Stochastic Integer Programming. An Algorithmic Perspective

Stochastic Integer Programming
An Algorithmic Perspective
Shabbir Ahmed
sahmed@isye.gatech.edu
www.isye.gatech.edu/~sahmed

School of Industrial & Systems Engineering

Shabbir Ahmed 2
SPX Tucson AZ Oct. 9, 2004

Outline
Two-stage SIP
• Formulation
• Challenges
• Simple Integer Recourse
• Structural results
• General integer recourse
• A few Decomposition algorithms

Multi-stage SIP
• Formulation
• Algorithms
• Scenario decomposition
• Polyhedral results

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Two-Stage SIPs
• Decisions in two stages
Stage 1 decision → Observe uncertainties → Stage 2 decision
(“here and now”) (“recourse”)

• Known distribution
The probability distribution of the uncertainties is known

• Exogenous uncertainties
Stage 1 decisions do not affect the distribution

• Discrete/Combinatorial decisions

• Goal: Minimize cost of stage 1 decisions + Expected cost of stage 2
decisions

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Examples

• Resource acquisition (Dempster et al.,1981,1983)
Acquire machines → Observe processing times → Schedule jobs.

• Location-Routing (Laporte et al., 1994)
Locate depots → Observe demand → Route vehicles.

• Ground Holding in Airline Operations (Ball et al., 2003)
Schedule arrival/departure → Observe delays → Decide optimal
holding pattern.

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General Formulation

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Difficulty 1

• Evaluating the second-stage cost for a fixed first-stage
decision and a particular realization of the uncertain
parameters.

• Involves solving (possibly) NP-hard integer program

• E.g. Second stage: schedule jobs after observing processing
requirements.

• Most SIP research assumes away this difficulty.

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Difficulty 2
• Evaluating the expected second-stage cost for a fixed first-stage
decision.
– If the uncertain parameters have a continuous distribution:

involves integrating the value function of an integer program and
is in general impossible.
– If the uncertain parameters have a discrete distribution:

if and each component has 3 independent realizations,
then !!
involves solving a huge number of similar integer programs.

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Difficulty 3
• is the value function of an integer program
• Is non-convex and discontinuous (lower-semicontinuous)

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Difficulty 3
• Optimizing , with respect to .

Theorem (Stougie 1985; Schultz 1993,1995)
If for all and , and then
is real-valued and lower-semicontinuous on .
If , in addition, has an absolutely continuous density, then
is continuous on .

• In general, is non-convex and often discontinuous, and
therefore, so is .

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Non-convexity and Discontinuity

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Simple Integer Recourse
(Stochastic RHS)

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Dealing with the difficulties (SIR)

• No Difficulty 1

where .

• Let and then

• Dealing with Difficulty 2: if we know how to evaluate the univariate
functions and , we are done.

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Dealing with Difficulty 2 (SIR)

Theorem (Louveaux and van der Vlerk, 1993)

and

• In many cases, the above sums are easy to evaluate.
• Separability allows for the easy evaluation of .
• Difficulty 2 resolved.

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Dealing with Difficulty 3 (SIR)
Let then

In general, is not convex.

Theorem (Klein Haneveld et al., 1995)
There exists a random variable such that

Here denotes the convex hull of a function over its entire
domain.

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Example

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Convexification
• Klein Haneveld et al (1995) give an algorithm for constructing the
convex hull in case of discrete distributions.

Theorem (Klein Haneveld, 1995)
If is of full row rank then

• In some cases

• Then, we only need to solve a problem with continuous simple
recourse

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Example

Shabbir Ahmed Klein Haneveld et al., 1995 18

Approximations

• Similar results: by perturbing the distribution, convex (continuous
simple recourse type) approximations/lower bounding functions for
the SIR function can be obtained.
– Uniform error bound on the approximation

• Solve SIPs with SIR approximately or obtain lower bounds via
solving continuous recourse models.
– Can we use these within a branch and bound scheme?
– Can we get convex hull/lower bound restricted to a subset of the
domain?

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General Mixed-Integer Recourse

• Assume that the recourse function is well-defined.

• Dealing with Difficulty 1
– Assume that the second stage MIPs are “easily” solvable
exactly.
– Some literature on using approximations (e.g. Dempster 1983).

• Dealing with Difficulty 2
– Approximate the distribution by a discrete distribution with a
“manageable” number of realizations, e.g., by sampling.
– Still need to solve several similar MIPs.

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(One way of) Dealing with Difficulty 2
The Sample Average Approximation Method
• Need to solve

• Let be the set of -optimal solutions and be the optimal
objective value.

• Generate i.i.d samples and solve the Sample Average
Approximating (SAA) Problem:

• Let be the set of -optimal solutions (with ) and be the
optimal objective value.

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Convergence of the SAA Method

Theorem (Kleywegt et al. 2001)

If is finite then and .

Moreover the convergence is exponentially fast.

The sample size needed to obtain an -optimal solution to the true
problem with probability is

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Convergence of the SAA Method

Theorem (Ahmed and Shapiro, 2002)

If is bounded (not necessarily finite) then the sample size
needed to obtain an -optimal solution to the true problem with
probability is

For pure integer recourse, only right hand-side uncertainty, and
discrete distribution with scenarios, the sample size needed to
obtain an -optimal solution to the true problem with probability
is

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Practical SAA Method
• Select a sample size , solve independent SAA problems.

• Let and be the optimal value and optimal solution of the I-th
SAA problem.

• A point estimate of a lower bound on is given by
(Mak et al, 1999)

• A point estimate of the objective value of a candidate solution is
where is a large sample.

• Then a point estimate of the optimality gap of the candidate solution
is obtained.

• Variability in the point estimates can be used to obtain confidence
intervals.
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Sample SAA Computation

Two-stage SIP with 108 scenarios.

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Solving Similar IPs
• Still need to solve many “similar” MIPs.

• In case of stochastic linear programming, this problem is tackled
using various warm-start strategies arising from exploiting LP duality.

• Unfortunately, a computationally useful IP duality theory is not yet
mature.

• Two of the approaches in the SIP literature for pure integer second-
stage:
– Gröbner Basis (Schultz et al., 1998)
– Value function construction (Kong et al., 2004)

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Dealing with Difficulty 3
• Assume a finite (“manageable”) number of scenarios.

where

and

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Deterministic Equivalent

• Large-scale MIP.
• If not too many scenarios, use, e.g., CPLEX.
• Otherwise decompose

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Two-Stage Decomposition

Solve
is computationally
tractable lower bounding
approximation of

Evaluation is via
Evaluate decomposition

Refinement

No

Yes
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STOP

Binary First-stage
Integer L-Shaped Method (Laporte & Louveaux, 1993)

Here Let
Denote
Let

Note

Then

and

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Integer L-Shaped Method
Master
Problem

Cuts

Evaluation is via
Evaluate decomposition

Refinement

No

Yes
Shabbir Ahmed 31
SPX Tucson AZ Oct. 9, 2004 STOP

Integer L-Shaped (Remarks)
• Master problem is a 0-1 MIP … solve by branch & bound.
• (Decomposed) MIP subproblems in the evaluation steps.
• Implementation: Do not B&B to optimality … branch-and-cut.
– Add cuts whenever a binary solution is encountered in the
B&B search.
• Cut quality depends on the quality of the lower bound.
– Can be improved if more information is available on the
value function.
• Other types of valid cuts can also be added.
– Standard LP Benders cuts are valid, but weak.
• Finite termination guaranteed.
• Application: Stochastic vehicle routing (Laporte et al. 2002)

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Disjunctive Decomposition
Sen and Higle (2000)
• Binary first-stage, Mixed-binary second-stage and fixed
recourse.
• Goal
– Avoid solving MIP subproblems during evaluation.
– Exploit similarity of subproblems.

• Given , for each solve the LP-relaxation. If solution is
fractional, find a valid inequality for the LP.

Only valid for
current subproblem
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The C3 Theorem
Sen and Higle (2000)
• There exists a function such that the cut can be translated
to be valid for the subproblem corresponding to any scenario
and any first-stage solution.

• The cut-coefficients do not change (“common”).
• However, is piece-wise linear and concave in .

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Convexification
• Since only binary first-stage solutions are relevant, convexify
(linearize) the concave function .

• RHS linear in first-stage variables. Pass Benders cuts to
master.

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The D2 Algorithm (Remarks)

• As long as we have a separation scheme for finding “proper”
valid inequalities for the subproblems, the algorithm
terminates in a finite number of steps with the optimum.

• Application: Server location (Ntaimo and Sen, 2003)

• Sen and Sherali (2004) extend the approach to when the
second-stage problems are partially solved by Branch-and-
cut.

Shabbir Ahmed 36

Mixed-Integer First Stage

• Binary first-stage ⇒ Need to evaluate/approximate value function
only at binary solutions (finiteness inherent).

• Mixed-integer first-stage ⇒ Optimize a non-convex discontinuous
objective over a (semi)-continuous domain.

• If Pure integer Second-stage:

Theorem (Schultz et al.1998)
The expected second-stage value function is piece-wise constant (over
polyhedral regions), and an optimal solution to the problem lies at an
extreme point of one of these polyhedra.

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Pure Integer Second Stage

• Inherent finiteness.

• Algorithm: Check all these
extreme points.

• Difficulties
– Polyhedra not easy to
characterize.
– May be too many.

• Alternative: Use
continuous branch and
bound.

Shabbir Ahmed 38

Continuous Branch & Bound
Objective P Objective P

R R
U

L L

Variable Variable

(a) Lower bounding (b) Upper bounding

Objective P
R

U R
R2
L R1 R1 R2

fathom

subdivide
Variable
Shabbir Ahmed 39
(c) Domain subdivision (d) Search tree

Finiteness Issue

• The most common branching scheme is
rectangular partitioning - branching along
a variable axis.

• The resulting partitions are rectangular.

• The polyhedral shaped discontinuous
pieces cannot be isolated by a finite
number of rectangular partitions.
Finite sequences
• There will be some partitions containing
discontinuities.

A potentially infinite sequence • These will require infinite partitioning for
bounds to converge.

Shabbir Ahmed 40

Pure Integer Second Stage
B&B in the tender space (Ahmed et al., 2004)
• Fixed technology matrix.
• Solve the problem in the space of the tender variables.
• Discontinuous are orthogonal to the tender axes.

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The B&B Algorithm

• Lower bounding: Second-stage value
function is non-decreasing and lsc.

• Upper bounding: Function evaluation. χL χB χU

• Branching: Partition along the
discontinuities. Maintains rectangular
partitions.

χL χB
• Finite convergence to global optima. χB χU

• Improved lower bounding methods.

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Sample Computation

• Test set: Capacity acquisition-assignment problems.
• Ahmed and Garcia, 2003.

Shabbir Ahmed 43

More on Pure Integer Second Stage
• Hemmecke and Schultz, 2003
– Rhs uncertainty. Pure integer first stage.
– IP Test Sets (Computational Algebra).

• van der Vlerk, 2004
– Rhs uncertainty.
– Convex lower bounds by changing the distribution.

• Kong et al., 2004
– Rhs uncertainty. Pure integer first stage.
– Construct and optimize value function.

• Kong et al. 2004
– Conditions for total unimodularity.
– Benders: MIP master may be better than LP master.

Shabbir Ahmed 44

Multi-Stage SIPs

• Decisions in multiple (but, a finite number of) stages

Stage 1 decision → Observe uncertainties → Stage 2 decision →
Observe uncertainties → Stage 3 decision → …..

• Example Applications:
– Unit commitment (Takriti et al. 1996, Caroe & Schultz, 1999).
– Capacity/Production planning (Ahmed et al. 2001, Lulli and Sen,
2002 ).
– Asset liability management (Drijver et al., 2000).

• Challenges: Same as before … but now in multiple-folds!

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The Scenario Tree
• Assuming finite support, the evolution of the uncertain parameters
can be modeled as a scenario tree.

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A Formulation

• Deterministic equivalent formulation.
• Solution at a node depends on the solutions in ancestor nodes.
• Index according to node – a tree formulation.

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Scenario Decomposition

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The Scenario Formulation

Non-anticipativity constraints

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Lagrangian Relaxation

• Any feasible solution to the Lagrangian dual provides a lower bound
to the true optimal value.

• Evaluating the dual function requires solving one deterministic
problem per scenario (Decomposition).

• Dual involves maximizing a concave non-smooth function … can be
solved using non-smooth optimization techniques.

• Difficulty: Many dual multipliers.

• Difficulty: Duality Gap
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Dual Decomposition
Caroe and Schultz (1999)

• Use Lagrangian dual as the lower bounding scheme within a branch
and bound algorithm.

• Branch to enforce non-anticipativity.

• Finite termination in case of pure integer solutions.

• Applicable to two-stage stochastic integer programs.

• Application: Unit commitment problem.

Shabbir Ahmed 51

Polyhedral Results

• Given valid inequalities for a
deterministic MIP, can we find a valid
inequality for the stochastic counterpart?

• Generating “tree” inequalities from
“path” inequalities.

• Branch and cut schemes for
deterministic equivalent.

• Tighten subproblems within
decomposition based branch and cut
schemes.

Shabbir Ahmed 52

Example: Uncapacitated Lot-sizing

• Common substructure in
many production
planning problems

• (l,S) inequalities sufficient
to describe convex hull.

• Exponential family …
polynomially separable.

Shabbir Ahmed 53

Stochastic Uncapacitated Lot-sizing

Guan et al. (2004)

• Given any subset of the
nodes, the corresponding
(l,S) inequalities are valid.
• These inequalities can be “combined” to generate a new family of
inequalities.
• Necessary and sufficient conditions for the inequalities to be facet-
defining.
• Excellent performance within branch & cut.
• The combining idea is quite general and can be applied to other SIPs.

Shabbir Ahmed 54

Concluding Remarks

• Would have liked to talk about Approximation Algorithms for SIP.

• Survey articles on SIP:
– Klein Haneveld and van der Vlerk (1998)
– Louveaux and Schultz (2003)
– Schultz et al. (1995)
– Sen (2004)

• WWW Resources:
– SP Community Page: http://www.stoprog.org
– Bibliography (2003): http://mally.eco.rug.nl/biblio/SIP.HTML
– Test Problems: http://www.isye.gatech.edu/~sahmed/siplib/

Shabbir Ahmed 55

References
S. Ahmed and A. Shapiro. The sample average approximation method for stochastic programs with integer recourse.
Optimization Online, http://www.optimization-online.org, 2002.
S. Ahmed and R. Garcia. Dynamic Capacity Acquisition and Assignment under Uncertainty. Annals of Operations
Research, 124:267-283, 2003
S. Ahmed, M. Tawarmalani, and N. V. Sahinidis. A finite branch and bound algorithm for two-stage stochastic integer
programs. Mathematical Programming, 100:355-377, 2004.
M. O. Ball, R. Hoffman, A. R. Odoni, and R. Rifkin. A stochastic integer program with dual network structure and its
application to the ground-holding problem. Operations Research, 51(1):167-171, 2003.
C. C. Carøe and R. Schultz. Dual decomposition in stochastic integer programming. Operations Research Letters,
24(1-2):37-45, 1999.
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of hierarchical planning systems. Operations Research, 29:707-716, 1981.
M.A.H. Dempster, M.L. Fisher, L. Jansen, B.J. Lageweg, J.K. Lenstra, and A.H.G. Rinnooy Kan. Analysis of heuristics
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Research, 8:525-537, 1983.
R. Hemmecke and R. Schultz. Decomposition of test sets in stochastic integer programming. Mathematical
Programming, 94:323-341, 2003.
Y. Guan, S. Ahmed and G.L. Nemhauser. A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing
problem. Stochastic Programming E-Print Series, 2004.
W.K. Klein Haneveld, L. Stougie and M.H. van der Vlerk. On the convex hull of the simple integer recourse objective
function , Annals of Operations Research, 56:209-224, 1995.
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Oper. Res., 85:39-57, 1999.
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Discrete Optimization. SIAM Journal on Optimization, 12:479-502, 2001.
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Superadditive Dual Approach, Technical Report, University of Pittsburgh, 2004.
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Shabbir Ahmed 56

References
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Operations Research Letters, 13:133-142, 1993.
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algorithm. Transportation Science, 28(2):95-103, 1994.
G. Laporte, F.V. Louveaux, and L. van Hamme. An integer L-shaped algorithm for the capacitated vehicle routing
problem with stochastic demands. Operations Research, 50:415-423, 2002.
F. Louveaux and M.H. van der Vlerk. Stochastic programming with simple integer recourse, Mathematical
Programming, 61(3):301-325, 1993.
W.K. Mak, D.P. Morton and R.K. Wood. Monte Carlo Bounding Techniques for Determining Solution Quality in
Stochastic Programs. Operations Research Letters, 24:47-56, 1999.
L. Ntaimo and S. Sen. The Million-Variable “March” for Stochastic Combinatorial Optimization. To appear in the Journal
of Global Optimization, 2004.
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Research, 18(3):578-589, 1993.
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recourse. Mathematical Programming, 70:73-89, 1995.
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Neerlandica, 50(3):404-416, 1996.
R. Schultz, L. Stougie and M.H. van der Vlerk. Solving stochastic programs with complete integer recourse: a
framework using Groebner Bases , Mathematical Programming, 83(2):229-252, 1998.
S. Sen and J. Higle. The C3 Theorem and a D2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set
Convexification. Technical Report, University of AZ. 2003.
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Programming. Technical Report, University of AZ. 2004.
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L. Stougie. Design and analysis of algorithms for stochastic integer programming, PhD Thesis, Center for Mathematics
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99(2):297-310, 2004.

Shabbir Ahmed 57

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