SLIDE 1 The Effects of Transformations on the Optimal Set in Interval Linear Programming
Elif Garajová1 (with Milan Hladík1 & Miroslav Rada2)
1Department of Applied Mathematics, Faculty of Mathematics and Physics,
Charles University
2Faculty of Informatics and Statistics, University of Economics, Prague
The 14th International Symposium on Operations Research in Slovenia (SOR’17), Bled
SLIDE 2
Interval Linear Programming
Consider a linear programming problem. . . minimize cTx subject to Ax ≤ b
estimating the future €25 6 c €27 1 inexact measurements a 5 0 05g approximation and rounding b 3 14159 1
SLIDE 3
Interval Linear Programming
Consider a linear programming problem. . . minimize cTx subject to Ax ≤ b
estimating the future €25.6 ≤ c ≤ €27.1 inexact measurements a = 5 ± 0.05g approximation and rounding b ≈ 3.14159 1
SLIDE 4
Interval Linear Programming
Consider an interval linear programming problem. . . minimize cTx subject to Ax ≤ b
estimating the future c = [25.6, 27.1] inexact measurements a = [4.95, 5.05] approximation and rounding b = [3.141592, 3.141593] 1
SLIDE 5 Interval Linear Programming: Definitions
- An interval linear program is a family of linear programs
minimize cTx subject to x ∈ M(A, b), where A ∈ A, b ∈ b, c ∈ c and M(A, b) is the feasible set.
- A linear program in the family is called a scenario.
- A vector x is a (weakly) feasible/optimal solution to the
interval program, if x is a feasible/optimal solution for some scenario with A ∈ A, b ∈ b, c ∈ c.
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SLIDE 6 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
- What are the possible feasible solutions?
- Which solutions are optimal for some scenario?
- What is the set/range of all optimal values?
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SLIDE 7 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 −1 3
SLIDE 8 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 −0.5 3
SLIDE 9 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 −0.33 3
SLIDE 10 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 −0.25 3
SLIDE 11 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 3
SLIDE 12 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 0.25 3
SLIDE 13 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 0.33 3
SLIDE 14 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 0.5 3
SLIDE 15 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Let's traverse through this! −1 1 1 3
SLIDE 16 Interval Linear Programming: Example
maximize x2 subject to [−1, 1]x1 + x2 ≤ 0 x2 ≤ 1
1 2 3 4
1 x2 x1
Optimal values: {0, 1} 3
SLIDE 17 Overview of Basic Results
Feasible solutions:
- Oettli–Prager: x solves Ax = b ⇔ |Acx − bc| ≤ A∆|x| + b∆
- Gerlach: x solves Ax ≤ b ⇔ Acx − A∆|x| ≤ b
Optimal values:
- Best optimal value (inequalities):
For each s ∈ {±1}n solve min (cc − Dsc∆)Tx s. t. (Ac − A∆Ds)x ≤ b, Dsx ≥ 0
- Worst optimal value (inequalities):
max bTy s. t. A
Ty ≤ c, ATy ≥ c, y ≤ 0
Optimal solutions:
- Special cases, approximations, …
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SLIDE 18 Dependency Problem (I)
max x1
[0, 1]x1 − x2 = 0, x2 ≤ 1, x1, x2 ≥ 0. max x1
x1 x2 x1 x2 x2 1 x1 x2 Optimal set: {(x1, x2) ∈ R2 : x1 ∈ [1, ∞) and x2 = 1} The solution 0 0 is now optimal, too! ...but the feasible set is the same! (Li, 2015)
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SLIDE 19 Dependency Problem (I)
max x1
[0, 1]x1 − x2 = 0, x2 ≤ 1, x1, x2 ≥ 0. max x1
[0, 1]x1 − x2 ≤ 0, [0, 1]x1 − x2 ≥ 0, x2 ≤ 1, x1, x2 ≥ 0. Optimal set: {(x1, x2) ∈ R2 : x1 ∈ [1, ∞) and x2 = 1} The solution 0 0 is now optimal, too! ...but the feasible set is the same! (Li, 2015)
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SLIDE 20 Dependency Problem (I)
max x1
[0, 1]x1 − x2 = 0, x2 ≤ 1, x1, x2 ≥ 0. max x1
1x1 − x2 ≤ 0, 0x1 − x2 ≥ 0, x2 ≤ 1, x1, x2 ≥ 0. Optimal set: {(x1, x2) ∈ R2 : x1 ∈ [1, ∞) and x2 = 1} The solution (0, 0) is now optimal, too! ...but the feasible set is the same! (Li, 2015)
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SLIDE 21 Dependency Problem (I)
max x1
[0, 1]x1 − x2 = 0, x2 ≤ 1, x1, x2 ≥ 0. max x1
[0, 1]x1 − x2 ≤ 0, [0, 1]x1 − x2 ≥ 0, x2 ≤ 1, x1, x2 ≥ 0. Optimal set: {(x1, x2) ∈ R2 : x1 ∈ [1, ∞) and x2 = 1} The solution (0, 0) is now optimal, too! ...but the feasible set is the same! (Li, 2015)
Wei Li. A note on dependency between interval linear systems (2015).
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SLIDE 22
Dependency Problem (II)
Example (Hladík, 2012) [1, 2]x ≤ 2 → [1, 2]x+ − [1, 2]x− ≤ 2, x+, x− ≥ 0 Original feasible set: (−∞, 2] Consider the new scenario 1x+ − 2x− ≤ 2, x+, x− ≥ 0… All real numbers are now feasible solutions, because we can express any real x as x = x+ − x−, where x+ = max(2x, 0) and x− = |x|.
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SLIDE 23
Transformations: The General Case
min cTx:
Ax = b, x ≥ 0
min cTx:
Ax ≤ b
min cTx:
Ax ≤ b, x ≥ 0
transitivity
What if A is fixed?
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SLIDE 24
Transformations: The General Case
min cTx:
Ax = b, x ≥ 0
min cTx:
Ax ≤ b
min cTx:
Ax ≤ b, x ≥ 0
transitivity
What if A is fixed?
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SLIDE 25
Splitting Equations into Inequalities
Theorem 1 The optimal solution set of the interval linear program min cTx: Ax = b, x ≥ 0 is equal to the optimal solution set of the program min cTx: Ax ≤ b1, −Ax ≤ −b2, x ≥ 0 with b1 = b2 = b. Proof idea: x optimal for scenario min cTx b2 Ax b1 x Ax b3 b x optimal for min cTx Ax b3 x
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SLIDE 26
Splitting Equations into Inequalities
Theorem 1 The optimal solution set of the interval linear program min cTx: Ax = b, x ≥ 0 is equal to the optimal solution set of the program min cTx: Ax ≤ b1, −Ax ≤ −b2, x ≥ 0 with b1 = b2 = b. Proof idea: x∗ optimal for scenario min cTx: b2 ≤ Ax ≤ b1, x ≥ 0 ⇒ Ax∗ = b3 ∈ b ⇒ x∗ optimal for min cTx: Ax = b3, x ≥ 0
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SLIDE 27 Imposing Non-negativity
Theorem 2 Let S denote the optimal solution set of min cTx: Ax ≤ b and let S′ be the optimal solution set of the program min cT
1x+ − cT 2x− : Ax+ − Ax− ≤ b, x+, x− ≥ 0
with c1 = c2 = c. Then, the following properties hold:
- If x ∈ S, then there is (x+, x−) ∈ S′ with x = x+ − x−.
- If (x+, x−) ∈ S′, then x+ − x− ∈ S.
Proof idea: For an optimal x , we have by dual feasibility some y with ATy c3 c2 c1
- c. Then, x is optimal for min cT
3x Ax
b.
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SLIDE 28 Imposing Non-negativity
Theorem 2 Let S denote the optimal solution set of min cTx: Ax ≤ b and let S′ be the optimal solution set of the program min cT
1x+ − cT 2x− : Ax+ − Ax− ≤ b, x+, x− ≥ 0
with c1 = c2 = c. Then, the following properties hold:
- If x ∈ S, then there is (x+, x−) ∈ S′ with x = x+ − x−.
- If (x+, x−) ∈ S′, then x+ − x− ∈ S.
Proof idea: For an optimal x∗, we have by dual feasibility some y∗ with ATy∗ = c3 ∈ [c2, c1] ⊆ c. Then, x∗ is optimal for min cT
3x: Ax ≤ b. 9
SLIDE 29
Transformations: The Special Case
min cTx:
Ax = b, x ≥ 0
min cTx:
Ax ≤ b
min cTx:
Ax ≤ b, x ≥ 0
transitivity
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SLIDE 30 Transformations: The Special Case
min cTx:
Ax = b, x ≥ 0
min cTx:
Ax ≤ b
min cTx:
Ax ≤ b, x ≥ 0
transitivity T h e
e m 1
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SLIDE 31 Transformations: The Special Case
min cTx:
Ax = b, x ≥ 0
min cTx:
Ax ≤ b
min cTx:
Ax ≤ b, x ≥ 0
transitivity transitivity T h e
e m 1 Theorem 2
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SLIDE 32 Conclusion
- In interval linear programming, the basic transformations
may change the feasible/optimal set and other properties of a program.
- We have shown that the transformations do not affect the
- ptimal set for problems with a fixed coefficient matrix (they
may still change other properties!).
- Thus, we can directly generalize results concerning the
- ptimal set of a particular type of programs to other types.
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SLIDE 33 References
Linear Optimization Problems with Inexact Data (2006). Authors: M. Fiedler, J. Nedoma, J. Ramík,
Interval linear programming: A survey (2012).
- M. Hladík. Linear Programming – New Frontiers
in Theory and Applications.
Thank you for your attention!
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SLIDE 34 References
Linear Optimization Problems with Inexact Data (2006). Authors: M. Fiedler, J. Nedoma, J. Ramík,
Interval linear programming: A survey (2012).
- M. Hladík. Linear Programming – New Frontiers
in Theory and Applications.
Thank you for your attention!
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