A)

For (iii) we also proceed as in

A)

. Here the MFCQ will not hold in general (see Fig. 5).

C) In view of (5.2.31) and (5.2.32) in 5], it su ces indeed to check that sign

0

@

;DtXp j⁼¹ jgj(z)

1

A< 0:

3.1 Illustrative examples 13

For this purpose we have Dt

0

@

p

X

j⁼¹ jgj(z)

1

A= u^{^}_jg^{^}_j(x^o) ^j = s + 1 s + 3:

Since z^2P4_gc, the MFCQ is not satised, we have ^{^}_j > 0. From g^{^}_j(x^o) > 0 we obtain:

Dt

0

@

p

X

j⁼¹ujgj(z)

1

A> 0:

Hence = sign(^;Dt^L(z)) < 0. This completes the proof.

Remark 4.

Letz = (x t)^2P_gc be a degenerate vertex and let

s + 3²Jo(z)(for the Problem P¹) or s + 1²Jo(z)( for the Problem P²), then it holds that(f G)^62F:

From the practical point of view, we follow a path in^P_gc, starting at a point of Type 1.

Theorem 3.

Assume that (C4)for Pi(t),i = 1 2. Then it holds:

(x^o to)^2P1_gc ⁸to< 0:

Proof. Follows immediately from the fact that gs⁺³(x^o to) < 0, Dxf(x^o to) = 0 and D_x²f(x^o to) = 2D for all to< 0.

3.1 Illustrative examples

Example 3.1.

min^ff(x¹ x²) := (x^1;x^1o)²+ (x^2;x^2o)^2j(x¹ x²)²M^g M := ^f(x¹ x²)^2R2jgj(x¹ x²)0 j = 1 7^g g¹(x¹ x²) := x^1;6x²+ 8

g²(x¹ x²) := 3x^1;x^2;27 g³(x¹ x²) := x¹+ 3x^2;29 g⁴(x¹ x²) :=^;6x¹+ x²+ 22

g⁵(x¹ x²) :=^;(x^1;7)^2;(x^2;4:5)²+ 9 g⁶(x¹ x²) := (x^1;7)²+ (x^2;5)^2;16 g⁷(x¹ x²) := (x^1;3)²+ (x^2;5:5)^2;9

x^1o= 10:5 x^2o= 5

g⁷(x¹ x² t) := g⁷(x¹ x²) + (t^;1)g⁷(x^1o x^2o)

3.1 Illustrative examples 14

0 1 2 3 4 5 6 7 8 9 10 11 01

23 45 67 89

x^o

g¹ g²

g³ g⁴

g⁷

g⁶ g⁵ t = 0

g⁷(x 0) = 0

Figure 6: Feasible Set for the Example 3.1

0 1 2 3 4 5 6 7 8 9 10 11 01

23 45 67 89

x^o

g¹

g² g³

g⁴

g⁵

Figure 7: Feasible Set for the Example 3.2

3.2 On the role of the MFCQ 15 Example 3.2.

min^ff(x¹ x²) := (x^1;x^1o)²+ (x^2;x^2o)^2j(x¹ x²)²M^g M :=^f(x¹ x²)^2R2jgj(x¹ x²)0 j = 1 7^g g¹(x¹ x²) := x^1;6x²+ 8

g²(x¹ x²) := 3x^1;x^2;27 g³(x¹ x²) := x¹+ 3x^2;29 g⁴(x¹ x²) :=^;6x¹+ x²+ 22

g⁵(x¹ x²) :=^;(x^1;10)^2;(x^2;4)²+ 9 x^1o= 10 x^2o= 5

g⁵(x¹ x² t) := g⁵(x¹ x²) + (t^;1)g⁵(x^1o x^2o)

Remark 5.

The Figures 8, 9 and 10 show the dependence of the solution on the dierent starting points. If we start with the point x^o = (10:5 5), we will not be successful, but starting withx^o= (9:55 2:90)and x^o= (4:5 7:5)leads us to the solution. Figure 11 shows the solution of Example 3.2.

3.2 On the role of the MFCQ

In this section we give an answer to the questionunder which conditions the path- following methods are successfulwith a specially chosen starting point, when we assume that (C4) holds. Under the assumption that the MFCQ is satised at each x²Mi(t), points of Type 4 are excluded. Hence, the case II in Section 2 may not appear. We ask for a condition on the original problems (P_i) i = 1 2.

We discuss the so-called Enlarged Mangasarian-Fromovitz Constraint Qualic- ation (briey EnMFCQ) introduced by Gfrerer et al. 10].

Denition 2 (EnMFCQ).

The EnMFCQ is satised forP¹ if, for all x^2P (polyhedron), there exists a vector ^2Rnwith

gj(x) + Dgj(x) < 0 j²J⁺(x) :=^fj²J ^jgj(x) 0 j⁶= s + 2^g Dxgs⁺²(x) < 0 if gs⁺²(x) = 0:

The EnMFCQ is satised forP² if, for allx^2P, there exists a vector ^2Rn with:

gj(x) + Dgj(x) < 0 j²J⁺(x) :=^fj²J^jgj(x) 0^g: The answer to the question above is given by the following theorem:

Theorem 4.

Assume the EnMFCQ is satised forPi i = 1 2. Then the MFCQ is satised for allx²Mi(t) i = 1 2

3.2 On the role of the MFCQ 16

9.4 9.6 9.8 10.0 10.2 10.4 10.6

2.5 3.0 3.5 4.0 4.5 5.0

x1

x2

type 2 type 3 type 4 type 5

9.4 9.6 9.8 10.0 10.2 10.4 10.6

2.5 3.0 3.5 4.0 4.5 5.0

x1

x2

-0.5 -0.4 -0.3 -0.2 -0.1 -0.0 0.1 0.2

9.4 9.6 9.8 10.0 10.2 10.4 10.6

t

x1

type 2 type 3 type 4 type 5

-0.5 -0.4 -0.3 -0.2 -0.1 -0.0 0.1 0.2

9.4 9.6 9.8 10.0 10.2 10.4 10.6

t

x1

-0.5 -0.4 -0.3 -0.2 -0.1 -0.0 0.1 0.2

9.4 9.6 9.8 10.0 10.2 10.4 10.6

t

x1

Figure 8: Example 1 in (x¹ x²) and (t x¹)-space

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

7.49 7.50 7.51 7.52 7.53 7.54 7.55 7.56

t

x2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

7.49 7.50 7.51 7.52 7.53 7.54 7.55 7.56

t

x2

5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50

7.49 7.50 7.51 7.52 7.53 7.54 7.55 7.56

x1

x2

5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50

7.49 7.50 7.51 7.52 7.53 7.54 7.55 7.56

x1

x2

Figure 9: Example 1 in (x¹ x²) and (t x¹)-space

3.2 On the role of the MFCQ 17

4.20 4.25 4.30 4.35 4.40 4.45 4.50

2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85

x1

x2

4.20 4.25 4.30 4.35 4.40 4.45 4.50

2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85

x1

x2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85

t

x2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85

t

x2

Figure 10: Example 1 in (x¹ x²) and (t x¹)-space

7.5 8.0 8.5 9.0 9.5 10.0

4.5 5.0 5.5 6.0 6.5 7.0 7.5

x1

x2

type 2 type 3 type 4 type 5

7.5 8.0 8.5 9.0 9.5 10.0

4.5 5.0 5.5 6.0 6.5 7.0 7.5

x1

x2

7.5 8.0 8.5 9.0 9.5 10.0

4.5 5.0 5.5 6.0 6.5 7.0 7.5

x1

x2

-0.5 0.0 0.5 1.0 1.5 2.0

7.5 8.0 8.5 9.0 9.5 10.0

t

x1

type 2 type 3 type 4 type 5

-0.5 0.0 0.5 1.0 1.5 2.0

7.5 8.0 8.5 9.0 9.5 10.0

t

x1

Figure 11: Example 2 in (x¹ x²) and (t x¹)-space

3.2 On the role of the MFCQ 18

Proof. Without loss of generality we consider the problem P¹. The proof for P² is readily the same as the proof for P¹. Therefore, we omit it. Set (J¹)o(x t) :=

fj²J ^jgj(x t) = 0 j⁶= s+3^g. Then, for all j²Jo(x t) = (J¹)o(x t)^fs+3^g, we have

gj(x t) = 0⁽⁾

(gj(x) = 0 if j⁶= s + 3 g_j(x) =^;(t^;1)g_j(x^o) if j = s + 3 Dxgj(x t) = Dxgj(x):

Let x²M¹(t), j ²J_o(x t). In view of Denition 2, it su ces indeed to check that the given vector is also an MF-vector in (x t). For j ⁶= s + 3 we have Dxgj(x t) = Dxgj(x) <^;gj(x) = 0 and for j = s + 3 we obtain

Dxgj(x t) = Dxgj(x) <^;gj(x) = (t^;1)gj(x^o)0 ⁸t²(^;1 1]:

Hence is an MF-vector.

Remark 6.

(a) In case, the MFCQ is satised for allx²Mi(t) t²(^;1 1] i = 1 2the setMi(t¹)is homomorphic toMi(t²)for allt¹ t²²0 1] i = 1 2. (b) Assume (C4) and that the MFCQ is satised for allx²M_i(t) t²(^;1 1].

Then we have^P_gc^\P4_gc=.

Remark 7.

Assumez^2P_gcwithx²H¹ands+3²Jo(z)respectivelyx²H² and s + 1²Jo(z). Then it holds that

z^2P4_gc^P5_gc and the MFCQ is not satised.

According to the propositions^A)(i) (v) and^Cof Theorem 2 and the example in Fig. 5, we know that points of Type 4 and 5, where the MFCQ is not satised, may appear.

Denition 3 (Gomez 23]).

We assumePi(t),i = 1 2, to beJJT-regular with respect to(^;1 1]and z = (x t)^2P_gc. A g.c. point zis called a turning point in negative direction (positive direction) if there exists a neighbourhood V (z), such that⁸(x t)^2P_gc^\V (z)holds fortt (t t).

The Figures 12 (a) and (b) show turning points in negativ direction and the Figures (c) and (d) those in positive direction.

Theorem 5.

Assume^Pi(t),i = 1 2, to beJJT-regular with respect to(^;1 1]. Then every turning point without possibilities to jump is a turning point in negative direction.

Proof. The proof relies on Theorem 1 in Gmez 23]. We will divide the proof into two steps. Without loss of generality we consider the problem P¹.

Step 1 (Type 5).- Let Jo(z) = ^f1 p^g and p = s + 3. We deduce from the

3.2 On the role of the MFCQ 19

(a) Type 4

z

(b) Type 5

z

(c) Type 5

z

(d) Type 4

z

Figure 12: 5 Turning points in negativ and positiv direction

properties of a point of Type 5 that the set^fDxgj(z) j = 1 p^;1^gis linearly independent. Then we obtain a uniquely determined solution u^p= (u^p1 u_pp^;1) of the system

Dxf(z) +^pX;1

j⁼¹ pjDxgj(z) = 0 with ^p_j ⁶= 0, j = 1 p^;1.

Next, consider the system

2

6

6

6

4

Dxf(x t) +^Pp_j^=1;1 _pjDxgj(x t) g¹(x t)

gp^;1...(x t)

3

7

7

7

5

= 0:

(3)

By the Implicit Function Theorem, there exists a uniquely determined function (x^p(t) u^p(t)) : (t^; t+ )^!Rn+p;1

(4) such that

(a) x^p(t) = x u^p(t) = u^p,

(b) for each t²(t^; t+ ) the vector (x^p(t) u^p(t) t) is a solution of (3).

Recall that (x^p(t) t) is a g.c. point if the inequality gp(x^p(t) t) = gs⁺³(x^p(t) t)0 (5)

holds. Hence, the curve (x^p(t) t) belongs to^P_gc either for t²(t t+ ) or for t²(t^; t). It su ces to show that

dtgd ^s+3(x^p(t) t) > 0

since gp becomes strictly positive, i.e., (x^p(t) t) is not feasible.

We calculate this quantity:

dtgd ^p(x^p(t) t) = Dxgp(x t)_x^p(t) + Dtgp(x t):

(6)