TECHNICAL NOTE
On a Gap between Multiobjective Optimization and Scalar Optimization
B. A
GHEZZAF1ANDM. H
ACHIMI2Communicated by W. B. Gong
Abstract.
We give a suitable example to show a gap between multi- objective optimization and single-objective optimization, which solves a problem proposed in Refs. 1–2.
Key Words.
Multiobjective optimization, scalar optimization, opti- mality, constraint qualifications.
1. Introduction
In this note, we are concerned with a gap between multiobjective optim- ization and scalar optimization. In nonlinear single-objective programming, constraint qualifications play an extremely important role in deriving first- order (Ref. 3) and second-order (Ref. 4) necessary conditions of the Kuhn–
Tucker type. Research in the past years has been directed toward generaliz- ing the original Kuhn–Tucker constraint qualification (e.g., Ref. 5) and interrelating them (Ref. 3). It seems that the Guignard constraint qualifi- cation is the weakest possible constraint qualification guaranteeing that necessary conditions of the Kahn–Tucker type hold. On the other hand, in multiobjective optimization problems, many authors have derived the first- order (Refs. 6–7) and second-order (Ref. 8) necessary conditions under the Abadie constraint qualification, but never under the Guignard constraint qualification. Naturally, an interesting question will arise: Do the necessary
1Professor, De´partement de Mathe´matiques et d’Informatique, Faculte´ des Sciences Aı¨n Chock, Universite´ Hassan II, Casablanca, Morocco.
2Graduate Student, De´partement de Mathe´matiques et d’Informatique, Faculte´ des Sciences Aı¨n Chock, Universite´ Hassan II, Casablanca, Morocco.
431
0022-3239兾01兾0500-0431$19.50兾02001 Plenum Publishing Corporation
conditions hold under the same weaker constraint qualifications as used in nonlinear single-objective programming?
In Ref. 1, the authors give an example to illustrate a gap between multi- objective optimization and single-objective optimization. But in the corre- sponding example, there is a mistake. In this note, we give another suitable example which solves a problem proposed in Refs. 1–2.
2. Main Result
Consider the following multiobjective optimization problem:
(P) min f(x), (1a)
s.t. x∈X, (1b)
where X is a nonempty subset of
⺢nand f:
⺢n→⺢
mis differentiable at x
0∈X.
We first state the following unified theorem of single-objective and mul- tiple-objective optimization problems.
Theorem 2.1. Let X be a nonempty set in
⺢n, and let x
0∈X. Further- more, suppose that f:
⺢n→
⺢mis differentiable at x
0. If x
0is locally weak Pareto minimal of f on X, then F∩ T G ∅, where F G {d: ∇ f(x
0)d F 0} and T is the cone of tangents of X at x
0.
Proof. See Ref. 3, Theorem 5.1.2 for the single-objective optimization case (m G 1) and Refs. 6–7 for the multiobjective optimization case
(m H 1).
䊐For illustrating the gap between single-objective and multiple-objective optimization problems, the following proposition is useful. Its proof is easy and is omitted here.
Proposition 2.1. Let S and T be sets in
⺢n. Suppose that S is the open half space. Then,
S ∩ T G ∅ ⇒ S∩ coT G ∅, (2)
where coT is the closure of the convex hull of T.
It is trivial that (2) hold if S G ∅. If m G 1, then F may be either the
empty or open half space. Consequently, we can prove Theorem 2.1 by
replacing the assertion that F∩ T G ∅ with the assertion F∩ co T G ∅. Nat- urally, an interesting question will arise: Does the relation (2) hold true even if S is an intersection of open half spaces?
3. Discussion of an Example
Let us illustrate in some detail why Proposition 2.1 is not true if m H 1.
When m G 1, F reduces to the open half space (Ref. 3) and F
cis then a closed convex set, where F
cis the complement of F; hence, Proposition 1.1 holds. However in general, if m H 1, F may not be the open half space.
Hence, F
cis not convex, as will be shown in the following example.
Example 3.1. We take n G m G 2. We set
X G {(x
1, x
2)
t兩(x
1C ax
2)(ax
1C x
2) ⁄ 0}, (3a) f
1(x
1, x
2) G x
1, f
2(x
1, x
2) G x
2, x
0G (0, 0)
t, (3b) where a is a positive number. It is easily verified that:
(i) x
0is a Pareto minimal solution to the vector minimization prob- lem
min( f
1(x
1, x
2), f
2(x
1, x
2)), s.t. (x
1, x
2)
t∈X;
(ii) both f
1and f
2are differentiable at x
0; (iii) F G {(x
1, x
2)
t∈⺢
2兩x
1⁄ 0, x
2⁄ 0};
(iv) T G X and
co T G
冦{(x
⺢2,
1, x
2)
兩x1C x
2G 0}, if if a a G ≠ 1. 1, Hence,
F ∩ coT ≠∅, (4)
when the positive number a ≠1.
Remark 3.1.
(i) It is very important to observe here that, in Example 3.1, X is not convex, if not, T is convex, and hence (4) is not possible.
(ii) If we assume that F ≠∅, then Theorem 2.1 shows that, at the
optimal solution (m G 1), the cone T lies always in a half space. However,
at the Pareto minimal solution (m H 1), this is not the case, as shown sche-
matically in Fig. 1.
Fig. 1. Cone of tangents ofXat the origin,TGT1∪T2.
