Realization of Step 2

Let (x^o x_on⁺¹) be specially choosen. Then Step 2 can be realized by nding a g.c. point of

min^ff(x xn⁺¹) :=^k(x xn⁺¹)^;(x^o x_on⁺¹)^k2j(x xn⁺¹)²M_I^g

P

I: where

M_I:=^f(x xn⁺¹)²M_I^jgs⁺³(x xn⁺¹) := fI(x xn⁺¹)^;fI(^x ^xn⁺¹) + 0^g: We introduce the following notations:

~A¹:=

;Q² 0 0 0

~A²:=

2

6

4

0 ::: 0 ... ... ...

0 ::: 1

3

7

5 A³:=

Q¹ 0 0 0

I :=^f1 n^g

~aj:=

aj

0

~am⁺i :=^;

ei

0

~as⁺¹:= en⁺¹ j = 1 mi²I

~as⁺²:=0 0

as⁺³:= c

;1

~bj := bj j ^2f1 m^g

~bm⁺i = 0 i = 1 n ~bs⁺¹:= 0 ~bs⁺²=^;q ~bs⁺³:=^;fI(^x ^xn⁺¹) +

5.1 The general quadratic optimization problem 36

where ei is the unity vector in^Rn+1and s as dened above. Then, the problem P_I is equivalent to

min^ff(x) =^kx^;x^ok2jx²M_I^g

P

I:

~M_I:=^fx^2Rn+1jgj(x)0 j²J^g J :=^f1 s^gfs + 1 s + 2 s + 3^g g_j(x) := ~a_Tjx + ~b_j j^2f1 s^g gs⁺¹(x) := x^T ~A¹x + ~a_Ts⁺¹x

gs⁺²(x) := x^T ~A²x + ~a_Ts⁺²x + ~bs⁺²

g_s⁺³(x) := x^T ~A³x + ~a_Ts⁺³x + ~b_s⁺³:

We easily verify that P_Iis a double quadratic problem. Therefore we consider min^ff(x xn⁺¹ t)^jgj(x xn⁺¹ t)0 j²J =^f1 s + 3^g

P

I(t) : where

f(x xn⁺¹ t) :=^k(x xn⁺¹)^;(x^o xn⁺¹o)^k2 g_j(x x_n⁺¹ t) := g_j(x x_n⁺¹) i^2f1 s^g gs⁺¹(x xn⁺¹ t) := gs⁺¹(x xn⁺¹)

gs⁺²(x xn⁺¹ t) := gs⁺²(x xn⁺¹)

g_s⁺³(x x_n⁺¹ t) := g_s⁺³(x x_n⁺¹) + (t^;1)g_s⁺³(x^o x_on⁺¹):

We note that (x^o x_on⁺¹) has to be chosen as in (C2).

Remark 10.

We see that ^x solves ^PI if and only if M_I = , for > 0 suf-ciently small. Here, we do not discuss the question: How can it be checked whetherM_Iis empty or not? This is why we assume

M_I⁶=: (^C8)

Now we have to show whether the choice of the starting point is possible.

According to Section 1, we dene

P:=^f(x xn⁺¹)^2Rn+1jgj(x xn⁺¹)0 (j = 1 s)^g G_i:=^fx^2Rnjgs⁺i(x xn⁺¹)0^g i = 1 3 H¹=^fx^2Rn+1jDxgs⁺³(x) = 0^g:

Theorem 8 (Choice of a starting point).

Let > 0 be suciently small, and let ^z = (^x ^xn⁺¹) be the solution ofIQOPc and cl intM_cI = M_cI. Then we have

int^P\intG^1\intG²ⁿG³⁶=:

5.1 The general quadratic optimization problem 37

Proof. Let z := (x xn⁺¹) and suppose that

int^P\intG^1\intG²ⁿG³=: Then we have

8z^2Rn+1 z⁶²int^P\intG^1\intG²or z²G³: (35)We distinguish 2 cases:

Case 1: ^z²intP^\intG^1\intG² Then, we have

^z⁶²G³ (36)since gs⁺³(^z) = > 0. From (35) we obtain:

(a) ^z⁶²int^P\intG^1\intG²(contradiction to ^z²M_cI).

(b) ^z²G³ (contradiction to (36)).

Case 2 : ^z²@^P@G¹@G²

Since cl intM_cI= M_cI, there exists a sequence ^fzk^gwith

fzk^gintM_cI zk ^;1^!

^z:

Hence, there is a zk^o such that gs⁺³(zk^o) = fI(zk^o)^;fI(^z) + > 0. Thus zk^{o 2}=G³. Arguing as in case 1, the proof is complete.

Theorem 9.

It holds (a) H¹=:

(b) The choice of the starting point (x^o x_on⁺¹) for P_I is, by the assumptions made in Theorem8, always possible.

According to the results above, we propose the following algorithm, which solves (IQOPc).

Algorithm

2

Step 0 Transform (QOP) into (^IQOP)^c

Step 1 Choose an > 0 su ciently small.

Compute a ^z²stat(IQOPc)(gc(IQOP) Set k := 0, x_ok := ^z.

Step 2 If M_I^k =, stop (x_ok is the solution). Else, go to step 3.

Step 3 If ^z satises (C2), set x_ok = ^z. Else, compute a ~z close to ^z which satises (C2) and set x_ok= ~z. Go to step 4.

Step 4 Call the algorithm 1:

Compute a g.c. point p for P_I^k(x_ok ^z). If the algorithm 1 is successful, set k = k + 1 ^z := p x_ok := ^z and go to step 2. Else, stop.

5.1 The general quadratic optimization problem 38 5.1.2 The concave quadratic optimization problem

In view of the realization of Step2 we have:

min^ff(x) =^kx^;x^ok2jx²M_c^g

P

c: where

M_c:=^fx^2Rnjgj(x)0 j²J^g J :=^f1 m m + 1 m + n + 1^g gj(x) := a_Tjx + bj j^2f1 m^g gm⁺i(x) :=^;xi i^2f1 n^g

gs⁺¹(x) := fc(x)^;fc(^x) + s := m + n and fc(x) = x^TQx + c^Tx. P_cis a double quadratic problem.

For the choice of the starting point we assume:

cl int(^PnGs⁺¹) =^PnGs⁺¹and H^2#int(^PnGs⁺¹):

Then we have the same results as in Theorem 8.

5.1.3 Illustrative examples

We consider Example 5.1 and try to nd an approximate solution using the algorithm 2. After transformation we have

min^fx^21;x^3jx^2R3 gj(x¹ x² x³)0 j = 1 6^g

IQOP

c: where

g¹(x¹ x² x³) := x^1;10 g²(x¹ x² x³) :=^;x¹ g³(x¹ x² x³) :=^;x² g⁴(x¹ x² x³) := x^2;10:0

g⁵(x¹ x² x³) := x^3;x²² g⁶(x¹ x² x³) := x^23;10000 (compactification) Step 1: Take the stationary point ^z = (10:0 10:0 100).

Step 2: Find a g.c. point of

min^f(x^1;x^o1)²+ (x^2;x^o2)²+ (x^3;x^o3)^2jg_j(x¹ x² x³)0 j = 1 7^g

P

I: where

g⁷(x¹ x² x³) := x^21;x^3;(^z^1;^z³) + :

In the Figures 16, 17, 18 and 19 we have sketched and presented solutions of P_I^k, considering the iterations k = 1 7 10 (see also Table 1). The algorithm stops at the iteration 11 and more precisely, at a point of Type 5, where, locally the set^P_gc becomes empty (see the Fig. 19).

5.2 Multiobjective optimization 39

Iteration points

k ^z x^o Solution of P_I^k

1 0:08 (10:0 10:0 100:0) (9:99 9:99 50:0) (7:08 9:99 50:24) 2 0:08 (7:08 9:99 50:2) (7:08 9:99 50:24) (7:07 9:99 50:2) 3 0:08 (7:07 9:99 50:2) (7:07 9:99 50:24) (7:06 9:99 50:2) 4 0:5 (7:06 9:99 50:2) (7:06 9:99 50:24) (7:02 9:99 50:2) 5 0:8 (7:02 9:99 50:2) (7:02 9:99 50:26) (6:96 9:99 50:2) 6 ¹⁰:⁰ (6:96 9:99 50:2) (6:96 9:99 50:26) (6:19 9:99 50:26) 7 ⁵⁰:⁰ (6:19 9:99 50:26) (6:19 9:99 50:26) (0:25 9:99 61:97) 8 0:8 (0:25 9:99 61:97) (0:25 9:99 61:97) (0:10 9:99 62:72) 9 0:8 (0:10 9:99 62:72) (0:10 9:99 62:72) (0:04 9:99 63:51) 10 ²⁰:⁰ (0:04 9:99 63:51) (0:04 9:99 63:51) (0:0001 9:99 83:51) 11 0:8 (0:0001 9:99 83:51) (0:0001 9:99 83:51) (⁰:^{0 10}:^{0 100})

Remark 11.

The considered Example 5:1, computed with PAFO 11], shows that, although we can reach t = 1for k20 and compute the solution, we are not able to check whether the computed solution is global, since we cannot decide whether M_I =. At the iteration k = 10 we cannot reach t = 1, and PAFO stops at a point of Type 5, which is the solution. At this point M_I becomes locally empty.

5.2 Multiobjective optimization

min^ff(x)^jx^2Pg f = (f¹ fl)^T (MOP)

where

f¹(x) = x^TQx + c^T1x

fj(x) = c_Tjx + dj j = 2 l

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

P is a convex polyhedron and the matrix Q is negative semi-denite.

The main information consists in estimating the objective value at the point xⁱ in comparison with fk

;

= inf^ffk(x)^jx^2Pg.

Telescreen:

5.2 Multiobjective optimization 40

5.2 Multiobjective optimization 41

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

9.4

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

9.4

0.00000 0.00002 0.00004 0.00006 0.00008 0.00010

9.988

0.00000 0.00002 0.00004 0.00006 0.00008 0.00010

9.988

5.2 Multiobjective optimization 42

f¹

;

f¹(x^o) f¹(x^o)^;f¹

jf^1j :100

... ... ...

fk

;

fk(x^o) fk(x^o)^;fk

jfk^j :100

... ... ...

f_l

;

f_l(x^o) fl(x^o)^;fl

jf_l^j :100 We consider

M( ¹) :=^fx^{2P j}f_k(x) ¹_l k = 1 l^g:

1is the goal of the decision maker. In order to nd the goal realizer ^x²M( ¹), we propose to nd a g.c. point of

min^fkx^;x^ok2jM( ¹)^\E(p)^g where (37)

E(p) :=^fx^2Rnjkx^k2p^g , and p^2Ris su ciently large.

M( ¹)^\E(p) =

8

<

:

x^2Rn a_Tix + bi 0 i = 1 s fj(x)^{; 1}_j 0 j = 1 l

kx^k2;p 0

9

=

Since (37) is a double quadratic problem, we can apply the results obtained above in order to nd a goal realizer.

REFERENCES 43

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Im Dokument On double one-parametric optimization problems and some applications (Seite 35-44)