19
Consider
G
E x
1
∈
. Since f and
t q
q
g g
,...,
1 1
γ γ
are convex functions, there exist vector
, μ
λ
such that x is a solution to problem
, μ
λ
fug
P
. Because
, μ
ξ
we put
k k
λ ξ =
,
k k
k k
ξ γ
μ α =
, since
, μ
ξ
, therefore we obtain that x is also a solution to the problem
ξ
α
P
. □
From Proposition 2.1 and Proposition 2.2, if
f
and
t q
q
g g
,...,
1 1
γ γ
are convex functions and if
k k
sign sign
γ α =
,
. t
t
k k
γ α
, for every
} ,...,
1 {
q k
∈
, the sets of properly efficient solutions to problem 2.3 and 2.6 verify the following
properties: a.
Every properly efficient solution to problem 2.6 is properly efficient for problem 2.3;
b. Setting
q
R ∈
γ
, with nonnull components, the set of properly efficient solutions to problem 2.3 is a subset of the union in
α
of the set of properly efficient solutions for problem 2.6.
2.4 Some relation between expected-value efficient solution, minimum-
variance efficient solution and expected-value standard deviation efficient solution
Consider a problem 2.1 and sets efficient solution expected value
PE
E
minimum variance
2
σ PE
E
, and expected value standard deviation
σ PE
E
associated with the problem. Let
w PE
w PE
w PE
E E
E
σ σ
, ,
2
be the sets of weakly efficient solutions associated with the problems in Definitions 2.1-2.3, respectively.
If we consider
, x
x g
x z
x f
k k
k k
σ =
=
And if we choose
1 =
γ
, given that, for
} ,...,
1 {
q k
∈
, its verified that
+
→ R R
n
: σ
, then the relations between these efficient sets are deduced directly from Theorem
20
2.5 Multi-criteria problems
Consider the following model of a multi-criteria optimization problem:
,..., min
1
x F
x F
q
2.8
D x
∈
2.9 where D is a nonempty set of all feasible solution,
m
D R
⊂
;
R →
D F
F
q
: ,...,
1
. Stated briefly, a multi-criteria optimization problem consists in the choice of a particular
solution
D x
∈
for which all of the utility functions
q k
x F
k
, 1
, =
, simultaneously approach bigger values or at least do not decrease.
Let us recall some concepts of multi-criteria optimization problem solutions; Zeleny [217] and Urli and Nadeau [196], Salukavadze and Topchishvili [166].
Definition 2.6 The solution
D x
P
∈
is called Pareto-optimal or efficient for the problem 2.8-2.9 if and only if, for every
D x
∈
, the system of inequalities
P k
k
x F
x F
,
q k
, 1
=
,where at least one inequality is strict, is inconsistent.
Definition 2.7 The solution
D x
w
∈
, is called weakly efficient or Slater-optimal for the problem 2.8-2.9 if and only if, for every
D x
∈
, the system of strict inequalities
w k
k
x F
x F
,
q k
, 1
=
, is inconsistent.
Definition 2.8 The solution
D x
G
∈
, is called proper efficient or Geoffrion-optimal for the problem 2.8-2.9 if and only if it is a Pareto-optimal solution for the problem
2.8-2.9 and there exists a positive number
θ
such that, for each
p k
, 1
=
, we have
θ
≤ −
− ]
[ ]
[ x
F x
F x
F x
F
j G
j G
k k
, for some j such that
G j
j
x F
x F
where
D x
∈
and
G k
k
x F
x F
q k
, 1
=
, is inconsistent.
Let
,
w j
E
, E
G j
E
denoted the sets of weakly-efficient, efficient, and proper efficient solutions, respectively, for the problem 2.8-2.9. It is obvious that
G j
E
⊂ E ⊂
w j
E
.
21
Next we will studied some relations between the efficient sets of several problems of deterministic multi-objective programming.
Let
f
and
g
be vectorial functions defined on the same set
n
H R
⊆
, with
q
R H
f →
:
and
q
R H
g
+
→ :
. Let us consider the following multi-objective problems: P
1
,..., ,
,..., min
1 1
1
x g
u x
g u
x f
x f
q q
q D
x ∈
2.10 P
2
,..., min
1
x f
x f
q D
x
∈
2.11 P
3
,..., min
1 1
x g
u x
g u
s q
q s
D x
∈
2.12 with,
H D
⊆
,
q
u R
R →
+
:
,
,...,
1 q
u u
u =
and
s
a real number.
2.6 Relations between classes of solutions for P
