Optimal value range in interval linear
programming
Milan Hladı́k
Department of Applied Mathematics
Charles University, Prague
EURO XXII, Prague
July 8–11, 2007
Introduction
Definition
I
An interval matrix
AI = [A, A] = {A ∈ Rm×n | A ≤ A ≤ A},
Introduction
Definition
I
An interval matrix
AI = [A, A] = {A ∈ Rm×n | A ≤ A ≤ A},
I
Midpoint and radius of AI
1
Ac ≡ (A + A),
2
1
A∆ ≡ (A − A).
2
Introduction
Definition
I
An interval matrix
AI = [A, A] = {A ∈ Rm×n | A ≤ A ≤ A},
I
Midpoint and radius of AI
1
Ac ≡ (A + A),
2
1
A∆ ≡ (A − A).
2
Consider the interval linear program
f (A, b, c) ≡ inf c T x subject to Ax = b, x ≥ 0,
where A ∈ AI , b ∈ bI , c ∈ c I .
Lower and upper bounds of the optimal value
f ≡ inf f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I ,
f ≡ sup f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I .
Lower and upper bounds of the optimal value
f ≡ inf f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I ,
f ≡ sup f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I .
Theorem (Rohn 2006)
We have
f = inf c T x subject to Ax ≤ b, Ax ≥ b, x ≥ 0.
Let f be finite or let the right-hand side of (1) be positively
infinite. Then
T
T
y
−
A
f = sup bcT y + b∆
|y | subject to AT
c
∆ |y | ≤ c.
(1)
Unified approach
Consider the general interval linear program
f (A, b, c) ≡ inf c T x subject to x ∈ M(A, b),
where M(A, b) is described by a linear system.
Unified approach
Consider the general interval linear program
f (A, b, c) ≡ inf c T x subject to x ∈ M(A, b),
where M(A, b) is described by a linear system.
Lower and upper bounds of the optimal value
f ≡ inf f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I ,
f ≡ sup f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I .
Unified approach
Consider the general interval linear program
f (A, b, c) ≡ inf c T x subject to x ∈ M(A, b),
where M(A, b) is described by a linear system.
Lower and upper bounds of the optimal value
f ≡ inf f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I ,
f ≡ sup f (A, b, c) subject to A ∈ AI , b ∈ bI , c ∈ c I .
Dual problem
max bT x subject to x ∈ N(A, c).
Solution sets of M(A, b) and N(A, c), respectively, are
M ≡ {x ∈ M(A, b) | A ∈ AI , b ∈ bI },
N ≡ {y ∈ N(A, c) | A ∈ AI , c ∈ c I }.
Solution sets of M(A, b) and N(A, c), respectively, are
M ≡ {x ∈ M(A, b) | A ∈ AI , b ∈ bI },
N ≡ {y ∈ N(A, c) | A ∈ AI , c ∈ c I }.
Theorem
We have
T
f = inf ccT x − c∆
|x| subject to x ∈ M.
Let f < ∞ or let the right-hand side of (2) be positively infinite.
Then
T
f = sup bcT y + b∆
|y | subject to x ∈ N.
(2)
Algorithm
Definition (Strong solvability)
The system describing M(A, b), A ∈ AI , b ∈ bI , is strongly
solvable if M(A, b) is nonempty for all A ∈ AI , b ∈ bI .
Algorithm
Definition (Strong solvability)
The system describing M(A, b), A ∈ AI , b ∈ bI , is strongly
solvable if M(A, b) is nonempty for all A ∈ AI , b ∈ bI .
Algorithm (Optimal value range [f , f ])
1. Compute
T
|x| subject to x ∈ M.
f := inf ccT x − c∆
Algorithm
Definition (Strong solvability)
The system describing M(A, b), A ∈ AI , b ∈ bI , is strongly
solvable if M(A, b) is nonempty for all A ∈ AI , b ∈ bI .
Algorithm (Optimal value range [f , f ])
1. Compute
T
|x| subject to x ∈ M.
f := inf ccT x − c∆
2. Compute
T
ϕ := sup bcT y + b∆
|y | subject to y ∈ N.
Algorithm
Definition (Strong solvability)
The system describing M(A, b), A ∈ AI , b ∈ bI , is strongly
solvable if M(A, b) is nonempty for all A ∈ AI , b ∈ bI .
Algorithm (Optimal value range [f , f ])
1. Compute
T
|x| subject to x ∈ M.
f := inf ccT x − c∆
2. Compute
T
ϕ := sup bcT y + b∆
|y | subject to y ∈ N.
3. If M(A, b), A ∈ AI , b ∈ bI , is strongly solvable, then set
f := ϕ, otherwise set f := ∞.
Examples
Example (simple case)
Let M(A, b) = {x | Ax ≤ b, x ≥ 0}.
Examples
Example (simple case)
Let M(A, b) = {x | Ax ≤ b, x ≥ 0}. Then
N(A, c) = {y | AT y ≥ c, y ≥ 0},
M = {x | Ax ≤ b, x ≥ 0},
T
N = {y | A y ≥ c, y ≥ 0}.
Examples
Example (simple case)
Let M(A, b) = {x | Ax ≤ b, x ≥ 0}. Then
N(A, c) = {y | AT y ≥ c, y ≥ 0},
M = {x | Ax ≤ b, x ≥ 0},
T
N = {y | A y ≥ c, y ≥ 0}.
Strong solvability of M(A, b) is equivalent to solvability to
Ax ≤ b.
Examples
Example (simple case)
Let M(A, b) = {x | Ax ≤ b, x ≥ 0}. Then
N(A, c) = {y | AT y ≥ c, y ≥ 0},
M = {x | Ax ≤ b, x ≥ 0},
T
N = {y | A y ≥ c, y ≥ 0}.
Strong solvability of M(A, b) is equivalent to solvability to
Ax ≤ b.
Optimal value bounds
f = inf c T x subject to x ∈ M,
T
f = sup b y subject to y ∈ N.
Example (with dependences)
Let the feasible set be described as follows
M(A, (b1 , b2 )) = {x, y | Ax = b1 , Ay = b2 , x, y ≥ 0}.
Example (with dependences)
Let the feasible set be described as follows
M(A, (b1 , b2 )) = {x, y | Ax = b1 , Ay = b2 , x, y ≥ 0}.
The dual feasible set and solution sets
N(A, (c 1 , c 2 )) = {u, v | AT u ≤ c 1 , AT v ≤ c 2 },
Example (with dependences)
Let the feasible set be described as follows
M(A, (b1 , b2 )) = {x, y | Ax = b1 , Ay = b2 , x, y ≥ 0}.
The dual feasible set and solution sets
N(A, (c 1 , c 2 )) = {u, v | AT u ≤ c 1 , AT v ≤ c 2 },
n
1
2
M = x, y | Ax ≤ b , Ax ≥ b 1 , Ay ≤ b , Ay ≥ b 2 , x, y ≥ 0,
o
1 T
2 T
b∆
y + b∆
x + A∆ |xy T − yx T | ≥ |(Ac x − bc1 )y T − (Ac y − bc2 )x T | ,
n
1
T
2
T
T
c
,
A
c
,
v
−
A
|v
|
≤
N = u, v | AT
u
−
A
|u|
≤
c
∆
c
∆
o
T
1
T
2
T
(c − Ac u)|vk | + (c − Ac v )|uk | + A∆ |vk u − uk v | ≥ 0 ∀k : uk vk < 0 .
Example (with dependences)
Let the feasible set be described as follows
M(A, (b1 , b2 )) = {x, y | Ax = b1 , Ay = b2 , x, y ≥ 0}.
The dual feasible set and solution sets
N(A, (c 1 , c 2 )) = {u, v | AT u ≤ c 1 , AT v ≤ c 2 },
n
1
2
M = x, y | Ax ≤ b , Ax ≥ b 1 , Ay ≤ b , Ay ≥ b 2 , x, y ≥ 0,
o
1 T
2 T
b∆
y + b∆
x + A∆ |xy T − yx T | ≥ |(Ac x − bc1 )y T − (Ac y − bc2 )x T | ,
n
1
T
2
T
T
c
,
A
c
,
v
−
A
|v
|
≤
N = u, v | AT
u
−
A
|u|
≤
c
∆
c
∆
o
T
1
T
2
T
(c − Ac u)|vk | + (c − Ac v )|uk | + A∆ |vk u − uk v | ≥ 0 ∀k : uk vk < 0 .
Optimal value bounds
1 T
2 T
f = inf(cc1 )T x − (c∆
) |x| + (cc2 )T y − (c∆
) |y | s.t. (x, y ) ∈ M,
2 T
1 T
) |v | s.t. (u, v ) ∈ N.
) |u| + (bc2 )T v + (b∆
f = sup(bc1 )T u + (b∆
Extension
Extension
Consider quadratic programming problem
min x T Cx + d T x subject to Ax ≤ b, x ≥ 0,
Extension
Consider quadratic programming problem
min x T Cx + d T x subject to Ax ≤ b, x ≥ 0,
Its dual is
max −x T Cx − bT u subject to 2Cx + AT u + d ≥ 0, u ≥ 0.
Extension
Consider quadratic programming problem
min x T Cx + d T x subject to Ax ≤ b, x ≥ 0,
Its dual is
max −x T Cx − bT u subject to 2Cx + AT u + d ≥ 0, u ≥ 0.
Theorem
If C is fixed and A ∈ AI , b ∈ bI , d ∈ d I , then
f = inf x T Cx + d T x subject to Ax ≤ b, x ≥ 0,
T
f = sup −x T Cx − bT u subject to 2Cx + A u + d ≥ 0, u ≥ 0.
The End.