Modification of Recourse Data for Integer Recourse Models

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Modiﬁcation of Recourse Data for

Integer Recourse Models

Maarten H. van der Vlerk

jointly with Ward Romeijnders & Willem K. Klein Haneveld www.rug.nl/feb/mhvandervlerk

University of Groningen, Netherlands (est. 1614)

Charles University, Prague (est. 1348) November 8, 2012

Outline

(Mixed-)Integer recourse: deﬁnition Motivation

Modiﬁcation of Recourse Data

Convexα-approximations for SIR & TU recourse Error bounds

Decision making under uncertainty isthe real problem

we should all be working on [G.B. Dantzig 2001]

Stochastic Programming on the Web:

SP Community Home Page http://stoprog.org

SP E-Print Series

SP Bibliography + Books on Stochastic Programming

(Mixed-)Integer recourse models

Coping with m random constraints T (ω)x ≥ h(ω) Only right-hand side random:

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Why include integer variables? Modeling power: natural integrality of decision variables

e.g. The Allocation of Aircraft to Routes [Ferguson & Dantzig ’56]

yes/no, on/oﬀ decisions

artiﬁcial indicator variables for conditional linear constraints (LP formulation of CO problems)

0≤ x ≤ Mz, x ∈ R, z ∈ {0, 1}

satisfy k out of n constraints, e.g. discrete Chance Constraints Pr{Tx ≥ ω} ≥ α ∈ (0, 1)

with Pr{ω = ωs} = ps, s = 1, . . . , S

Why not?

continuous SLP is already diﬃcult enough complexity: 2nd-stage problems NP-hard

[Dyer & Stougie ’06]continuous SLP is #P complete

−→ SMIP not harder (. . . )

Main issue: integer recourse functionQ is non-convexin general

(example from[Schultz et al. ’98])

How to solve SMIP?

Borrow fromsolution approaches for deterministic MILP: e.g. (LP + rounding)

Branch & Bound with LP relaxation Benders’ decomposition

Polyhedral theory: valid inequalities Lagrangian relaxation

Combine with SLP algorithms−→ algorithms for SMIP?

Various authors (+ co-authors): Schultz

Louveaux Sen Ahmed

. . . see e.g. S(I)P Bibliography and SPePS

Our perspective: SMIP is a battlebetween randomness:Good

integrality:Bad

Usually, result is Ugly: non-convex, . . .

−→

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Modiﬁcation of Recourse Data (VdV 2003)

Recourse data (q, W , Y , Fω)

structure:

(q, W ) complete / suﬃciently expensive recourse, . . .

y∈ Y : simple bounds,integrality

distribution

Nice properties for recourse models

Convexity

Discrete distribution

Continuous decision variables

Idea of approach

Modify data (q_{, W , Y , F}ω)_{→ (˜q, ˜}W_{, ˜}Y_{, ˜F}ω) so that easy to solve

good approximation

MRD for integer recourse

Expected value functionQ(x) Q(x ) =E_ω min y {qy : Wy ≥ω− Tx, y ∈Z n2 +}

withω acontinuousrandom vector

Approximation of VdV (2004) _{Q(x ) =}ˆ _E ξ min y {qy : Wy ≥ξ− Tx, y ∈R n2 +}

withξ adiscreterandom vector

Special case: (one-sided) Simple Integer Recourse (W = I )

Special case: Simple integer recourse (

W = I )

SR: Modeling linear penalty costs for individual surpluses (shortages): v (s) = min y {qy : y ≥ s,y∈ Z m +} = m i=1 min yi {qiyi: yi≥ si, yi∈ Z+} = m i=1 q_is_i+ (assuming q≥ 0) withx+:= max{0, x}, x ∈ R SIR value functionv is separable

Assume T deterministic−→Q separablein tender variables Tx

Special case: Simple integer recourse (

W = I )

Genericone-dimensionalexpected value function Q(z) =Eωω − z+, z ∈ R

Q is generallynon-convex

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SIR function Q is non-convex in general. However: Theorem[Klein Haneveld, Stougie, VdV ’06]

SIR function Q is convexif and only if_{ω ∼ pdf f} with f(s) =G(s + 1)−G(s), s ∈ R whereG is an arbitrary cdfwith ﬁnite mean

−→ Idea forMRD:

Approximateoriginal pdf f_ωwith

apdf ˆf that isgenerated by some cdf G

α-approximations

Fixα ∈ [0, 1)

LetG be cdf of adiscrete r.v. with support inα + Z:

1. f_α(s) = G (s + 1)− G(s)isconstanton C_αl := (α + l − 1, α + l], l ∈ Z 2. Forω_α∼ f_α,P{ω_α∈ C_αl} = P{ω ∈ C_αl} −3 −2 −1 0 1 2 3 0.0 0.1 0.2 0.3 0.4 density

For later use: analogous for dimension m≥ 2

α-approximations (2)

−→ α-approximationQ_α:

Q_α(z) :=E_ω_αωα− z+, z ∈ R withωα∼ fαa continuous random variable. It turns out that

Q_α(z) :=E_ω_αωα− z+= _∞

z (1−G(x )) dx

⇒ Qα(z) =Eξ(ξ− z)+, z ∈ R That is: Q_αis acontinuousexpected surplus function (SR) withdiscrete random variableξhavingcdf G.

Theorem[Klein Haneveld, Stougie, VdV ’93]−→

Everyconvex SIRfunction Q withcontinuousω, can be represented as ancontinuous SRfunction withdiscreteξ

α-approximations (3): MRD for SIR

Drop integrality in second stage

Replaceω ∼ f by α-approximation ξ = ω − α + α

⇒

1. continuousSR models withdiscrete_{ω − α + α}

can be solved eﬃciently

2. For SIR models auniform error boundis available for

α-approximations

Theorem [Klein Haneveld, Stougie, VdV 2006] ForSIR function Qand for allα ∈ [0, 1),

sup z∈R|Q(z) − Qα(z)| ≤ min |Δ|fω 4 , 1

with|Δ|f_ω:= total variation of pdf f_ω

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MRD /

α-approximation for more general integer recourse

Expected value functionQ(z) Q(z) =_Eω min y {qy : Wy ≥ω− z, y ∈Z n2 +}

MRD: (i) drop integrality, (ii) substituteα-approximation of ω Forα ∈ Rm,α-approximation Qα(z) Q_α(z) =Eω min y {qy : Wy ≥ω − α + α− z, y ∈R n2 +} Q_α is a continuous recourse function, convex polyhedral ConsiderW complete, Totally Unimodular (TU)

second-stage problem: IP(s) = LP(s)provided s∈ Zm

Convex hull

Claim Van der Vlerk (Math Prog, 2004)

Q_αis the convex hull of Q if W is TU (_αdepending on f_ωonly)

Counterexamplewhere the convex hull is not polyhedral_{= Q}_α

Q(z) =_Eω_{ω − z}+ ω follows a triangular distribution on [0, 1] with mode 1_/2 −0.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0

Conclusion: Claim needs stronger assumptions

Known results

α-approximations of TU integer recourse

Claim of VdV (2004) holds foruniform distributions

[Romeijnders & VdV ’12]

Q(z) = Q_α(z) for z∈ α + Zm

Recall: Error boundonly for SIR[KH et al. ’06]

Forα ∈ R, sup z∈R|Q(z) − Qα(z)| ≤ min |Δ|f 4 , 1 |Δ|f is the total variation of pdf f

Goal

Obtain a similar error bound forTU integer recourse

Error bound for

α-approximations

New approach for SIR

Let f₀be a pdf with|Δ|f₀= B.

We are interested in sup_α,z∈R|Q(z) − Q_α(z)| when ω ∼ f₀

Key insight: Instead, consider

sup α∈R sup z∈R sup f ∈F{|Q(z) − Qα (z)_{| :}_{|Δ|f ≤ B}_}

F is set of ‘nice’ density functions f (bounded variation, . . . )

Analysis

1. Round-upfunctions R(z) :=Eω[ω − z], z ∈ R

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Property of total variation

Lemma (‘ﬂattening’)

Letpdf f ∈ Fbe given.

Let I be a bounded interval and deﬁneg∈ F as

g (x ) = f (x ), x /∈ I K_I, x ∈ I with K_I:=|I |−1_If (u)du Then|Δ|g ≤ |Δ|f

Round-up functions

Expected round-up function andα-approximation R(z) :=Eω[ω − z] R_α(z) :=Eω[ω − α + α− z] Error R(z)− R_α(z) = Eω ω − z − Eω ω − α + α− z = E_ω φα,z(ω) withdiﬀerence function

φα,z(x ) :=_{x − z + z}₋_{x − α + α}

The diﬀerence function

φ

_α,z

φα,z(x) := x − z + z−x − α + α Solve max α,z,f{|Ef[φα,z(ω)]| : |Δ|f ≤ B} 0.0 0.5 1.0 1.5 2.0 2.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Properties ofφα,z periodic in x, α, z with period 1 _φ_α,z(x ) =_−φ_z,α(x ) Consequences Restrict maximizationw.r.t. α and z to [0, 1) MaximizeEf[φ_α,z(ω)] instead of|E_f[φ_α,z(ω)]|

The diﬀerence function

φ

_α,z

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The diﬀerence function

φ

_α,z φα,z(x ) := x − z + z−x − α + α Solve max α,z,f{|Ef[φα,z(ω)]| : |Δ|f ≤ B} 0.0 0.5 1.0 1.5 2.0 2.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Properties ofφα,z _Iφα,z(x )dx = 0for any interval I of length|I | = 1 Consequences Optimal f are

(piecewise constant and)

alternating(high – low)

The diﬀerence function

φ

_α,z

φα,z(x ) := x − z + z−x − α + α Solve max α,z,f{|Ef[φα,z(ω)]| : |Δ|f ≤ B} 0.0 0.5 1.0 1.5 2.0 2.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Properties ofφα,z _Iφα,z(x )dx = 0for any interval I of length|I | = 1 Consequences

Optimal f are pc and

alternating

The diﬀerence function

φ

_α,z

alternating

The diﬀerence function

φ

_α,z

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Round-up functions

This allows to determine, forα, z ∈ R, max

f ∈F{|Ef[φα,z(ω)]|:|Δ|f ≤ B} = min{γα,z, γα,z(1− γα,z)

B

2} whereγ_α,z := z + 1− (z − α + α)∈ [0, 1]

Finally, maximization w.r.t._{α and z yields} desired uniform error bound on|R(z) − Rα(z)|

Round-up functions

Uniform error bound: forα ∈ R sup

z∈R|R(z) − Rα(z)| ≤

|Δ|f

8 In fact, for |Δ|f ≥ 4, sup

z∈R|R(z) − Rα(z)| ≤ 1 −

2 |Δ|f Simple integer recourse

The same error bound holds for SIR: Q(z) =Eω[ω − z+] sup

z∈R|Q(z) − Qα(z)| ≤ |Δ|f

8

Bound is sharp; improvement of

[KH et al. ’06]by factor 2 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 New bound Old bound

Similar approach for recourse with TU matrix

W

Second-stage value functionv: for s ∈ Rm

v (s) := min y {qy : Wy ≥s, y ∈Z n2 +} = min y {qy : Wy ≥s, y ∈Z n2 +} = min y {qy : Wy ≥s, y ∈R n2 +} (because W is TU) = max λ {λs:λW ≤ q, λ ∈R m

+} (by strong LP duality)

Dual feasible region: Λ :={λ ∈ Rm₊:λW ≤ q}

v (s) is ﬁnite for all s∈ Rm⇒ Λ is non-empty and bounded

Extreme pointsof Λ:λ_k, k = 1, . . . , K

v (s) = max k=1,...,Kλ

k_s

TU recourse matrix

W

Expected recourse functionQ: for z ∈ Rm

Q(z) =Eω[v (ω − z)] = E_ω

max k=1,...,Kλ

k_{ω − z} MRD: (i) drop integrality, (ii) substituteα-approximation of ω For_{α ∈ [0, 1)}m Q_α(z) =Eω max k=1,...,Kλ k₍_{ω − α + α}_{− z)}

Q(z)similarto expectedround-upfunctions_λk_Eω[_{ω − z]}

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First error bound for TU integer recourse

Error bound when components ofω areindependent

sup z∈Rm|Q(z) − Qα(z)| ≤ m i=1 λ∗ i|Δ|f₈ i withλ∗_i := max k=1,...,Kλ k i

Special case: simple integer recourse

Approximation is good when all total variations are small

First error bound for TU integer recourse

Notation: x_(i)= (x_{1, . . . , x}_i−1_{, x}_i+1_{, . . . , x}_n)

Error bound when components ofω aredependent

Example:

Bivariate normal distribution with correlationρ. If variances are suﬃciently large and|ρ| ≤ .4 then EB(dep.) _{≤ 1.1 EB(indep.)}

Future research

Construct convex hull of integer recourse models

Extending MRD /α-approximations to General integer recourse (non TU) Mixed-integer recourse

Multi-stage recourse Binary recourse variables ...