Robust combinatorial optimization with variable uncertainty Michael - - PowerPoint PPT Presentation
Robust combinatorial optimization with variable uncertainty Michael Poss Heudiasyc UMR CNRS 7253, Universit e de Technologie de Compi` egne 17th Aussois Combinatorial Optimization Workshop M. Poss (Heudiasyc) Variable uncertainty Aussois
Michael Poss
Heudiasyc UMR CNRS 7253, Universit´ e de Technologie de Compi` egne
17th Aussois Combinatorial Optimization Workshop
Variable uncertainty Aussois 1 / 31
1
Robust optimization
2
Variable budgeted uncertainty
3
Cost uncertainty
Variable uncertainty Aussois 2 / 31
1
Robust optimization
2
Variable budgeted uncertainty
3
Cost uncertainty
Variable uncertainty Aussois 3 / 31
min
n
cixi s.t.
n
aijxj ≤ bi, i = 1, . . . , m x ∈ {0, 1}n Suppose that the parameters (a, b, c) are uncertain: They vary over time They must be predicted from historical data They cannot be measured with enough accuracy ... Let’s do something clever (and useful)!
Variable uncertainty Aussois 4 / 31
Stochastic programming A lot ⇔ Robust programming A little
Variable uncertainty Aussois 5 / 31
Stochastic programming A lot ⇔ Robust programming A little Robust pr. Uncertain parameters are merely assumed to belong to an uncertainty set U ⇒ one wishes to optimize some worst-case
Stochastic pr. Uncertain parameters are precisely described by probability distributions ⇒ one wishes to optimize some expectation, variance, Value-at-risk, . . . Intermediary models exist: distributionally robust optimization, ambiguous chance-constrained
Variable uncertainty Aussois 5 / 31
Now All decisions must be taken before the uncertainty is known with precision ⇒ probability constraints, (static) robust
Delayed Some decisions may be delayed until the uncertainty is revealed ⇒ multi-stage stochastic programming, adjustable robust optimization
Variable uncertainty Aussois 6 / 31
min
n
cixi s.t.
n
aijxj ≤ bi, i = 1, . . . , m, ∀ai ∈ Ui x ∈ {0, 1}n, The linear relaxation of this problem is tractable if Ui is defined by conic constraints: Ui = {ai ∈ Rn : u ai − v ∈ K}. In particular, polyhedrons and polytopes are nice (K = Rn
+).
Variable uncertainty Aussois 7 / 31
∀a ∈ U ⇔
αj ≥ 0 The feasibility set of the constraint is a polyhedron (thus, convex) !
Variable uncertainty Aussois 8 / 31
∀a ∈ U ⇔
αj ≥ 0 The feasibility set of the constraint is a polyhedron (thus, convex) ! A (very) popular polyhedral uncertainty set is (Bertsimas and Sim, 2004): UΓ :=
ai, −1 ≤ δi ≤ 1,
Main reasons for popularity: Nice computational properties for MIP and combinatorial problems. Intuitive interpretation. Probabilistic interpretation.
Variable uncertainty Aussois 8 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5.
Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5. Consider two robust-feasible routes x1 and x2 : x11 = 3 and x21 = 10.
Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5. Consider two robust-feasible routes x1 and x2 : x11 = 3 and x21 = 10. Because x1 and x2 are robust-feasible:
i =1
ti ≤ T, ∀a ∈ UΓ, and
i =1
ti ≤ T, ∀a ∈ UΓ
Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5. Consider two robust-feasible routes x1 and x2 : x11 = 3 and x21 = 10. Because x1 and x2 are robust-feasible:
i =1
ti ≤ T, ∀a ∈ UΓ, and
i =1
ti ≤ T, ∀a ∈ UΓ which becomes
i =1
ti ≤ T, and
i =1
ti ≤ T, ∀a ∈ UΓ
Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5. Consider two robust-feasible routes x1 and x2 : x11 = 3 and x21 = 10. Because x1 and x2 are robust-feasible:
i =1
ti ≤ T, ∀a ∈ UΓ, and
i =1
ti ≤ T, ∀a ∈ UΓ which becomes
i =1
ti ≤ T, and
i =1
ti ≤ T, ∀a ∈ UΓ For any probability distribution for t: P
i:x1
i =1
ti > T = 0.
Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T. For simplicity, we suppose t ∈ UΓ for Γ = 5. Consider two robust-feasible routes x1 and x2 : x11 = 3 and x21 = 10. Because x1 and x2 are robust-feasible:
i =1
ti ≤ T, ∀a ∈ UΓ, and
i =1
ti ≤ T, ∀a ∈ UΓ which becomes
i =1
ti ≤ T, and
i =1
ti ≤ T, ∀a ∈ UΓ For any probability distribution for t: P
i:x1
i =1
ti > T = 0. If x2 is not robust-feasible for Γ = 10, there exists probability distributions: P
i:x2
i =1
ti > T > 0
Variable uncertainty Aussois 9 / 31
1
Robust optimization
2
Variable budgeted uncertainty
3
Cost uncertainty
Variable uncertainty Aussois 10 / 31
Let ˜ ai be random variables and ǫ > 0. The chance constraint P
aixi > b
(1) leads to very difficult optimization problems in general.
Variable uncertainty Aussois 11 / 31
Let ˜ ai be random variables and ǫ > 0. The chance constraint P
aixi > b
(1) leads to very difficult optimization problems in general. In some situations, we know that (1) can be approximated by
∀a ∈ U (2) for a properly chosen U. These approximations are conservative: any x feasible for (2) is feasible for (1). We must balance conservatism and protection cost ⇒ devise good protection sets U.
Variable uncertainty Aussois 11 / 31
What about UΓ ?
Variable uncertainty Aussois 12 / 31
What about UΓ ? Let ˜ ai be random variables independently and symmetrically distributed in [ai − ˆ ai, ai + ˆ ai]. Bertsimas and Sim (2004) prove that if a vector x satisfies the robust constraint
∀a ∈ UΓ, then it satisfies also the probabilistic constraint P
aixi > b
2n
Variable uncertainty Aussois 12 / 31
From P
aixi > b
2n
we see that choosing Γ = (−2 ln(ǫ))1/2 n1/2 yields P
aixi > b
Variable uncertainty Aussois 13 / 31
From P
aixi > b
2n
we see that choosing Γ = (−2 ln(ǫ))1/2 n1/2 yields P
aixi > b
For many problems, x1 < n1/2 for optimal (or feasible) vectors x (network design, assignment, ...) ⇒ Γ > n1/2 already for ǫ = 0.5 ⇒ for these problems, protecting with probability 0.5 yields protection with probability 0! ⇒ overprotection !
Variable uncertainty Aussois 13 / 31
It is easy to see that the bound from Bertsimas and Sim can be adapted to P
aixi > b
Γ2 2x1
Variable uncertainty Aussois 14 / 31
It is easy to see that the bound from Bertsimas and Sim can be adapted to P
aixi > b
Γ2 2x1
Γ can be reduced when x is small Let’s use multifunctions !
Variable uncertainty Aussois 14 / 31
It is easy to see that the bound from Bertsimas and Sim can be adapted to P
aixi > b
Γ2 2x1
Γ can be reduced when x is small Let’s use multifunctions ! Define αǫ(x) = (−2 ln(ǫ)x1)1/2 . Consider x∗ be given. If
i ≤ b
∀a ∈ Uαǫ(x∗) , then P
aix∗
i > b
2x∗1
Variable uncertainty Aussois 14 / 31
Let γ : {0, 1}n → R+ be a non-negative function. Uγ(x) :=
ai, −1 ≤ δi ≤ 1,
Variable uncertainty Aussois 15 / 31
Let γ : {0, 1}n → R+ be a non-negative function. Uγ(x) :=
ai, −1 ≤ δi ≤ 1,
We have shown that the new model
∀a ∈ Uαǫ(x), should be considered instead of the classical model
∀a ∈ UΓ.
Variable uncertainty Aussois 15 / 31
The previous bound is bad. Bertsimas and Sim propose a better bound: P
i > b
2n (1 − µ) n ⌊ν⌋
n
n l , where ν = (Γ + n)/2, µ = ν − ⌊ν⌋. We can make this bound dependent on x by considering B(x1, Γ). βǫ(x) is the solution of the equation B(x∗1, Γ) − ǫ = 0 (3) in variable Γ. We solve (3) numerically.
Variable uncertainty Aussois 16 / 31
max
n
cixi s.t.
n
aixi ≤ b, a ∈ Uγ x ∈ {0, 1}n, which can be rewritten as max
n
cixi s.t.
n
aixi + max 0 ≤ δi ≤ 1 δi ≤ γ(x)
n
δiˆ aixi ≤ b, x ∈ {0, 1}n,
Variable uncertainty Aussois 17 / 31
Using the dualization approach: max
n
cixi s.t.
n
aixi + zγ(x) +
n
pi ≤ b, z + pi ≥ ˆ aixi, i = 1, . . . , n z, p ≥ 0, x ∈ {0, 1}n. Non-convex reformulation. x binary may help.
Variable uncertainty Aussois 18 / 31
Theorem
Consider robust constraint aTx ≤ b, ∀a ∈ Uγ(x), x ∈ {0, 1}n, (4) and suppose that γ = γ0 + γixi is an affine function of x, non-negative for x ∈ {0, 1}n. Then, (4) is equivalent to
n
aixi + γ0z +
n
γiwi +
n
pi ≤ b z + pi ≥ ˆ aixi, i = 1, . . . , n, wi − z ≥ − maxj(ˆ aj)(1 − xi), i = 1, . . . , n, p, w, z ≥ 0, x ∈ {0, 1}n.
Variable uncertainty Aussois 19 / 31
10 20 30 40 50 60 70 80 100 200 300 400 500 600 700 800 900 1000 β0.01 γ1 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 β0.01 min(γ1, γ2)
Variable uncertainty Aussois 20 / 31
Objective
1 Is there a benefit in using Uβ instead of UΓ ? 2 Computational “complexity” of solving the robust counterparts.
Variable uncertainty Aussois 21 / 31
Objective
1 Is there a benefit in using Uβ instead of UΓ ? 2 Computational “complexity” of solving the robust counterparts.
Models We compare the following at ǫ = 0.01: UΓ The classical robust model with budget uncertainty. Uγ1 Our new model with variable budget uncertainty: γ1
Uγ1γ2 Our new model with variable budget uncertainty: min(γ1, γ2) over-approximates β.
Variable uncertainty Aussois 21 / 31
Instances from Bertsimas and Sim (2004)
0.5 1 1.5 2 2.5 3 3.5 100 200 300 400 500 600 700 800 900 1000 Deterministic cost increase in % Number of items n Uγ1γ2 + + + + + + + + + + + Uγ1
Variable uncertainty Aussois 22 / 31
model Uγ1 Uγ1γ2 Uγ1γ2γ3 time model/time UΓ 1.7 3.4 6.1 gap model/gap UΓ 0.87 0.98 1.1 Fixing M to maxj(ˆ aj) affects the LP relaxation. If M = 1000, gap Uγ1/gap UΓ → 3.9 !
Variable uncertainty Aussois 23 / 31
1
Robust optimization
2
Variable budgeted uncertainty
3
Cost uncertainty
Variable uncertainty Aussois 24 / 31
Suppose that only cost coefficient are uncertain min max
c∈U n
cixi s.t.
n
aijxj ≤ bi, i = 1, . . . , m x ∈ {0, 1}n, which can be rewritten COΓ ≡ min
x∈X max c∈UΓ cTx.
The previous probabilistic approximation leads to a relation between COΓ and min
x∈X VaRǫ cTx.
Variable uncertainty Aussois 25 / 31
Definition: VaRǫ(cTx) = inf{t|P(cTx ≤ t) ≥ 1 − ǫ}. We see easily that COΓ provides an upper bound of the optimization of VaR
Variable uncertainty Aussois 26 / 31
Definition: VaRǫ(cTx) = inf{t|P(cTx ≤ t) ≥ 1 − ǫ}. We see easily that COΓ provides an upper bound of the optimization of VaR The upper bound is very bad for small cardinality vectors
Variable uncertainty Aussois 26 / 31
Definition: VaRǫ(cTx) = inf{t|P(cTx ≤ t) ≥ 1 − ǫ}. We see easily that COΓ provides an upper bound of the optimization of VaR The upper bound is very bad for small cardinality vectors Model COγ overcomes this flaw COγ ≡ min
x∈X max c∈Uγ cTx.
Variable uncertainty Aussois 26 / 31
10 15 20 25 30 35 40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Cost reduction in % Value of ε NE1 AL1 MN1 IA1
Variable uncertainty Aussois 27 / 31
Theorem
When γ is an affine function, COγ can be solved by solving n + 1 problems CO and taking the cheapest optimal solution.
Variable uncertainty Aussois 28 / 31
Theorem
When γ is an affine function, COγ can be solved by solving n + 1 problems CO and taking the cheapest optimal solution.
Theorem
When γ is a non-decreasing function of x1, COγ can be solved by solving n cardinality constrained problems COΓ and taking the cheapest
Variable uncertainty Aussois 28 / 31
We use the notation Γ′ = min(n, max
k=0,...,n γ(k)).
Theorem
Consider a combinatorial optimization problem that can be solved in O(τ) by using dynamic programming. If γ(k) ∈ Z for each k = 0, . . . , n, then COγ can be solved in O(nΓ′τ). Otherwise, COγ can be solved in O(n2Γ′τ).
Variable uncertainty Aussois 29 / 31
We use the notation Γ′ = min(n, max
k=0,...,n γ(k)).
Theorem
Consider a combinatorial optimization problem that can be solved in O(τ) by using dynamic programming. If γ(k) ∈ Z for each k = 0, . . . , n, then COγ can be solved in O(nΓ′τ). Otherwise, COγ can be solved in O(n2Γ′τ).
Theorem
Consider a combinatorial optimization problem that can be solved in O(τ) by using dynamic programming. Then, COΓ can be solved in O(Γτ). If Γ ∼ n1/2, we get O(n1/2τ), improving over the O(nτ) from Bertsimas and Sim.
Variable uncertainty Aussois 29 / 31
We introduce a new class of uncertainty models. They correct the flaw of Bertsimas and Sim model. The tractability of the new model is often comparable (or equal) to the traditional model. Remark: The model can be extended to non-combinatorial problems but tractability becomes an issue.
Variable uncertainty Aussois 30 / 31
Variable uncertainty Aussois 31 / 31