The above problem is equivalent to the following problem.
max F (x) =
N
X
i=1
β
i
,
subject to F
i
(x) ≥ β
i
, i = 1, ..., N,
g
i
(x) ≤ 0, i = 1, ..., m,
x ∈ R
n
Then we can obtain the same result as the Lemma 2.1: If (¯x,
¯
β) is the solution of the above
maximization problem , then there exist ¯u
i
, i = 1, ··· , N such that ¯x is a solution of the
following problem for u = ¯u and β =
¯
β.
max
N
X
i=1
u
i
(f
i
(x) − β
i
h
i
(x)),
subject to g
i
(x) ≤ 0, i = 1, ..., m,
x ∈ R
n
And ¯x also satisfies the system of equations (2.4) and (2.5) for u = ¯u and β =
¯
β.
The above problem is convex programming for given parameters β ≥ 0 and u > 0. In
what follows, all the results for the minimization problem hold true for the maximization
problem.
Let α = (β, u) denote parameter vector and let
Ω = {α = (β, u) ∈ R
2N
|0 ≤ β ≤ β
u
, 0 < u
l
≤ u ≤ u
u
},
where
β
u
= (β
u
1
, ··· , β
u
N
), u
l
= (u
l
1
, ··· , u
l
N
), u
u
= (u
u
1
, ··· , u
u
N
), β
u
i
= max
x∈X
f
i
(x)
h
i
(x)
, u
u
i
= max
x∈X
1
h
i
(x)
,
and
u
l
i
= min
x∈X
1
h
i
(x)
, i = 1, ··· , N.
Remark 2.2. The β
u
, u
l
and u
u
involved in the definition of Ω are needed only for theoretical
consideration and they are not needed in solving the problem (2.1) at all.
5
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