Sensitivity analysis in linear optimization:
Invariant active constraint set and invariant
partition intervals
Alireza Ghaffari Hadigheh
Tabriz University, Tabriz, Iran
Joint work with
Kamal Mirnia and Tamás Terlaky
MOPTA04, July 28-30.
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Outline
• Linear Optimization Problem
• Perturbed Linear Optimization Problem
• Motivation: Three Types of Sensitivity Analysis
• Invariant Support Set Interval (ISS)
• Invariant Active Constraint Set Interval (IACS)
• Invariant Partition Interval (IP)
• Relation between the ISS, IACS, IP and Invariancy Intervals
• Concluding Remarks
• References
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The Linear Optimization Problem
Primal Linear Optimization Problem:
min
s.t.
cT x
Ax = b
x ≥ 0,
Dual Linear Optimization Problem:
max
s.t.
bT y
AT y + s = c
s ≥ 0,
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Perturbed Linear Optimization Problem
Primal Perturbed Linear Optimization Problem:
min
s.t.
(c + 4c)T x
Ax = b + 4b
x ≥ 0,
Dual Perturbed Linear Optimization Problem:
min
s.t.
(b + 4b)T y
AT y + s = c + 4c
s ≥ 0.
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Type I Sensitivity Analysis (Basis Invariancy)
• Goal: The given optimal basic solution remains optimal.
• Realm: Simplex methods (classic sensitivity analysis).
• Draw Back: Having multiple optimal (degenerate) solutions
−→ Different methods lead to different optimal basic solutions
−→ Confusing optimality ranges.
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Type II Sensitivity Analysis (Support Set
Invariancy)
• Support set: σ(v) {i|vi > 0, i = 1, 2, . . . , n}.
• Primal-dual optimal solution: (x∗, y ∗, s∗).
• Property of the solution: σ(x∗) = P .
• Invariant Support Set Partition : (P, Z)
P = {i : x∗i > 0} and Z = {1, 2, . . . , n}\P.
• Goal: Having a primal-dual optimal solutions (x∗(), y ∗(), s∗())
with
σ(x∗) = σ(x∗()).
• Invariant Support Set (ISS) Interval:
ΥL(4b, 4c).
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Type III of Sensitivity Analysis (Optimal Partition
Invariancy)
• Optimal Partition: π = (B, N )
B = {i : x∗i > 0 for an optimal solution x∗},
N = {i : s∗i > 0 for an optimal solution (y ∗, s∗)}.
• Goal: The optimal partition is invariant.
• Invariancy Interval.
• Actual Invariancy interval.
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The ISS Partition and The Optimal Partition
The primal-dual optimal solution (x∗, y ∗, s∗) is not strictly complementary;
The ISS Partition:
P
Z
The Optimal Partition:
B
N
P ⊆ B and
Z ⊇ N.
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How to Find the ISS Interval
• ΥL(4b, 4c) = [`, u]
`(u) = min(max) s.t.
AP xP − 4b = b,
ATP y − 4cP = cP ,
ATZ y + sZ − 4cZ = cZ
xP ≥ 0, , sZ ≥ 0,
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Type II Sensitivity Analysis for Dual Problem
• Primal-dual optimal solution: (x∗, y ∗, s∗),
• Property of the solution : σ(s∗) = P ,
• Invariant Active Constraint Set Partition: (P , Z),
P = {i : s∗i > 0} and Z = {1, 2, . . . , n} \P ,
• Goal: Having a primal-dual optimal solution (x∗(), y ∗(), s∗()) with
σ(s∗()) = σ(s∗) = P ,
• Invariant Active Constraint Set Interval: ΓL(4b, 4c),
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How to Find the IACS Interval
• ΓL(4b, 4c) = [γ`, γu]
•
γ`(γu) = min(max)
s.t.
AZ xZ − 4b = b
ATZ y − 4cZ = cZ
APT y + sP − 4cP = cP
xZ ≥ 0, sP ≥ 0,
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The ISS, IACS and Optimal Partitions
Primal-dual optimal solution (x∗, y ∗, s∗) is not strictly complementary;
The ISS Partition:
P
Z
The Optimal Partition:
B
N
The IACS Partition:
Z
P
P ⊆ B ⊆ Z and
Z ⊇ N ⊇ P.
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Type II Sensitivity Analysis for Primal and Dual
Problems Simultaneously
• Primal-dual optimal solution: (x∗, y ∗, s∗);
• Property of the solution: σ(x∗) = P and σ(s∗) = P .
• Invariant Partition: (P, Z̃, P )
P = {i : s∗i > 0} and Z = {1, 2, . . . , n} \P .
• Goal: Having a primal-dual optimal solution (x∗(), y ∗(), s∗()) with
the property:
σ(x∗) = σ(x∗()) = P and σ(s∗) = σ(s∗()) = P
• IP interval : ΘL(4b, 4c).
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The ISS, IACS and IP and Invariancy Partitions
Primal-dual optimal solution (x∗, y ∗, s∗) is not strictly complementary;
The ISS Partition:
P
Z
The Optimal Partition:
B
N
The IACS Partition:
Z
P
The Invariant Partition:
P
Z̃
P
Z = Z̃ ∪ P and Z = Z̃ ∪ P,
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Specialization of Methods (1)
• The IACS interval ΓL(4b, 0) is always a closed interval but the ISS
interval ΥL(4b, 0) is always an open interval.
• The IACS interval ΓL(0, 4c) is always a open interval but the ISS
interval ΥL(0, 4c) is always an closed interval.
• We have
ΥL(4b, 0) ⊆ ΓL(4b, 0).
and ΥL(4b, 0) = int(ΓL(4b, 0)) iff (x∗, y ∗, s∗) is strictly complementary.
• We also have:
ΓL(0, 4c) ⊆ ΥL(0, 4c),
and ΓL(0, 4c) = int(ΥL(0, 4c)) iff (x∗, y ∗, s∗) is strictly complementary.
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Specialization of Methods (2)
• We always have:
ΘL(4b, 4c) ⊆ ΥL(4b, 4c) and ΘL(4b, 4c) ⊆ ΓL(4b, 4c),
• Equalities hold when (x∗, y ∗, s∗) is strictly complementary.
• ΘL(4b, 0) = ΥL(4b, 0)
• ΘL(0, 4c) = ΓL(0, 4c)
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Comments
• Interior of these intervals are subsets of the actual invariancy interval
with the expectation that they may include the adjacent breakpoints
(transition points).
• When the given primal-dual optimal solution is nondegenerate (both
primal and dual): Type I and Type II sensitivity analysis are identical
• Type II and Type III sensitivity analysis are not identical even for
strictly complementary solutions.
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Illustrative Example 1-I
Primal:
max
s.t.
2x1
x1
x1
2x1
x1
x1 ,
+ x2
+ x2 +x3
+2x2
+x4
+ x2
+x5
x2 ,
x3 ,
x4 ,
=4
=6
=6
+x6 = 3
x5, x6 ≥ 0.
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Illustrative Example 1-II
Dual:
min
s.t.
4y1 +6y2 +6y3 +3y4
y1 + y2 +2y3 +y4 −s1
y1 +2y2 +y3
−s2
y1
−s3
y2
−s4
y3
−s5
y4
−s6
s1, s2, s3, s4, s5, s6,
=2
=3
=0
=0
=0
=0
≥ 0.
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Illustrative Example 1-III
• Strictly complementary solution:
5 1 3 1
x∗ = ( , 1, , , 0, )T , y ∗ = (0, 0, 1, 0)T and s∗ = (0, 0, 0, 0, 1, 0)T .
2 2 2 2
• Optimal Partition : π = (B, N )
B = {1, 2, 3, 4, 6} and N = {5}.
• 4b = (1, −1, 0, 0)T and 4c = 0
• The actual invariancy interval (−1, 3);
• the IACS interval [−1, 3].
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Illustrative Example 2-I
max
s.t.
min
s.t.
2x1 +4x2 +6x3
x1 +x2 +2x3 +x4
= 10
x1 +4x2 +5x3
+x5 = 10
x1 ,
x2,
x3, x4, x5 ≥ 0,
10y1 +10y2
y1
+y2 −s1
=2
y1 +4y2
−s2
=4
2y1 +5y2
−s3
=6
y1
−s4
=0
y2
−s5 = 0
s1, s2, s3, s4, s5 ≥ 0.
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Illustrative Example 2-II
• x∗ = (10, 0, 0, 0, 0)T and multiple dual optimal solutions.
• Optimal partition: π = (B, N )
B = {1} and N = {2, 3, 4, 5}.
• 4c = (3, 2, 1, 0, 0)T and 4b = (1, 1)T
• The actual invariancy interval (− 27 , ∞);
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Illustrative Example 2-III
• Case 1. Dual optimal degenerate basic solutions,
y (1) = ( 43 , 32 )T with s(1) = (0, 0, 0, 34 , 23 ).
– ΓL(4b, 4c, 0) = ΘL((4b, 4c, 0) = {0}
• Case 2. Dual optimal nondegenerate basic solutions,
y (2) = (0, 2)T with s(2) = (0, 4, 4, 0, 2)T .
– ΓL(4b, 4c, 0) is (− 27 , ∞).
– ΘL((4b, 4c, 0) = (− 27 , 0]
– ΘL((4b, 4c, 0) ⊂ ΓL(4b, 4c, 0)
• Case 3. Dual optimal nonbasic (strictly complementary) solutions,
y (3) = (1, 1)T with s(3) = (0, 1, 1, 1, 1)T .
2
ΥL(4b, 4c, 0) = ΓL(4b, 4c, 0) = ΘL(4b, 4c, 0) = (− , ∞).
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Conclusions
Type II sensitivity analysis is investigated for:
• Primal linear optimization problem =⇒ ISS Intervals;
• Dual linear optimization problem =⇒ IACS intervals;
• Primal and dual linear optimization problem =⇒ IP intervals;
• Relation between them and actual invariancy interval;
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Selected References
• Koltai and Terlaky T. Koltai and T. Terlaky, The difference between
managerial and mathematical interpretation of sensitivity analysis results in linear programming, International Journal of Production Economics 65, 257-274, 2000.
• A.R. Ghaffari Hadigheh and T. Terlaky, Sensitivity analysis in linear
optimization: Invariant support set intervals. Submitted to European
Journal of Operation Research, 2003.
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Thank You!
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