fi : Rn R i = 1, . . . , k k e" 2 S ‚" Rn
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S.
fi i = 1, . . . , k
f(x) = [f1(x), f2(x), . . . , fk(x)]T S
Rn S = "

x = [x1, x2, . . . , xn]T " S
S S
fi : Rn R i = 1, . . . , k k e" 2 S ‚" Rn
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S.
Z := f(S)
Rk
z " Z z = f(x) x " S
fi : Rn R i = 1, . . . , k k e" 2 S ‚" Rn
Å„Å‚
ôÅ‚
òÅ‚max{f1(x), f2(x), . . . , fk(x)}
ôÅ‚
ółx " S
Å„Å‚
ôÅ‚- min{-f1(x), -f2(x), . . . , -fk(x)}
òÅ‚
ôÅ‚
ółx " S.
Å„Å‚
ôÅ‚ - 1, x2}
òÅ‚min{x
ôÅ‚
ółx " [0, 1].
x = 0
Å„Å‚
ôÅ‚ - x, x2}
òÅ‚min{1
ôÅ‚
ółx " [0, 1].
x = 0
x = 1
Rk k e" 2
R [1, 2]T
[3, 3]T [1, 4]T [2, 3]T Rk
Á ‚" X × X
Á
" x " X x Á x
" x, y, z " X (x Á y '" y Á z) Ò! x Á z
" x, y " X (x Á y '"y Á x) Ò! x = y
Á
Á X
" x, y " X (x Á y ("y Á x)
Á X
Á
 Rk k e" 2 d"
" x, y " Rk x d" y Ò! " i = 1, . . . , k xi d" yi .
x d" y Ô! x - y " Rk := {w " Rn : wi d" 0, i = 1, . . . , k}.
-
Rk
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S.
x " S
<" " x " S f(x) d" f(x ) '" f(x) = f(x ) Ô!

<" " x " S (" i " {1, . . . , k} fi(x) d" fi(x ) '" " i0 " {1, . . . , k} fi (x) < fi (x ) .
0 0
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S.
 x
f(S) )" (Rk + {f(x )}) = {f(x )} (Z )" (Rk + {f(x )}) = {f(x )}).
- -
R2
Rk k > 2
w = (w1, . . . , wk) " Rk k wi = 1
+
i=1
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S.
Å„Å‚
k
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚ wifi(x) min
òÅ‚
i=1
(P (w))
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ółx " S.
x w wi > 0
i = 1, . . . , k x w
Å„Å‚
ôÅ‚ - x, x2}
òÅ‚min{1
ôÅ‚
ółx " [0, 1].
 w = (w1, w2) w1, w2 > 0 w1 + w2 = 1
w1 = 1 - w2 w2 " (0, 1)
Å„Å‚
ôÅ‚ - w2)(1 - x) + w2x2 min
òÅ‚(1
P (wMOP -b)
ôÅ‚
ółx " [0, 1].
1-w2
f(x) = w2(x - )2 -
2w2
(1-w1)(1-5w2)
4w2
1-w2
w2 " [1, 1)
2 2w2 3
xw =
min
1
1 w2 " (0, )
3
(1 - y, y2), y " (0, 1]
(P (wMOP -b))
(1, 0)
x w
x w
S ‚" Rk fi
S x
Å„Å‚
ôÅ‚
òÅ‚min{f1(x), f2(x), . . . , fk(x)}
(MOP )
ôÅ‚
ółx " S,
k
w = (w1, . . . , wk) wi e" 0 i = 1, . . . , k wi = 1 x
i=1