New embeddings for nonlinear multiobjective optimization problems I
Jurgen GUDDAT
Humboldt-Universitat zu Berlin Institut fur Mathematik
Francisco GUERRA
Universidad de las Americas, Puebla Departamento Fisica y Matematicas
Dieter NOWACK
Humboldt-Universitat zu Berlin Institut fur Mathematik
February 12, 1998
Abstract
In a dialogue procedure the decision maker has to determine in each step the aspiration and reservation level expressing his wishes (goals). This leads to an optimization problem which is not easy to solve in the nonconvex case (the known starting point is not feasible). We propose a modied standard embedding (one parametric optimization). This problem will be discussed from the point of view of parametric optimization (non-degenerate critical points, singularities, pathfollowing methods to describe numerically a connected component in the set of stationary points and in the set of generalized critical points, respectively, and jumps (descent methods) to other connected components in these sets). This embedding is much better for computing a goal realizer or replying that the goal was not realistic than the embeddings considered in the literature before, but in the worst case we have to nd all connected components and this is an open problem.
Keywords: Multiobjective optimization, non-degenerate critical points, singularities, pathfollowing methods
1
2 Multiobjective optimization: embeddings
1 Introduction
We consider the following multiobjective optimization problem (MOP) minf(f1(x) ::: fl(x))jx2Mg
where
M := fx2IRn jgj(x)0 j 2Jg
J := f1 ::: sg K :=f1 ::: lg fk gj 2Cq(IRn IR) k 2K j 2J q 2f2 3g We assume
(A1) M 6=
Let fk k 2 K, be the global minimum of fk(x) subject to M or, in the nonconvex case, a suciently good lower bound. We assume that fk k 2 K, to be known. In the following we describe a well-known dialogue procedure (cf. e.g. 9], 12]) which constitutes the basis of our investigation. Let xi be the currently computed feasible point. Then we consider a telescreen picture as described in Table 1.1.
The main information consists in estimating the objective values at the point xi in comparison to fk = infffk(x)jx2Mg k 2K.
f1 f1(xi) f1(xi);f1
jf1j 100
... ...
fk fk(xi) fk(xi);fk
jfkj 100
... ...
fl fl(xi) fl(xi);fl
jflj 100 Table 1.1
The third column contains the percentage of deviations of the current valuesfk(xi) from the lower bounds fk. Clearly, we have to assign suitable values to these quantities in case offk = 0 andfi =;1. The decision maker should answer the following questions by means of the telescreen pictures:
(i) Which fk (k 2 K) do you wish to improve? Let K1 K be the corresponding index set.
(ii) Which goals 1k do you wish for fk k2K1?
J. Guddat, F. Guerra, D. Nowack 3 (iii) Which upper bound 1k do you accept for fk k2KnK1 ?
1 is the goal (the wishes of the decision maker), where 1k k 2 K1, represent the so-called aspiration level and 1k k 2KnK1, the reservation level.
We consider
M(1) :=fx2M jfk(x)1k k 2Kg
and a point ^x2M(1) is called a goal realizer. We will discuss the following questions:
Question 1:
How useful are pathfollowing methods with the known starting point x0 2M for nding a point ^x2M(1) in case M(1)6= ?Question 2:
How to obtain information that M(1) =?Of course, the two questions depend on each other. They were discussed for instance in 8], 9], 10], 7]. Here we consider various one-parametric optimization problems that are motivated by (locally) ecient points, (locally) ecient points with boundary
", and (locally) weakly ecient points for the problem (MOP). We denote these sets by Meff (Mloceff),M"eff (M"loceff), and Mweff (Mlocweff), respectively.
First, we consider the following multiparametric optimization problem P1() : min
8
<
: X
k2K0kfk(x)jx2M fk(x)k k2K
9
=
2IRl
where 0 2 := f2 IRl jk > 0 k2 Kg is arbitrarily xed. Byglob()(loc()) we denote the set of all global (local) minimizers for P1(). Then the following relation is known
Meff =
2IRlglob() (Mloceff =
2IRlloc()) (see e.g. 1], 3], 9]).
Then we obtain a one-parametric optimization problem by choosing a starting parameter 0 with
0k fk(xi) k 2K
and considering the connecting line
n 2IRl j = 0+t(1;0) t20 1]o: Then we obtain
P1(t) := P1(t 0 1) : min
8
<
: X
k2K0kfk(x)jx2M1(t)
9
=
t20 1]
where
M1(t) := M1(t 0 1) :=nx2M jfk(x)0k +t(1k;0k) k 2Ko:
4 Multiobjective optimization: embeddings Of course, using pathfollowing methods starting with x0 and attainingt = 1 at a point x provides ^x^ 2 M(1) because ofM1(1) =M(1) (here we assume that M(1)6=).
Unfortunately, we obtain such a point only for convex (MOP) with certainty. We have the same situation for some other parametrizations (see the references above).
Regarding P1(t) in the non-convex case, M1(t) could be empty for all t 2 (t1 t2) and (t1 t2) 0 1] (see Example 4.1 in Chapter 4). Then the decision maker does not know whether his goal1 is a realistic one or not. In Example 4.1,M(1) is not empty.
From this point of view we propose a completely dierent parametrization, which is not motivated by the solution sets Meff etc., but which has the advantage that the parameter-depending feasible set is non-empty and compact for allt 20 1] under the assumption that
(A2) M(1)\E(p)6= and x0 2IRn arbitrarily chosen with kx0k2 < p where
E(p) := fx2Rnjkxk2 pg p > 0 suciently large:
Now we consider the optimization problem
(P) minfkx;x0k2 jx2M(1)\E(p)g and the following modied standard embedding
P2(t) := P2(t x0 1) : minfkx;x0k2 j x2M2(t)g t 20 1]
M2(t) :=fx2IRnj tgj(x) + (t;1)g0j 0 j 2J t ~fk(x) + (t;1)f0k 0 k 2K
kxk2;p0g t 20 1]
where
~fk(x) := ~fk(x 1k) :=fk(x);1k k 2K
p suciently large, f0k > 0 k 2K and g0j > 0 j 2J with f0k1 6=f0k2 k1 k2 2K k1 6=k2
and g0j1 6=g0j2 j1 j2 2J j1 6=j2 .
2 Theoretical Background
and the Program Package PAFO.
First, we present a very short version of 2.5, 2.6 in 12]. We consider the general one-parametric problem:
P(t) minff(x t)jx2M(t)g t2IR (2.1)
J. Guddat, F. Guerra, D. Nowack 5 where M(t) = fx 2 IRn j hi(x t) = 0 i 2 I gj(x t) 0 j 2 Jg, and f hi gj 2
Cq(IRnIR IR) i2I j 2J q 2.
Furthermore, we introduce the following notations:
gc := f(x t)2IRnIRjx is a generalized critical point1 (g.c. point) ofP(t)g stat := f(x t)2IRnIRjx is a stationary point of P(t)g
loc := f(x t)2IRnIRjx is a local minimizer of P(t)g H := (h1 ::: hm)T G := (g1 ::: gs)T:
The Linear Independence Constraint Qualication (briey LICQ) is satised at x 2 M(t) if the vectors Dxhi(x t), i 2 I, Dxgj(x t), j 2 J0(x t), are linearly independent (J0(x t) :=fj 2J jgj(x t) = 0g).
The Mangasarian-Fromovitz Constraint Qualication (briey MFCQ) is satised at x2M(t) if:
(MF1) Dxhi(x t), i2I, are linearly independent, (MF2) there exists a vector 2Rn with
Dxhi(x t) = 0 i2I 2 Dxgj(x t) < 0 j 2J0(x t):
The KKT-system for a given problem P(t) is fullled at a point (x t) if there exists a point y 2 IRm+s such that H(x y t) = 0, where H : IRn+m+s+1 ! IRn+m+s is dened by
H(x y t) =
8
>
>
>
>
>
<
>
>
>
>
>
:
Dxf(x t) +iP
2IyiDxhi(x t) +jP
2Jy+m+jDxgj(x t)
;hi(x t) i2I y;m+j;gj(x t) j 2J
9
>
>
>
>
>
=
>
>
>
>
>
(2.2) (for 2 IR let + = maxf 0g and ; = minf 0g). Obviously, the so-called Kojima-mappingHin (2.2) is piecewise continuously dierentiable. In 17] the classical denition of a regular value of a continuously dierentiable function is generalized for piecewise continuously dierentiable functions. Furthermore, it is shown that if 02IRn+m+s is a regular value ofH, then the setH;1(0) is a piecewise one-dimensional C1-manifold (briey PC1-manifold).
Next, we cite our short characterization from 2.5 in 12] of the class F introduced by Jongen, Jonker and Twilt (15, 16]). In 16] the local structure of gc is completely
1
Forthede nitionwerefer to16],see 12],to o
2
WeconsiderthegradientD
x h
i (x
t)as arowvector.
6 Multiobjective optimization: embeddings described if (f H G) belongs to a C3s-open and dense subsetF ofC3(IRnIR IR)1+m+s, where C3s denotes the strong (or Whitney-) C3-topology (see 12], too).
If (f H G)2F, then gc can be divided into 5 types.
Type 1: A point z = (x t)2gc is of Type 1 if the following conditions are satised:
There exist i j 2IR, i2I, j 2J0(z) with
Dxf +X
i2I iDxhi+ X
j2J0(z)jDxgj jz=z = 0 (2.3)
LICQ is satised at x2
M
(t), (2.4a)(therefore i, j, i2I, j 2J0(z) are uniquely dened)
j 6= 0 j 2J0(z), (2.4b)
D2xL(x t)jT(z) is nonsingular, (2.4c)
where D2xL is the Hessian of the Lagrangian L(x t) = f(x t) +X
i2I ihi(x t) + X
j2J0(z)jgj(x t)
and the uniquely determined numbers i j are taken from (2.3). Furthermore, T(z) =f 2IRnjDxhi(z) = 0 i2I Dxgj(z) = 0 j 2J0(z)g
is the tangent space atz. D2xL(z)jT(z) representsVTD2xLV , 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 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 gc. The points of the Types 2{5 represent basic degeneracies:
Type 2 { violation of (2.4b) Type 3 { violation of (2.4c)
Type 4 { violation of (2.4a) and jIj+jJ0(z)j;1< n Type 5 { violation of (2.4a) and jIj+jJ0(z)j=n + 1.
The full curve stands 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.
For each of these ve types Figure 2.1 illustrates the local structure of gc in the neighbourhood of stationary points. Let gc, 2f1 ::: 5gbe the set of g.c. points of Type . Figure 2.2 illustrates the local structure of F in loc and stat. The class F is dened by
F =n(f H G)2C3(IRnIR IR)1+m+s jgc 5
=1gco:
The full curve stands for a curve of local minimizers and the dotted curve in Fig. 2.2(c), (d), (e), (f) represents a curve of stationary points not being local minimizers. The dotted curve in Fig. 2.2(g), (h) stands for a curve of stationary points in case J0(x t) = .
J. Guddat, F. Guerra, D. Nowack 7
MFCQ holds(k) z
MFCQ violated(l) z
MFCQ violated(m) z
Type 5
J0(z)6= (g)
z
J0(z)6= (h) 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 2.1
8 Multiobjective optimization: embeddings
Type 5(i) Type 5(j) Type 5(k) t
x
Type 3(e) Type 3(f) Type 4(g) Type 4(h) Type 1(a) Type 2(b) Type 2(c) Type 2(d)
Figure 2.2
The following theorem provides a special perturbation of (f H G) with additional parameters that can be chosen arbitrarily small such that the perturbed function vector belongs to the class F.
Theorem 2.1
(cf. 18]). Let (f H G)2C3(IRnR R1+m+s). Then, for almost all (b A c D e F)2IRnIRn(n+1)=2IRmIRmnIRsIRsn, we have(f(x t) + bTx + xTAx H(x t) + c + Dx G(x t) + e + Fx)2F: Here "almost all" means: each measurable subset of
f(b A c D e F)j(f(x t) + bTx + xTAx H(x t)+ c + Dx G(x t) + e + Fx) =2Fg
has the Lebesgue-measure zero. 2
Remark 2.2
(cf. 18]). Considering stat we note that the condition (f H G) 2 F implies that zero is a regular value of the Kojima-mappingH. 2Denition 2.3
LetK IRf1g.(i) The problemP(t) is called regular in the sense of Jongen-Jonker-Twilt { briey JJT-regular { (with respect to K) if (f H G)2F(IRnK)\gc S5
=1gc .
J. Guddat, F. Guerra, D. Nowack 9 (ii) The problem P(t) is called regular in the sense of Kojima-Hirabayashi { briey KH-regular { (with respect to K) if 0 2 IRn+m+s is a regular value of H (HjIRnIRmIRsK).
Theorem 2.4
(cf. 17]). Let (f H G)2C3(IRnIR IR)1+m+s. Then, for almost all (b c d)2IRnIRmIRs, the problemP(bcd)(t) minff(x t) + bTx
hi(x t) + ci = 0 i2I gj(x t) + dj 0 j 2J
9
>
=
>
is KH-regular. 2
Now, we present two theorems that are essential for our analysis.
Theorem 2.5
(cf. 4]). We assume(C1) M(t) is non-empty and there exists a compact set C with M(t) C for all t 20 1].
(C2) P(t) is KH-regular with respect to 0 1].
(C3) There exists a t1 > 0 and a continuous function x : 0 t1) ! IRn such that x(t) is the unique stationary point for P(t) for t 20 t1).
(C4) MFCQ is satised for all x2M(t) for all t20 1].
Then there exists a PC1-path in stat that connects (x0 0) with some point (x 1).
2
Applying Remark 2.2 we obtain
Corollary 2.6
We assume (C1), (C3), (C4) and (D2) P(t) is JJT-regular with respect to0 1].Then there exists a PC2-path K(x0 0) in stat connecting (x0 0) with some point (x 1). Furthermore if (x t)2K(x0 0) then (x t) belongs to S
2f1235ggc. 2
Remark 2.7
Assume (C4). Now we have a look at Fig. 2.2. Since the MFCQ is satised, points of Type 5 in (j) and (k) are excluded. 2 Finally we present a consequence of a general topological stability result given in 13]:Theorem 2.8
(cf. 13]). We assume (C1) and (C4). Then M(t1) is homeomorphicwith M(t2) for all t1 t2 20 1]. 2
10 Multiobjective optimization: embeddings
On the program package PAFO
(this is a very short version of 4.5 and 5.2 in 12]) PAFO (cf. 19] and 5]) is based on a pathfollowing method (called PATH III in 4.5 12]) and jumps (called JUMP I in 5.2 12] and JUMP II in 5.3 12]).We explain the main ideas of PATH III and JUMP I, but not those of JUMP II as we do not need them here.
PATH III
This algorithm computes a numerical description of a compact connected component in
Pgc, i.e., in particular it nds a discretization of an interval tA tB] , tA < 0 < tB (not necessarily tA tB] 0 1]), and corresponding g.c. points starting at (x0 0) 2 Pgc
(cf. (A2)). 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 diculties walking around turning points of the Types 3 or 4.
More precisely: If there exists a PC2-path connecting (x0 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) 2Pgc in general, we propose to jump from one connected component inclPloc andPgc, respectively, to another one.
JUMP I
This algorithm works in the set cl Ploc. Starting at the known local minimizerx0 at t0 = 0, a connected component incl Ploc for increasingt will be numerically 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 Ploc will 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 a turning point of Type 2 or a point of Type 3 occur. Lett be near t, t < t, and letxm(t) and xs(t) be the local minimizer and a point ofPstatnPloc, respectively.
Then, as t tends to t, the vector u(t) := xs(t);xm(t)
kxs(t);xm(t)k
tends to a tangential vector, say u, which is a direction of (higher order) descent (cf.
Fig. 2.3 for a point of Type 3).
Hence, for t near t t < t, the vector xs(t);xm(t) provides an approximately tangential direction of descent (cf. Fig. 2.3).
A g.c. point of Type 4 is a quadratic turning point and, when passing z along cl Ploc, the local minimizer switches into a local maximizer. We have the following cases for
J. Guddat, F. Guerra, D. Nowack 11
t x
(xt) (xm(t)t)
(xs(t)t)
t x
t x
jump
Figure 2.3
t x
(xt) local minimizer
local maximizer
t x f decreases
case I t
x f increases
case II
Figure 2.4 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 branch of local minimizers. In fact, since the feasible setM(t) is compact, we compute a point on Pgc beyond the turning point, say (xmax(t) t) with t < t t close 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 mimimizer.
In Case II, there is no proposal for possible jumps (cf. Fig. 2.4).
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 component in the feasible set shrinks to one point and becomes empty for increasing t.
3 Properties of the standard embedding
The rst theorem includes basic properties of the standard embeddingP2(t). We have
Theorem 3.1
Assume (A1) and (A2). Then P2(t) has the following properties (i) M2(t) is nonempty and compact for all t20 1]12 Multiobjective optimization: embeddings (ii) M2(1) =M(1)\E(p)
(iii) x0 is a global minimizer, the only stationary point and non-degenerated forP2(0). If we assume, moreover, that
(A3) P2(t) is JJT-regular with respect to (0 1]
then all singularities may appear.
Remark 3.2
P2(t) is constructed in such a way that the starting point has nice prop- erties. We cannot come back to t = 0 in Pstat, but the following diculties may arise, where we do not attain t = 1.a) A point of Type 4 (Case II) appears and we do not have a jump in Pstat to another connected component (cf. Example 4.2)
b) A point of Type 5 (where the MFCQ is not satised) appears and we do not have a jump.
Under the following additional assumption:
(A4) MFCQ is satised for all x2M2(t) for all t20 1]
the diculties mentioned above are excluded.
Theorem 3.3
Assume (A1) - (A4). Then there exists aPC2-path inPstat connecting (x0 0) and (x 1) with x 2 M(1) and points of the Types 1,2,3, and 5 (MFCQ issatised) may appear. 2
Proof:
We check the assumptions of Corollary 2.6 2
Remark 3.4
In a nite number of predictor and corrector steps PAFO computes a point lying in the radius of Newton methods with respect to x, i.e. we have at least a superlinear rate of convergence.If we replace (A3) by the weaker assumption
(A3)' P2(t) is KH-regular with respect to (0 1], then we can use Theorem 2.5 and we obtain
Theorem 3.5
Assume (A1), (A2), (A3)', and (A4). Then there exists a PC1-path inPstat connecting (x0 0) and (x 1) with x 2M(1). 2
J. Guddat, F. Guerra, D. Nowack 13 The Examples 4.1 and 4.2. illustrate Theorem 3.3.
In the following we discuss the assumptions (A3) and (A3)' and ask how large the classes have to be for (A3) resp. (A3)' to be satised. Let F = (f1 ::: fl)T and G = (g1 ::: gs)T. Then we consider the mapping : C3(IRn IR)s+l+2 ! C3(IRn IR IR)s+l+2 dened by
(kx;x0k2 G F kxk2;p) =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
kx;x0k2 tg1(x) + (t;1)g01 tgs(x) + (t... ;1)g0s
t ~f1(x) + (t;1)f01 t ~fl(x) + (t... ;1)f0l
kxk2;p
3
7
7
7
7
7
7
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7
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7
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7
7
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5
by using the embedding P2(t) for the problem (P). In addition to the class FjK (JJT- regular with respect to K), we introduce the classKjK (KH-regular with respect toK).
Theorem 3.6
(i) Let (F G) 2 C3(IRn IR)s+l. Then ;1(Fj01]) is C3s-open and ;1(Fj(01]) is C3s-dense in C3(IRn IR)s+l+2(ii) Let (F G) 2 C2(IRn IR)s+l. Then ;1(Kj01]) is C2s-open and ;1(Kj(01]) is C2s- dense in
C2(IRn IR)s+l+2. 2
For the proof of Theorem 3.6 we study Theorem 2.1 specied by the embeddingP2(t), i.e., we dene
f(x t) := kx;x0k2
gj(x t) := tgj(x) + (t;1)g0j j = 1 ::: s gs+k(x t) := t ~fk(x) + (t;1)f0k k = 1 ::: l gs+l+1(x t) := kxk2;p
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
(3.1)
Let A be an nn symmetric matrix (A 2 IR12n(n+1)), bj 2 IRn j = 1 ::: s + l + 1, c2IRn dj 2IR j = 1 ::: s + l + 1
d := (d1 ::: ds+l)T 2IRs+l B := (b1 ::: bs+l+1 d)
14 Multiobjective optimization: embeddings
D:= (A C B)2 IR := 12n(n+1)+n(s+l+1)+s+l and we dene
f(x t A c) := f(x t) + xTAx + cTx
gj(x t bj dj) :=gj(x t) + tbjTx + tdj j = 1 ::: s + l:
gs+l+1(x t bs+l+1 ds+l+1) :=kxk2;p + (bs+l+1)Tx + ds+l+1
We consider the problem
PD(t) : minff(x t A c)jx 2M(B t)g t2IR where
M(B t) =fx2IRn jgj(x t bj dj)0 j = 1 ::: s + l + 1g:
Proposition 3.7
Let (F G) 2 C3(IRn IR)s+l. Then PD(t) is JJT-regular for almostall D2IR with respect to (0 1]. 2
The proof follows the same lines as the proof of Theorem 2.1 in 18]. 2 Using the special functions in (3.1) once more, we obtain
Proposition 3.8
Let(F G)2C2(IRn IR)s+l. Then, for almost all(b d) 2IRnIRs+l, the problem P(bd)(t) : minff(x t)+bTxjgj(x t)+tdj 0 j = 1 ::: s+l gs+l+1(x t)+ds+l+1 0g is KH-regular with respect to (0,1]. 2
For the proof see Theorem 2.4 and the hints for the proof in Chapter 8 in 17].
Proof of Theorem 3.6
:(i) a);1(Fj(01]) isC3s-dense inX := C3(IRn IR)s+l+2. LetH := (kx;x0k2 G F kxk2; p) 2X. We have to show that for any "-neighbourhood V3"H of H in the C3s (or strong C3) topology (cf. e.g. Chapter 2 in 12]) there exists an H0 2 V3"H \;1(F j(01]), in other words,
H02V3"H and (H0)2Fj(01]: (3.2)
Using the theorem on the partitioning of unity, we denote B% :=fx 2IRn jkxk < %g and consider the open covering of IRn given by fB3 IRnncl B2) and a corresponding partition of unityf 11 21g,
1i :IRn ;!f0 1g i = 1 2:
That means
J. Guddat, F. Guerra, D. Nowack 15
supp 11 B3, supp 21 IRnncl B2, where supp i1 =clfx 2 IRn j i1(x) > 0g i = 1 2:
11(x) + 21(x) = 1 for all x2IRn.
Then 11jcl B2 1 and 21jIRnnB3 0. This partition of unity exists independent of H and V3"H. Now, we construct H1:
H1(x) :=
2
6
6
6
6
6
6
6
6
6
6
6
6
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6
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6
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4
h10(x) h11(x) h1s...(x) h1s+1(x) h1s+l...(x) h1s+l+1
3
7
7
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7
7
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where
h10(x) := 11(x)hkx;x0k2+xTA1x + c1Txi+ 21(x)kx;x0k2 h1j(x) := 11(x)hgj(x) + (bj)1Tx + d1ji+ 21(x)gj(x) j = 1 ::: s h1s+k(x) := 11(x)h~fk(x) + (bs+k)1Tx + d1s+ki+ 21(x) ~fk(x) k = 1 ::: l h1s+l+1(x) := 11(x)kxk2;p +bs+l+1 1Tx + d1s+l+1+ 21(kxk2;p):
Now, by Propsition 3.7 we can choose D2IR suciently small to have 1. H1 2V3"H ,
2. Pgc((H1))\(cl B2(0 1])=1S5 Pgc(H1)):
Now consider the open covering of IRn given by fB4ncl B1 B2fIRnn cl B3gg and a corresponding partition of unity f 12 22g, which exists independent of H H1 and V3"H. Taking into account the rst part and Proposition 3.7, we can choose anotherD2 2IR suciently small to obtain
16 Multiobjective optimization: embeddings 1. H2 2V3"H, where
H2(x) :=
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h20(x) h21(x) h2s(x)...
h2s+1(x) h2s+l...(x) h2s+l+1
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h20(x) := 12(x)hh10(x) + xTA2x + c2Txi+ 22(x)h10(x)
h2j(x) := 12(x)hh1j(x) + (bj)2Tx + d2ji+ 22(x)h1j(x) j = 1 ::: s
h1s+k(x) := 12(x)hh1k(x) + (bs+k)2Tx + d2s+ki+ 22(x)h1k(x) k = 1 ::: l + 1:
2. Pgc((H2))\(cl B3 (0 1])=1S5 Pgc((H2)), 3. H2 jcl B1 H1 jcl B1:
In this way, we obtain a sequence of functions H% such that 1. H% 2V3"H
2. Pgc((H%))\(cl B%+1(0 1])=1S5 Pgc((H%)) 3. H% jcl B%;1 H%;1 jcl B%;1
For x 2 IRn we dene %(x) := minf% jx 2 B%g and H0(x) := H%(x)(x) and we obtain (3.2).
(i) b) ;1(F j01]is C3s-open in X.
Since Fj01] is C3s-open in Yj01] := C3(IRn 0 1] IR)s+l+2, it is sucient that : X ! Yj01] is continuous. We denote := (0 1 ::: s+l+1). It is sucient to consider ~ := (1 ::: s+l):
For (G0 F0)2X it holds
jj(G F);j(G0 F0)j = jt(gj(x);gj0(x))j j = 1 ::: s
js+k(G F);s+k(G0 F0)j = jt(fk(x);fk0(x))j k = 1 ::: l:
9
>
=
>
(3.3)
J. Guddat, F. Guerra, D. Nowack 17 Let (G F) 2 X be xed and let V%~(GF)3 be a %-neighbourhood of ~(G F) in Yj01]. Here %(x t) = (%1(x t) ::: %s+l(x t)), where %(x t) 2 C0(IRn 0 1] IR+) = 1 ::: s + l:
We have to prove that there exists an "-neighbourhood V3"(GF) of (G F) in X such that
~(V3"(GF))V%~(GF)3 (3.4)
Here, "(x) := ("1(x) ::: "s+l(x)), where " 2C0(IRn IR+) = 1 ::: s + l.
Put "(x) := min0t1%(x t) = 1 ::: s + l. Then " 2 C0(IRn IR+) and "(x)
%(x t) for all (x t) 2 IRn0 1] = 1 ::: s + l. Using (3.3), the inclusion (3.4) is fullled and, therefore, ~ and also are continuous due to continuity arguments of the functions G and F.
(ii) a);1(Kj01]) isC2s-dense inC2(IRn IRIRsIRlIR). We follow the same concept as in (i) a) using Proposition 3.8.
(ii) b) ;1(Kj(01]) being C2s-open in C2(IRn IRIRsIRlIR) follows by continuity
arguments with respect to . 2
We note that such a kind of theorem is proposed e.g. in 6] for another embedding with respect to justifying (A3).
Now we ask how we can enter the class F and K, respectively. We consider PB(t) : minn(x;x0)TA(x;x0) j tgj(x) + (t;1)(g0j+bjTx + dj)0 j 2J
t ~fk(x) + (t;1)(f0k+~bkTx + ~dk)0 k 2K
kxk2+cTx;p0o
B = (A x0 B d ~B ~d c p)
B := (b1 ::: bs) ~B := (~b1 ::: ~bl) d = (d1 ::: ds)T ~d= (~d1 ::: ~dl)T
Let A IR12n(n+1) be the set of all non-singular symmetric (n n)-matrices. Then A is open in IR12n(n+1) and IR12n(n+1)n A has the Lebesgue measure 0.
We have (see the proof of Theorem 3.6)
Corollary 3.9
Let (F G) 2 C3(IRn IR)s+l. Then, PB(t) is JJT-regular with respect to (0,1) for almost all (A x0 B d ~B ~d c p) 2AIRnIRnsIRsIRnlIRlIR .Remark 3.10
For the starting situation (t = 0) we have to choose A to be positive denite and bj dj j 2 J ~bk ~dk k 2 K, in such a way that bjTx0+dj < 0 j 2 J, and~bkTx0+ ~dk < 0 k 2 K. Then x0 is a global minimizer, the only stationary point, and non-degenerated.
18 Multiobjective optimization: embeddings Now we consider
PC(t) : minfkx;x0k2 j tgj(x) + (t;1)(g0j+dj)0 j 2J t ~fk(x) + (t;1)(f0k + ~dk)0 k 2K
kxk2;p0g where C := (x0 d ~d p).
Corollary 3.11
Let (F G) 2 C2(IRn IR)s+l. Then, PC(t) is KH-regular with respect to (0,1) for almost all C 2 IRnsl+1 with dj > g0j j 2 J and ~dk > f0k k 2 K and p >kx0k2.Remark 3.12
(i) x0 is a global minimizer, the only stationary point, and non-degenerated.
(ii) If we choose p >kx0ksuciently large, the feasible set of PC(t) is non-empty and compact for all t20 1):
Finally, we discuss the assumption (A4). This is a condition to the parameter-depending feasible set M2(t) for all t20 1].
First, we ask for a sucient condition with respect to the set M(1)\E(p), which we will call, as in other papers (cf. e.g. 4] 11], 2]), the Enlarged Mangasarian-Fromovitz Constraint Qualication (briey EnMFCQ).
Let (F G)2C1(IRn IR)s+l. The EnMFCQ for M(1)\E(p):
For all x2E(p) it holds: There exists a 2IRn with (i) gj(x) + Dgj(x) < 0 j 2fj 2J jgj(x)0g (ii) ~fk(x) + Dfk(x) < 0 k 2fk 2K j ~fk(x)0g (iii) 2xT < 0 if kxk=p
Theorem 3.13
Let (F G)2C1(IR IR)s+l. Assume (A2) and the EnMFCQ. Then the MFCQ is satised for all x2M2(t) for all t20 1].The proof runs along the lines of the proof of Theorem 10 in 2].
Remark 3.14
Using Theorem 2.8 we obtain that M2(0) is homeomorphic toM2(1) = M(1)\E(p) and M2(0) is a convex set. This shows how restrictive the assumption EnMFCQ is.Second, we ask for a necessary and sucient condition, where we follow the idea de- scribed for other embeddings in 11]. We know that the starting point x0 for P2(0) (the only stationary point, cf. Theorem 3.1) lies on a uniquely determined connected componentC(x0 0) inPstat. Furthermore, we know thatC(x0 0) is the only connected
J. Guddat, F. Guerra, D. Nowack 19
t t t
x x x
Figure 3.1
component in Pstat crossing the hyperplane f(x t)2IRnIRjt = 0g. We introduce the following condition for P2(t):
(F1) MFCQ is satised for all x 2 M2(t) with (x t) 2 cl C(x0 0) j01]
Theorem 3.15
Let (F G) 2 C3(IRn IR)s+l. Assume (A2) and (A3). Then there exists a PC2-path in Pstat connecting (x0 0) with some point (x 1), where x is a stationary point of (P) if and only if (F1) is satised. 2 Remark concerning the proof: Use the same concept as in the proof of Theorem 2.5.Remark 3.16
(i) If the condition (F1) is satised and if we do not attaint = 1, then M(1) is empty, i.e., 1 was not a realistic wish of the decision maker. The program package PAFO provides information whether (F1) is satised (a) ) or not (b) ), namely (a) if there are singularities of the Types 2,3, and 5 (where the MFCQ is satised, that means, there is a continuation in Pstat with the same orientation (cf. Fig.3. 1(a))).
(b) if there are points of the Types 4 or 5 (where the MFCQ is not fullled, that means, the path ends in Pstat and has a continuation in Pgc only with the op- posite orientation (cf. Fig. 3.1(b) and Example 4.2)).
If there appears a point (x t) of Type 4 when approaching (x t) by local minimizers, then, in Case I, we can jump to another connected component in Pstat and, in Case II, there is no jump (cf. Fig. 2.4). We have the same situation if a point of Type 5 appears, where the MFCQ is not satised. Therefore, using pathfollowing and jumps in the set Pstat we are not able to give an answer to the question whether M(1) is empty or not. then we can try to compute connected components in Pgc and possible
