Models and Algorithms for Stochastic Programming, Financial Optimization

8/4/2019 Models and Algorithms for Stochastic Programming, Financial Optimization

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Models and Algorithms for Stochastic

Programming

Jeff Linderoth

Dept. of Industrial and Systems EngineeringUniv. of Wisconsin-Madison

[email protected]

Enterprise-Wide Optimization MeetingCarnegie-Mellon University

March 10th, 2009

Jeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 1 / 82


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Mission Impossible

ExplainingStochastic

Programming in90 mins

I will try to givean overview please interrupt

with questions!



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What Ill Ramble On

Models

How to deal with uncertainty

Why modeling uncertainty is important

Who has used stochastic programming?

Why more people dont use stochastic programming

Algorithms

Extensive Form

Benders Decomposition (2-stage)

Sampling

Nested Benders Decomposition (multistage)



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Dealing with Uncertainty Definition of Stochastic Programming

Etymology

program:

(3) An ordered list of events to take place or procedures to be followed; ascheduleLate Latin programma, public notice, from Greek programma, programmat-, from

prographein, to write publicly

stochastic:

(1b) Involving chance or probabilityGreek stokhastikos, from stokhasts, diviner, from stokhazesthai, to guess at, from

stokhos, aim, goal.

Source: The American Heritage Dictionary of the English Language, Fourth

Edition.


D li i h U i S f U i


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Dealing with Uncertainty Sources of Uncertainty

Sources of Uncertainty

Houston, we have uncertainty!

What we anticipate seldom occurs; what we least expected

generally happens.Benjamin Disraeli (1804 - 1881)

FinancialMarket price movementsDefaults by a business partner

Operational

Customer demands,

Travel timesTechnology related

Will a new technology beready in time

Market Related

Shifts in tastes

Competition

What will your competitorsstrategy be next year?

Acts of God: Jeffs travelexperience yesterday!!!

WeatherEquipment failureBirds flying into planes


D li ith U t i t S f U t i t


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Stochastic Programming

A tool used in planning under uncertainty

More specifically: Mathematical Programming, or Optimization, inwhich some of the parameters defining a problem instance arerandom, or uncertain

Optimization

minxX

f(x)

x: Variables you control

Stochastic Optimization

minxX()

F(x, )

: Variables you dont control

Stochastic Optimization is UNDEFINED

You cant possibly choose an x that optimizes for all

More specifiation is required




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Jeffs Stochastic Programming Assumptions

In stochastic programming, we assume that a probability distribution

for the uncertainty is known or can be approximated.We also assume that probabilities are independent of the decisionsthat are taken.

Decision-dependent uncertainty

Decisions influence probabilitydistributions

Decisions influence knowledgediscovery

Want to know about stochasticprogramming withdecision-dependent uncertainty?

Talk to Ignacio!




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Probability Theory(?)

This notion of having to know a probability distribution for therandomness is troubling, since in reality, very few people exactly knowthat

Their customer demands follow a log-normal distribution with mean

17.26 and variance 2.88726Their plant will have forced shutdowns following a Weibull distributionwith parameters (100.25, 73.7916)

Instead, you might be able to

Estimate distributions from historial data (be careful!)

Have qualitative probability measures (low/medium/high)Create your own scenarios of interest




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The Journey is the Reward?

Business process people can argue/discuss amongst themselves whatthe various scenarios might be and the outcomes of those scenarios.

This process by itself can be very useful

There is a good amount of frightening-looking mathematical theoryand computational evidence that solutions obtained from stochasticprograms are often quite stable with respect to changes in theinput probability distribution

The Upshot

It doesnt matter too much if your numbers arent quite right

The insights you gain from considering the uncertainty can still bevaluable


Dealing with Uncertainty Related Decision Making Technologies


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A Concrete Example: An Uncertain LP

min cx

s.t. Ax b

T()x h()

x 0

T() and h() are uncertain: X() = {x | Ax b, T()x h()}We must choose x despite this uncertaintyExamples:

Decide production quantities before knowing demandsConstraint data includes imprecise measurements

Three Approaches

1 Robust optimization

2 Chance-constrained programming

3 Recourse-based stochastic programming




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g y g g

Robust Optimization

Uncertain data is assumed to lie in an uncertainty set

(T(), h()) U

Guarantee that constraints be satisfied for all possible realizations

min cx

s.t. Ax b

Tx h (T, h) U

x 0

Tractability depends on structure of U




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g y g g

Robust Optimization

To control conservatism, uncertainty set can be parameterized by abudget of uncertainty

Example 1: Tij() [lij, uij] (Bertsimas and Sim)

At most K of the components in each row can differ from the nominalvalueNature can choose which K will differK large

highly conservative (Soyster)

K = 0

No robustness

Can formulate this problem as a linear program

Example 2: U is ellipsoidal (Ben-Tal and Nemirovski)




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g y g g

Robust Optimization

Advantages:

Computationally tractable

Can yield extremely reliable solutions

Does not require stochastic model

Disadvantages:

Does not use a stochastic model

Although conservatism can be controlled, the control parameterdoesnt have meaning to decision makers




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Assume uncertain data are random variables with known distributionsTwo approaches to uncertain constraints:

1 Require constraint to be satisfied with high probability

min{cx : x

X, P{T()x

h()}

1 }

is a parameter, e.g. = 0.05 or = 0.01Linear program with probabilistic (chance) constraints

2 Penalize violations of constraints

min

cx + E[(h() T()x)+] : x X

Special case of a Two stage stochastic program




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Linear Programs with Probabilistic Constraints

Individual constraints:

min

cx : x X, P{T()ix h()i} 1 i i

Joint constraints:min{cx : x X, P{T()x h()} 1 }

Bad news: calculating probability is hard

Worse news: probabilistic constraints are generally non-convex!




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Non-convexity of the feasible region

Consider: P{x1 1, x2 2} 0.6

Each dot: a realization of which occurs with probability 1/10

x2

x1




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Two Stage Stochastic Programming

(SP) mincx + E[(h() T()x)+] : x XChoose x Observe (T(), h()) Pay penalty

Good news: (SP) is convex

Bad news: Calculating expectation is hard

Successful Approach: Sample Average Approximation

Generate (T()1, h()1), . . . , (T()N, h()N) and solve

(SPN) mincx +Ni=1

1

N (h()i

T()i

x)+

: x XxN is a often a good approximation to true optimal solution

Well see (a lot) more later!




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Stochastic Programming vs. Simulation

Simulation(Pro): Very flexibleSystem need not be mathematically defined(Pro): Fast(Con): If I run 100 what-ifs and get 100 different solutions, howdoes simulation help me plan for the future?

Stochastic Programming(Con): More challenging to build and solve models(Pro): SP helps you optimize over your what-ifs.

The Upshot!

Use simulation to generate scenarios. Input the scenarios to a stochas-tic program to show how to decide how to best hedge against thisuncertainty




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Multistage Decision Making

1

x1

2

x2

3

xT1

T

xT

Random vectors

1 Rn1 , 2 Rn2 , . . . , T R

nT

Make sequence ofdecisions x1 X1, x2

X2, . . . , xT XT.The evolution of information is of fundamental importance to thedecision-making progress.

We make a decision now (x1)

Nature makes a random decision 2: (stuff happens)

We make a second period decision x2 that attempts to repair thehavoc wrought by nature in (recourse).

Repeat as necessary...

We make decisions in stages, in between which uncertainty is revealed

to usJeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 19 / 82

Why use Stochastic Programming The Newsvendor


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Hot Off the Presses

A paperboy (newsvendor) needs to decide how many papers to buy inorder to maximize his profit.

He doesnt know at the beginning of the day how many papers he cansell (his demand).

Each newspaper costs c.

He can sell each newspaper for a price of q.He can return each unsold newspaper at the end of the day for r.(Obviously r < c < q).

The Newsvendor Problem

Given only knowledge of the probability distribution F of demand,how may papers should the newsvendor buy?




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Newsvendor Problem

Suppose that the newsvendors goal is to maximize the profits in thelong run. (In expectation)...

Intuitively, it seems that the newsvendors best strategy is to everypurchase the average demand

Take Away Message!

The optimal solution is NOT to use the mean demand.

In fact, the two solution can be far apart. (Depending on thedistribution, and parameters r,c,q




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ExampleThe Newsvendor

c = 50,q = 70,r = 5

Demand: (Truncated) Normal distributed. = 100, = 50

Mean Value Solution

Buy 100. (Duh!)Expect to profit: 2000

TRUE long run profit 650Stochastic Solution

Buy 75.Expect to profit: 1500TRUE long run profit 880

The difference between the two solutions (880 650) is called thevalue of the stochastic solution.

How much is it worth to you to plan using full uncertainty informationas opposed to mean-values for the uncertain parameters




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A Take Away Message

The Flaw of AveragesThe flaw of averages occurs when uncertainties are replaced bysingle average numbers planning.

Did you hear the one about the statistician who drowned fording ariver with an average depth of three feet.

Point Estimates

If you are planning with point estimates for demands, then you areplanning sub-optimally

It doesnt matter how carefully you choose the point estimate itis impossible to hedge against future uncertainty by consideringone realization of the uncertainty in your planning process


Stochastic Programming Success Stories Financial Optimization


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Russell-Yasuda Kasai

Yasuda Kasai: Seventh largest (worldwide) property and casualtyinsurer.

Assets of > 3.47 trillion

Liability structure is complex, but want a tool that will allow them tomaximize the revenue from these assets in the face of assetmanagement restrictions

Frank Russell Company hired to develop Asset-Liability ManagementModel based on (multistage) stochastic programming

Carino, Myers, Ziemba, Second place in Edelman prize competition ofINFORMS.




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Asset Allocation Model

Decisions:Investment amounts for various assets

Random Events:Return on investment for each asset.Liability payouts

Constraints:

Asset Allocation Constraints (Complex)Loan ModelLiability Model

Compared to a performance benchmark established at YasudaKasai at the beginning of the Fiscal Year to measure the value

added by their use of the model, the new model increased annual

income by9.5 billion.

Mr. Kunihiko Sasamoto, Director and Deputy President, Yasuda Kasai.Jeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 25 / 82


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Product Portfolio Planning

Decisions:

Invest in various projects (All or nothing investment).Complicated project prerequisite structure

Random Events: (HUGE impact)Design-win from customersTechnology failures

Market forcesConstraints:

ResourcesHire-fire costs




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Product Portfolio Management at Agere

We implemented a decision support tool for Agere

1 Optimization Model

2 Simulator of future conditions (random events were correlated!)

The muckety-mucks loved it!

They like the ability to talk about the different scenarios.

Focuses discussion in business planning meetings

Gives unbiased simulator view of potential outcomes of decisions


Stochastic Programming Success Stories Logistics


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SP in the Supply Chain

Decisions:

Regular supply chain decision: How much? where? and when?

Random Elements:

Demands, prices, resource capacity.Supply chains going global imply that companies are now more exposedto risky factors such as exchange rates and reliability of transferchannels.

Constraints:

Regular supply chain constraints: Flow balance, material availability,etc.




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A Case Study

T. Santoso, S. Ahmed, M. Goetschalckx, and A. Shapiro. A Stochastic ProgrammingApproach for Supply Chain Network Design under Uncertainty, European Journal of Operational Research, vol.167, pp.96-115, 2005.

Two real supply chains

One Domestic (Cardboard packages to breweries and soft drink

manufacturers...)One global

Sizes: Around 100 facilities. Around 100 customers,

In general, the (sampled) stochastic model was roughly 5% better

than using the mean value of demand, translating into millions ofdollars in potential savings.



S C


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Supply Chain Projects

Bulk Gas Production and DistributionUncertainty in customer demands,competitor drainBuilt (prototype) optimization modeland simulator.

They are now(?) doing a realimplementation

Lesson Learned

Having a (static) simulation of the production-disribution process is akey component to the project


Stochastic Programming Success Stories Other Industries

O h I d i l A li i f SP


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Other Industrial Applications of SP

Energy Industry

Unit Commitment Problem: Schedule production from powergeneration units

Telecommunication

Capacity/bandwidth planning: Invest in capacity for the network before

you know the true bandwidth demandsMilitary

Network Interdiction Problem: Where to place agent on a network tointerrupt evil-doers

It aint that rosy

As far as I know, mot implementations are built on a case-by-case basisand are fairly ad-hoc.


Why More People Dont Use SP

S h i P i Obj i Ri k P fil


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Stochastic Programming ObjectivesRisk Profile

What is your goal?

1

I want to do well on averageExpected Value

2 I want to limit my exposure in the worst case or cases

Value at Risk/Conditional Value at Risk

3 I want the probability that I achieve a goal to be sufficiently high?

Chance constraints

4 I want to achieve a steady return?

Dispersion-based objectives

Each of these imply a different notion of risk, and lead to differentstochastic optimization problems

Stochastic Programming isnt about getting a number, its aboutgetting a distribution that looks good to you



S SP Obj ti


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Some SP Objectives

min F(x, ) Mean-Value Problem

minEF(x, ) Risk Neutral

minEF(x, ) (F(x, )) Risk Measures

(F(x, )) = VarF(x, ) Markowitz(F(x, )) = E [(EF(x, ) F(x, ))+] Semideviation



Thi P l W t


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Things People Want

Arbit

rary Distri

butio

ns

(Conditional)Value atRisk

Netw

orkPro

blems

Scenario Trees

StochasticDynamic Programming

Robust O

ptimization

(Joint)

Chan

ceCons

traints

Stochastic

Dominance

Stochastic Control

JointDistributions

Nonlinear problems

Intege

r prob

lems

Free Beer



S ti St h sti P s


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Supporting Stochastic Programs

I point out all these different flavors of SP to highlight what I thinkhas been one of the hinderances of having a modeling laguage for SP.

I dont know the key to success, but the key to failure is trying to

please everybody.

Bill Cosby (1937 - )

I believe the fact that a stochastic program is not a well-definedconcept is one of the fundamental reasons why more people dont use

stochastic programmingOther reasons people dont use stochastic programming?



Why Dont More People Use Stochastic Programming


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Why Don t More People Use Stochastic Programming

They dont start their training early enough!

Jacob Linderoth, age 4 months, reading Introduction to StochasticProgramming


Why More People Dont Use SP Barriers to Stochastic Programming

Why Dont More People Use Stochastic Programming


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Why Don t More People Use Stochastic Programming

Because they dont know the probability distribution?

Even crude approximations can help

Because they cant solve them?

Linderoth and Wright solve a 10-million scenario problem

Recent theory suggests that you dont need to include many scenariosto get an accurate solution to the true problem

Because they cant model them?

Modeling tools are on the way (more later)

Because it is hard to verify that the solution is better

The same could be said of Deterministic OptimizationUse simulation to verify that the solution is better


Why More People Dont Use SP Barriers to Stochastic Programming

Probability Management


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A true believer is Sam Savage (consulting professor at at Stanford).

He believes companies should have a comprehensive probabilitymanagement plan.


Simulations to generate distributionsInformation systems to hold distributions of key uncertain inputs

A Chief probability officer responsible for signing off on thedistributions

You can start small...

1 What are your scenarios and distributions?

2 Do you have models that can use this information?


Algorithms


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ALGORITHMS

I focus almost exclusively on two-stage recourse problems


Algorithms Two-Stage Stochastic Programs with Recourse



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A Stochastic Program

minxX{EF(x, )}

2 Stage Stochastic LP w/Recourse

F(x, )def= cTx + Q(x, )

cTx: Pay me now

Q(x, ): Pay me later

The Recourse Problem

Q(x, )def= minqTy

Wy = h() T()x

y 0

Expected Recourse Function:

Q(x)def= E[Q(x, )]

Two-Stage Stochastic LP

min cTx xJeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 41 / 82 Algorithms Extensive Form

Extensive Form


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Extensive Form

Assume = {1, 2, . . . S} Rr,

P( = s) = ps, s = 1 , 2 , . . . , S

Ts T(s), hs = h(s)Then can write extensive form:

cTx + p1qTy1 + p2q

Ty2 + + psqTys

s.t.Ax = b

T1x + Wy1 = h1T2x + Wy2 = h2

... +. . .

...TSx + Wys = hs

x X y1 Y y2 Y ys Y

The Upshot!

This is just a larger linear program

It is a larger linear program that also has special structure


Algorithms Extensive Form

Best-Known Solution Procedure


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Best Known Solution Procedure

M

E

T

H O D


Algorithms Extensive Form

Small SPs are Easy!


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Small SP s are Easy!

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

number of scenarios

Time

Cplex/Extensive Form

Lshaped


Algorithms The LShaped Method

Two-Stage Stochastic Linear Programming


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Two Stage Stochastic Linear Programming

We assume that the P has finite support, so has a finite number ofpossible realizations (scenarios):

Q(x) =

N

i=1piQ(x, i)

For a partition of the N scenarios into sets N1,N2, . . .Nt, let Q[j](x)be the contribution of the jth set to Q(x):

Q[j](x)def= iNjpiQ(x, i)

so then Q(x) =t

j=1Q[j]



Important (and well-known) Facts


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p ( )

Q(x, i), Q[](x), and Q(x) are piecewise linear convex functions of x.

If i is an optimal dual solution to the linear program correspondingto Q(x, i), then TTi i Q(x, i)

gj(x)def=

iNjpiT

Ti i Q[j](x).

Key IdeaRepresent Q[j](x) by an artificial variable j and find supportingplanes for j

j Q[j](xk) + gj(x

k)T(x xk) ()

Point of Decomposition

Evaluation of Q(x) is separable

We can solve linear programs corresponding to each Q(x, i)independently in parallel!



Worth 1000 Words?


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x

Q(x)



Worth 1000 Words


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x

Q(x)

xk



Worth 1000 Words


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x

Q(x)

x1x2



(Multicut) L-shaped method


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( )

M

s1 s2 s3 s4 s5

M

s1 s2 s3 s4 s5

1 Solve the masterproblem M with thecurrent approximation toQ(x) for xk.

2 Solve the subproblems,

(sj) evaluating Q(xk

) andobtaining subgradient(s)to update masterapproximation M

3 k = k+1. Goto 1.

Lets Get Parallel!

Of course, solution of sj can be carried out independently.



Warning!


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If Q(x) is not convex, then this algorithm doesnt work

If you have a integer recourse variables y Zp

Rnp

, the problembecomes significantly more difficult.

Your Options

Give your favorite solver the full extensive form (and pray)

Weak relaxation

Decomposition method: Care and Schultz, Sen

Spatial branch and bound:

Want to know about stochasticinteger programming/spatial branchand bound?

Talk to Nick!




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Does it Work? The Worlds Largest LP

Linderoth and Wright built a fancy decomposition-based solvercapable of running on the grid

Storm A stochastic cargo-flight scheduling problem (Mulvey andRuszczynski)

We aim to solve an instance with 10,000,000 scenarios

x R121, yk R1259

The deterministic equivalent LP is of size

A R985,032,88912,590,000,121



The Super Storm Computer


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Number Type Location184 Intel/Linux Argonne

254 Intel/Linux New Mexico

36 Intel/Linux NCSA

265 Intel/Linux Wisconsin88 Intel/Solaris Wisconsin

239 Sun/Solaris Wisconsin124 Intel/Linux Georgia Tech90 Intel/Solaris Georgia Tech13 Sun/Solaris Georgia Tech

9 Intel/Linux Columbia U.10 Sun/Solaris Columbia U.

33 Intel/Linux Italy (INFN)

1345



TA-DA!!!!!


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Wall clock time 31:53:37CPU time 1.03 Years

Avg. # machines 433

Max # machines 556Parallel Efficiency 67%

Master iterations 199CPU Time solving the master problem 1:54:37

Maximum number of rows in master problem 39647



Number of Workers


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0

100

200

300

400

500

600

0 20000 40000 60000 80000 100000 120000 140000

#workers

Sec.


Sampling

Why Sampling is Necessary


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ys y(s) is the recourse action to take if scenario s occurs.

Pro: Its a linear program.

Con: Its a BIG linear program.

Imagine the following (real) problem. A Telecom company wants toexpand its network in a way in which to meet an unknown (random)

demand.

There are 86 unknown demands. Each demand is independent andmay take on one of five values.

S = || = 86k=1(5) = 586 = 4.77 1072

The number of subatomic particles in the universe.How do we solve a problem that has more variables and moreconstraints than the number of subatomic particles in the universe?


Sampling

But Its Even Worse!


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The answer is we cant!If is not a countable set say if it is made up of continuous-valuedrandom variables, our deterministic equivalent would have variables and constraints. :-)

We solve an approximating problem obtained through sampling.

The Very Good News

Using Monte-Carlo methods (Sample Average Approximation), wecan obtain high-quality solutions

Even Better: Can obtain (statistical) bounds on the quality of thesolution


Sampling

Sample Average Approximation(SAA)


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The StorySolving two-stage SP exactly is often impossible

Solving two-stage SP approximately is often easy: Sample AverageApproximation (SAA)

I view SAA as the Jeff Linderoth of solution methods

It aint smartIt aint sexyBut it generally does work!


Sampling

SAA for Dummies


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Let v be the optimal solution to the true problem:

v

def= minxXf(x) def= EF(x, )

Take a sample (1,...,N) of N realizations of the vector , andform the sample average function

fN(x) def= N1Nj=1

F(x, j)

For Stochastic LP w/recourse, evaluate fN(x)

solve one LP for each

of N scenarios

Optimize sample average function:

vNdef= min

xX

fN(x)

def= N1

Nj=1

F(x, j)


Sampling

SAA for Dummies, Cont.


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Note that vN is a random variable, as it depends on the (random)sample of size N

From this information, we can get bounds on the optimal solution

valuev

All Good Talks Contain...

Thm. E(vN

) v f(x) x


Sampling

Making SAA Work


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Take a solution x from a SAA instanceWe are mostly interested in estimating the quality of a given solution

x. This is f(x) v.

1 Get upper bound on v from f(x). Estimate f(x) by solving N

(completely independent) linear programsrecourse LPs with x fixed.

fN(x) def= (N)1Nj=1

F(x, j)

2 Get a lower bound on v from E(vN). Estimate E(vN) by solving Mindependent stochastic LPs, giving optimal values v1N, v

2N, . . . v

MN

E(vN)def= M1

Mj=1

vjN

Independent no synchronization good for the GridJeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 61 / 82

Sampling

More Theory


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A very interesting result of Shapiro and Homem-de-Mello says thefollowing:

Suppose that x is the unique optimal solution to the true problemLet xN be the solution to the sampled approximating problemUnder certain conditions, the event (xN = x

) happens withprobability 1 for N large enough.The probability of this event approaches 1 exponentially fast asN !!There exists a constant such that

limN

N1 log[1 P(x = x)] .

This is a qualitative result indicating that it might not be necessary tohave a large sample size in order to solve the true problem exactly.For a problem with 51000 scenarios a sample of size N 400 isrequired in order to find the true optimal solution with probability95%!!!


Sampling

Does SAA Work on Real Problems?


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M = 10 times Solve a stochastic sampled approximation of size N.

Compute confidence interval on lower bound estimate E(vN)

Choose one x from solution to M SAA instances and computeconfidence interval on upper bound estimate fN(x), with N = 10000

Test InstancesName Application ||LandS HydroPower Planning 106

gbd Aircraft Allocation 6.46 105

storm Cargo Flight Scheduling 6 1081

20term Vehicle Assignment 1.1 1012

ssn Telecom. Network Design 1070


Sampling

20term Convergence


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251500

252000

252500

253000

253500

254000

254500

255000

255500

10 100 1000 10000

Value

N

Lower BoundUpper Bound


Sampling

ssn Convergence


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2

4

6

8

10

12

14

16

18

10 100 1000 10000

Value

N



Sampling

storm Convergence


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1.544e+06

1.545e+06

1.546e+06

1.547e+06

1.548e+06

1.549e+06

1.55e+06

1.551e+06

1.552e+06

1.553e+06

1.554e+06

1.555e+06

10 100 1000 10000

Value

N



Sampling

gbd Convergence


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1500

1550

1600

1650

1700

1750

1800

10 100 1000 10000

Value

N

Lower Bound

Upper Bound


Multistage SPs

Multistage Stochastic LP


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1

x1

2

x2

3

xT1

T

xT

Random vectors1

Rn1 , 2

Rn2 , . . . , T RnT

Make sequence ofdecisions x1 X1, x2 X2, . . . , xT XT.

Risk Neutral: We always aim to optimize the expected value of ourcurrent decision xt

Linear: Assume Xt are polyhedra

Discrete: Assume t are drawn from a discrete distribution.

The Hard Part

Decisions made at period t (xt) must only depend on events and decisionsup to period t


Multistage SPs

The Stickler. My Favorite Eight Syllable Word.


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We need to enforce nonanticipativity.

Other eight-syllable words...autosuggestibility, incommensurability, electroencephalogram,unidirectionality

At any point in time, different scenarios look the same

We cant allow different decisions for these scenarios.We are not allowed to anticipate the outcome of future random eventswhen making our decision now.

How to do it?

1 Use Tree Structure (Nested Decomposition)

2 Create (extra) variables for all possible scenarios, and enfroceequality between decisions that should be nonanticipative(Progressive Hedging)


Multistage SPs

Scenario Tree


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xnx(n)

x01

2

N: Set of nodes in the tree

(n): Unique predecessor of noden in the tree

S(n): Set of successor nodes of n

qn: Probability that the sequence

of events leading to node n occurs

xn: Decision taken at node n

Warning!

Scenario Trees can get bigThere are some tools that try and prune the tree while keepingsimilar statistical properties in the stochastic process


Multistage SPs

Multistage Stochastic Programming

E i F


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Entensive Form

zSP = minnN

qncTnxn Tnx(n) + Wnxn = hn n NValue Function of node n

Qn(x(n))def= min

xn

cTnxn +

mS(n)

qmnQm(xn) | Wnxn = hn Tnx(n)

qmn: conditional probability of node n given node m

Tree structure encodes nonanticipativity


Multistage SPs Algorithm

Nested Decomposition


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0: Root node of the scenario tree

x0: Initial state of the system

Recursive Formulation

zSP = Q0(x0)

Cost to go: Gn(x)def=

mS(n) qmnQm(x)

Mkn(x): Lower bound on Gn(x) in iteration k

Qn(x(n)) minxn

cTnxn + M

kn(xn)

Wnxn = hn Tnx(n)

((MLPn))



Building Mkn(x)


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Create a partition (or clustering Cn) of S(n)

A lower bound mkn[j]

for each element of the partition (each cluster)is created independently

Mkn(x)def= jCn m

kn[j](x)

mkn[j](x)def= inf

j

je Fkn[j]x + fkn[j]

Fk

n[j], fn[j] obtained from dual solutions (to form subgradients) oflinear programs of nodes within cluster [j]

Mkn(x)

Gn(x

)



Action Pictures


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x0

1

x0Jeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 74 / 82 Multistage SPs Algorithm

A small Multistage Telecom Problem


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A

B C D

E F

Set of stages T

Set J of links

Sets It of demands

Random demand dt() R|It|

Budget each period

Install capacity on links eachperiod to minimize the total

expected unserved demand




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Some (Limited) Computational Results

T = 5

K: Realizations/Period

N: Number of scenarios

DE: Size of deterministicequivalent

K N DE Size

30 0.81M 18M * 31M50 6.25M 140M * 236M60 12.9M 290M * 488M



Computational Results


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It: Number of iterations (Times MLP0 was solved)

E: Parallel efficiency.

Time machines solving MLPn

Time machines available

K It Avg Workers Wall Time CPU Time E30 9 62 2:34:21 6:15:15:10 6750 7 75 1:12:49:27 85:20:24:15 77

60 11 162 3:16:51:00 431:12:15:37 73


Multistage SPs Modeling Tools

Existing Modeling Tools


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Many stochastic programming implementations Im aware of have

been built from scratch

But there are some modeling tools on the way

Name Author(s) CommentAIMMS AIMMS Team CommercialGams Gams Team CommercialMPL Kristjensen Commercial

XPRESS-SP Verma, Dash Opt. Commercial, BetaSPiNE Valente, CARISMA

STRUMS Fourer and Lopes Prototype(?)

SUTIL Czyzyk and Linderoth C++ classesSLPLib Felt, Sarich, Ariyawansa Open Source C Routines

COIN-Smi, SP/OSL COIN, IBM C++ methods



Existing Solution Tools


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Most stochastic programming implementations of which Im aware,merely form and solve extensive form

Other software:

Name Author(s) CommentAIMMS AIMMS Team Commercial, LShaped method

SLP-IOR Kall, Mayer LShaped, Stochastic Decomposition, othersMSLiP Gassmann Nested LShapedSPInE Valente, CARISMA Commercial, LShaped method, may not exist anymoreBNBS Altenstedt Nested LShaped method, Open source

ATR Linderoth, Wright Design to run in parallel. Not simple to build and run



Conclusions



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A tool for decision making under uncertainty

Considers the impact of recourse decisionsIt may not be the answer, but it does help you hedge againstupcoming uncertaintyMore importantly, it gets people talking about the impact ofuncertainty in the decision making process

Planning with mean-value estimates will not lead to an optimalpolicy

Used with some success in industry

Financial Services (Many successes)Logistics and Supply Chain (Fewer successes, but coming!)

Tools and algorithms are on the way



We Want YOU!


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To consider using StochasticProgramming as a decisionsupport tool to help managein turbulent times!

Thanks!I am happy to help. email: [email protected]

http://www.stoprog.org/

Jeff Linderoth (UW-Madison) Models & Algs. for SP CMU-EWO 81 / 82 Multistage SPs Modeling Tools

Some Take Away Quotes
http://www.stoprog.org/http://www.stoprog.org/


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If a man will begin with certainty, he shall end in doubts, but if

he will be content to beign with doubts, he shall end incertainties

Francis Bacon

It is a good thing for the

uneducated person to read

books of quotations

Winston Churchill

Jeff Linderoth (UW-Madison) Models & Algs for SP CMU-EWO 82 / 82

Documents

Models and Algorithms for Stochastic Programming, Financial Optimization