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Selected Topics in Robust Optimization

Abstract

In this chapter, the authors give a brief introduction to important concepts of RO paradigm. The remainder of the chapter is organized as follows: Section 2 gives an introduction on optimization under uncertainty, and presents brief comparisons among the well-known sub-fields of optimization under uncertainty such as RO, stochastic programming (SP), and fuzzy optimization (FO). Section 3 presents important methodologies of RO paradigm. Section 4 gives insights about alternative ways of choosing the uncertainty set. Section 5 shows alternative methods of assessing the quality of a robust solution and presents miscellaneous topics. Finally, Section 6 summarizes conclusions and gives future research directions.


    ! 
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      $
%"

  &'(((% ) '(*+%)  '(*(%
  )"   '((,% & '(*+%   )   '((-
.#'(**%)"/#'(('%0
'((1%#23'(*'%.4#
%)"/#'(('%)'(**%)
5$'((6%7'(*8%7'(*,

9   :       
;% ;%    
:%
%%
        . <%       *,
%*8,*,,%%
      #   #   
         . <%
:= ; 
  :>?
.<%%%
+@*'+*,8* %
  A %   
.<%#
B
    :   >?  /  
#A)"
 /# '(((       ((*C   D 
,(C*+1(&B
E-*+*((C%'*(%(((C%
 %
$4

     3       
F
a
T
x b
*

a=a+ρ ζ
D
a
%
ζR2

   <
ζ
1
%
ρ
  :
/D
  %
a=a
%    

%
a
T
x=b
.*
[
a
T
x b
]

x<
1
.*F.
(
a+ρ ζ
)
T
x b
GH* *I'

.*
ζ

<
  
x ,
 
%<
  
.   
,(C%
ζ

$4%<

/<
.'FJ$        


E.'%
   "   %       .  
%      
$/%$
.*%.'
x .
2
   %%  <
5
x1
4
E
x1

     
x
3
/   

x1

x2
%%

x1

x2
E%#
x2

x1
%
$>#?

           %
%   %   5B%    . J
%.  $
%<
  %      )"   '((1% <%  5B  
F*           
#'
  + # 
    E 
         % 5B   
        .    5B   
BK#*11,%)&< '(**%5L#'((+%3 3
*1,1
% <!)"
'((1%<F
2
A>?F
>?
 #4 %
%
       4  >? M   # 

E    GA*I< ! 
AN  A  5 +'E %GA+I 
<    )"  '((1%3+ O **%   
<P<58
.%%#5B%
    %       
%#9 %.%  
      
 !
"== !<
E  % = =      (  *  
:    E
% =;  
%
 !     "    /  
5B%
  <  5 +*   /%      <
4%. !4
..%#
.
E%F
2=%5B%.
              5
#            
%           
$4
!"E5B%
Q"R%
5B#
/#
<9%
5BN<)"'((1%<"<%
3'((-%
N0S#TU'(*6%
<
!"  #      % 5B  .
2<
$F*% 5B   
N589%$
3
  %     
  4>#? J  
#   <%      4  
/%
$7N 4#%/#'(*'%
 5B  
'%
"  V#5.%
          E %    
%V
"%24
        N % J  U '(*, 
0S#T  R '(*+  %  V  0 '(*(% 7  5 '(*(%
2'(*8+ V$    5B% .  
      E % #   5B%    
 .  =    
<   %5B   
 %           
   %    $      

!"$
E%
 %% 4   %
=
    "" %       

.#<%%
 4     
>?
min
x
{
cTx:Ax b
}
(
c , A , b
)
U
'

cR
n
    = D%
AR
mxn
 <  
D% 
bRm
   R5 % 
U
   
 A <  5'%   
GA*I%GA'IGA+I%%
R
n
>?%
𝒰% !%%
 𝒰N  E     >?% 
      F *  =   "   
N'𝒰<N+"/%
<
%5=D
c
"
b

    
C

B
N % %     

4

min
x
max
cC
{
cTx:Ax b AU , bB
}

2             
4="F
min
x ,t
{
t:c
T
xt 0cC , Ax +b x
n+1
0
(
A , b
)
U × B
}
<
tR

x
n+1
=−1

%
U
<
conv (U)
%%<%

U
3
U
4#
:" 
U
%4<:"

conv (U)
.%)"'((1%*'M*+
%     $        
" J%         

b1

b2
F
x
1
+b
1
0, x
2
+b
2
0
%
U=
{
bR
2
:b
1
0, b
2
0, b
1
+b
2
1
}

E
i
th
(
i=1,2
)

𝒰 4       = 
U

bi
Ui,
%  
"#.%)"'((1%**M*'
/          %
<GJ'I
 &!"'
'3
F
min
x
{
c
T
x:A
(
ζ
)
x b ζ Ƶ
}
+

ζRL
 !%
ƵRmxn

% 
A
(
ζ
)
R
mxn
   D <    #   
A
(
ζ
)

D 
ζ
$ A 
xRn
         
[A
(
ζ
)
x b ]

ζƵ

A%"
%<+F

(
a+P ζ
)
T
x b ζƵ .
8
2           D 
a+P ζ
 %

ζƵ
%
aRn
%
bR

PRnxL
N"
%..*11+/
P ζ
:
ζ
%%
nL
/8
4%%"
#"
44%
5
%E%

3
8     /4
23
F
Ƶ=
{
ζ:D ζ +q 0
}
,

DRmxn, ζ RL

qRm
( &"  /  8  4    "
F
a
T
x+max
ζ: +q 0
(
P
T
x
)
T
ζ b .
,
( '2 #  < , <
=%
,4

a
T
x+min
w
{q
T
w:D
T
w=−P
T
x , w 0} b . (6)
( E-%
D
w
R%3


w:a
T
x+q
T
w b , D
T
w=−P
T
x , w 0
$6
/6
xRn

wRm.
*F
[
(
a+P ζ
)
T
x b ζƵ
]
W X 3 
)<
J
ζ
1
ζ
2
1
a
T
x+
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T
x
1
b
a
T
x+
P
T
x
2
b
&B
3YB
*
*<
  
R%&B5'
N*/
 * 
           3  $  
   %          
= !4
E   Q <  %
"%

S
i
Ƶ
i
 
i
.<% %
Si

   % %   

Ƶi
 
Si
%E%
6
E%
  #     % %       
< 2  
Si
%     
%    .   %    )#   '((@ E
:Z  

Si

Ƶi

%
%       
Si
     
%      5 )   '(*-  
A%#%
33'((,
          
 E   <
DN37[
'((@%*
 )"!"
GA*I%% ""#%<
!/%"#%
#""%%""
 :     . <%      
<""%""
#    %  
=%
"! A3
F

min
x , y
(
.
)
{
c
T
x:A
(
ζ
)
x+By
(
ζ
)
b ζƵ
}
,
(
8
)

xRn
""
ζRL
%
y
(
·
)
Rk

""#%
BR
mxk

D<%%</%A3@  <
     
y(ζ)
   
D9%
y(ζ)
:<

y
(
ζ
)
:=y0+Q ζ , (9)

y
0
R
k

QR
kxL
DD()
!3AA3
"N  %  )"   '((1% 3 *8    J% 
@F
min
x, y0,Q
{cTx:A(ζ)x+B y0+BQ ζ d ζƵ}
5+*%
  
ζ
%    
x , y0

Q
 AA34 3
7
Q=0
1AA3 ! 3
P<    =          
9%AA3$<
  3%%/
D      
)'(*(
&%A"%%)"
'((,%=)"'((1%J<*8'*%)"/#
'(('.A%0S#T'(*6
!*A3   <N
)"'((1%3*8+7'(*,
!*A#1 =!%
            
ζƵ
.  
= %    )  3 '((6% \  
'(**%)7'(*8%7'(*,
%+A%A
=
<#)"'((1$
=
min
x≥ 0, y 0
x:y 1
2ζx+1, x
(
2ζ
)
yζ: 0 ζ ρ (10)
2<%%
E<3A35+*%
%
ζ


 &!<<""%%

ζ[0, ρ]
/%
ζ=ρ


ζ=0
N

ζ[0, ρ]

34%%*(F
y 1
2ρx +1, x 2y ,(11)

x 2
1ρ(12)
**'%
x 2y ρx +2


4%=3

2
1ρ


ζ
%=3

ρ
*N%=3
2
(
1ρ
)
1
*'
/!A5
""%%
N
y
""=%%:
  
ζ
N   
y
 = 
ζ
 
8

y=y0+y1ζ

yo

y
1
D%A343
y1=0
5&V]
1+1
2ζx
%
y0=1

y
1
=1
2x

     &V /       
*(%
&V
y
F
x
(
2ζ
)
(
1+1
2ζx
)
ζ:1 ζ ρ . (13)
E
x=4
*+
ρ=1
F
4
(
2ζ
) (
2ζ+1
)
ζ: 1 ζ ρ=1.
%=A3
4
34%$=
%%
ρ=1
yields
=3
¿
=A3
4

, '(
E%
"$      
.<%*
        
E#"
<3
&B < %% 3 
"=   
 !#" 
"%
   "**%%<)
"<< $ 
#"
      "<<
 :     <
 3       "  53B   

"<<
     33*1,1A
F

Pr
ζP
{ζ:a
(
ζ
)
T
x b } ε ,
*8

ζR
L
  !  %
aRn
   
D%
xRn
%
ε[0,1]
%
 #!V$%
F
9

Pr
ζP
{ζ:a
(
ζ
)
Tx b } ε , PƤ(15)
 
     ^ E        

   %  7
A<%
<#%%*8

ζ
%
ζ
E%
:#"<

%
 <     9 %       
Uε



a
(
ζ
)
T
x b ζU
ϵ
,(16)
**8 )"'((1%3'
U
ε
4 "
**F
U
ϵ
coax
{ζR
L
:
ζ
2
Ω
ϵ
,
ζ
1}
*6
3*-F
z
j
+w
j
=−
[
a
j
]
T
x , j
{
1, , l
}

j=1
l
¿zj¿+Ωϵ=
j=1
l
wj
2 b
[
a0
]
x ,
(
18
)
¿

ε
%
(z , w )

%
a
(
ζ
)
=a0+
j=1
l
ζjaj
D
ζ , ε

4
exp (2/2)
%
ε
]
2 ln ε
%
E[ζ i]=0
¿ζ i1

ζi
!
i{1,. .. , L }.
A    )"  '((1% B '+8%  
+)5'((8F
Ƶϵ=
{
ζ:
ζ
1 Γ
ζ
1
}
,
(
19
)

ε=exp (Γ
2
/2L).
)5

"< <       )  5 <
ε
/%  3    )  5     
53B*@&B
))'(*'2
 
ζ
       
µ
 
σ2
% 
F
Ƶϵ=
{
ζ:
|
i=1
L
ζi
|
ρ
n σ
}
,(20)
10

ρ

1ε
))
'(%%
%&B
&BA
L>1
)"   '(*+  "    "  

p

q
F
I
ϕ
(
p ,q
)
=
i=1
m
q
i
ϕ
(
p
i
q
i
)
(21)

ϕ
'*<"N"%B
'((,&
p

q
4
N

p
W%
2N
ϕ''
(
1
)
Iϕ
(
p,q
)
d
χm1
2as N
F
Ƶ
ϵ
=
{
p:p 0, e
T
p=1, 2N
ϕ
''
(
1
)
I
ϕ
(
p ,
^
p
)
χ
m1;1ϵ
2
}
,(22)

^
p
''
p

N
%
χm1;1ε
2

1ε


χ2
 
m1
        
p

<
1ε
)" '(*+<"
 < 
0S#T   R '(*+   <     
A   "    ("'   
N%)"'(*,%J<*,
- ./
- 0'!"1(
A#:=
  /  
 # =# 
=        9%  =
 $      $   #  
(
%93
  !    
         
     =         = E 
%

7'(*,
Q5
 
n

(
Ri, Ni
)
(
i=1,2, , n
)
,

Ri

Ni


i
th
N
11
J
R
i

Ni


ζ
%
R
i

Ni
E
$%%
Var
(
RiNi
)
]
Var
(
Ri
)
+Var
(
Ni
)
2Cov
(
Ri, Ni
)
%
F

H
0
:m
R
=m
N

H
1
:m
R
<m
N

α
%
m
R

m
N

R
i

Ni
%
=
X

Y
(,&]

R
i
=N
i

Z

Ri Ni
W%
Z
H] 
($,=
Z

(1α)


 "    
H0:µR=µN

H1:µR<µN
,%

µx

µy

Ri

Ni
%=

R

N

Ri Ni
    &
Zi=RiNi,´
Z=(
i=1
n
Zi/n)

s2=
i=1
n
(
Zi´
Z
)
2/(n1)
%
T=
n
i=1
n
(Zi´
Z)/ s
 " 
n1
     

H0
=
T

α

t
"

n1

- !"'%2&$"1
%2
E  %  4    
 4   #    . <% # 
7'(*,%<
       <  )"   '((1%  *+ 3  
F
(
2+ζ
)
x11,
-4F
(
2+ζ
)
x
1
+s=1,
s 0
R%%

(
2+ζ
)
x1+s=1ζ:
|
ζ
|
1
'+
F

(
2+ζ
)
x
1
+s=1, s 0ζ:
|
ζ
|
1
'8

ζ

{
ζ:
|
ζ
|
1
}
%4E
'+F
x
1
1/3
%'(
x
1
=0
 '+'8  4   # % 4
 4      % 
F .%
!./$012334 %5
A<F
12
|
x1ζ
|
+
|
x2ζ
|
2,
4F
y1+y22
y1 x1ζ
y1 ζ x1
y2 x2ζ
y2ζ x2

R%%F

|
x
1
ζ
|
+
|
x
2
ζ
|
2ζ:
|
ζ
|
1,
',
F
y1+y22
y1 x1ζζ:
|
ζ
|
1

y1 ζ x1ζ:
|
ζ
|
1
'-
y
2
x
2
ζζ:
|
ζ
|
1
y
2
ζ x
2
ζ:
|
ζ
|
1
  4 E%     #         ', 
(
θ ,θ
)
,1 θ 1,
'-
x=
(
0,0
)
'-
%"%
$%   $    %
%4',F
x1ζ+x2ζ 2ζ:
|
ζ
|
1

x1ζ+ζx22ζ:
|
ζ
|
1
'6
ζ+x
2
x
1
ζ 2ζ:
|
ζ
|
1
ζx
1
+ζx
2
2ζ:
|
ζ
|
1
.$[%7
'(*,%_-
- $3
E#'(*+ > 
 "        "" 
?%>?#BDN%
=#
7%E
#BDE
%  =      %%
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V )  A 7 )      <" 
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14
V)A  A 9 %,8*F*,(M
*-@%'((-
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,(*%'(**
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16
... The ellipsoid uncertainty set introduces a dependence among different components of the uncertainty parameter so that all components cannot take their worst-case realizations at the same time because the total dispersion is bounded by the radius of the ellipsoid (i.e., 1 in our case, due to scaling of the parameters). It is known that the ellipsoidal uncertainty set is less conservative and yields more diversified solutions than the box; for more details, we refer the reader to [43] that gives a detailed overview on selection criteria of uncertainty sets. ...
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