Pareto Efficiency in Robust Optimization

Dan Iancu

Graduate School of BusinessStanford University

joint work with Nikolaos Trichakis (HBS)

1 / 26

Classical Robust Optimization

Typical linear optimization problem under uncertainty

maxxPRn

pTx

aTi x ď bi, @i P I

maxxPRn

minuPU

ppuqTx

aipuqTx ď bipuq, @u P U , @i P I

For many classes of U , results in tractable formulationsSoyster [1973], Ben-Tal and Nemirovski [1998, 2002], El-Ghaoui et al.[1998], Bertsimas and Sim [2003, 2004], ...

Framework successfully adopted in many applications

However, approach suffers from several shortcomings ...

2 / 26

Conservativeness

Worst-case focus Ñ limited potential upside.

Absolute or relative regret [Savage, 1972], “soft-robustness” [Ben-Talet al., 2010], “light-robustness” [Fischetti and Monaci, 2009],bw-robustness [Roy, 2010, Gabrel et al., 2011], α-robustness [Kalaiet al., 2012], ...

Typically same (or slightly decreased) modeling flexibility and same(or slightly increased) computational complexity

Trade off robustness (worst-case performance) for potential upside

3 / 26

Multiplicity of solutions

Classical RO framework can lead to multiple solutions

Typically seen as a benefit[Bertsimas et al., 2010], [Iancu et al., 2012]

4 / 26

Multiplicity of solutions

Classical RO framework can lead to multiple solutions

Typically seen as a benefit[Bertsimas et al., 2010], [Iancu et al., 2012]

Which solution to pick?

Minimax/maximin criteria can lead to Pareto inefficiencies

§ Idea recognized in other areas[Young, 1995], [Bertsimas et al., 2012], [Ogryczak, 1997], [Suh andLee, 2001]

§ Typical fix (lexicographic max-min fairness) only works for a finite setof scenarios

4 / 26

In this talk...

Adapt and formalize the property of Pareto Efficiency in RO

Illustrate that RO need not produce solutions with this property

Provide basic characterization of “Pareto” solutions

Extend RO framework to produce “Pareto” solutions, at essentially noextra computational cost

Illustrate benefits in three popular applications

5 / 26

Setup

maxxPX

minpPU

pTx

Feasible set of solutions X “

x P Rn : Ax ď b

(

Uncertainty set of objective coefficients U “ tp P Rn : Dp ě du

6 / 26

Setup

maxxPX

minpPU

pTx

Feasible set of solutions X “

x P Rn : Ax ď b

(

Uncertainty set of objective coefficients U “ tp P Rn : Dp ě du

Classical RO framework results in

§ Optimal value zRO

§ Set of robustly optimal solutions

XRO “

x P X : D y ě 0 such that DT y “ x, yTd ě zRO(

6 / 26

Set of robustly optimal solutions

XRO “

x P X : D y ě 0 such that DT y “ x, yTd ě zRO(

x P XRO guarantees that no other solution exists with higherworst-case objective value pTx

7 / 26

Set of robustly optimal solutions

XRO “

x P X : D y ě 0 such that DT y “ x, yTd ě zRO(

x P XRO guarantees that no other solution exists with higherworst-case objective value pTx

What if an uncertainty scenario materializes that does not correspondto the worst-case?

Are there any guarantees that no other solution exists that, apartfrom protecting us from worst-case scenarios, also performs betteroverall, under all possible uncertainty realizations?

7 / 26

Pareto Robustly Optimal solutions

maxxPX

minpPU

pTx (1)

Definition

A solution x is called a Pareto Robustly Optimal (PRO) solution forProblem (1) if

(a) it is robustly optimal, i.e., x P XRO, and

(b) there is no x P X such that

pT x ě pTx, @p P U, and

pT x ą pTx, for some p P U.

8 / 26

Pareto Robustly Optimal solutions

maxxPX

minpPU

pTx (1)

Definition

A solution x is called a Pareto Robustly Optimal (PRO) solution forProblem (1) if

(a) it is robustly optimal, i.e., x P XRO, and

(b) there is no x P X such that

pT x ě pTx, @p P U, and

pT x ą pTx, for some p P U.

XPRO Ď XRO: set of all PRO solutions

8 / 26

Some questions

Given a RO solution, is it also PRO?

How can one find a PRO solution?

Can we optimize over XPRO?

Can we characterize XPRO?

§ Is it non-empty?

§ Is it convex?

§ When is XPRO “ XRO?

How does the notion generalize in other RO formulations?

9 / 26

Finding PRO solutions

Theorem

Given a solution x P XRO and an arbitrary point p P ripUq, consider thefollowing linear optimization problem:

maximize pT y

subject to y P U˚

x ` y P X.

Then, either

the optimal value is zero and x P XPRO, or

the optimal value is strictly positive and x “ x ` y‹ P XPRO, for any

optimal solution y‹.

U˚ def“ ty P R

n : yT p ě 0, @ p P Uu is the dual of U

10 / 26

Remarks

Finding a point p P ripUq can be done efficiently using LP techniques

Testing whether x P XRO is no harder than solving the classical ROproblem in this setting

Finding a PRO solution x P XPRO is no harder than solving theclassical RO problem in this setting

11 / 26

Corollaries

If p P ripUq, all optimal solutions to the problem below are PRO:

maximize pTx

subject to x P XRO

If 0 P ripUq, then XPRO “ XRO

If p P ripUq, then XPRO “ XRO if and only if the optimal value ofthe LP below is zero:

maximize pT y

subject to x P XRO

y P U˚

x ` y P X

12 / 26

Optimizing over / Understanding XPRO

Secondary objective r: can we solve

maximize rTx

subject to x P XPRO?

Interesting case: XRO ‰ XPRO

13 / 26

Optimizing over / Understanding XPRO

Secondary objective r: can we solve

maximize rTx

subject to x P XPRO?

Proposition

XPRO is not necessarily convex.

X “ tx P R4` : x1 ď 1, x2 ` x3 ď 6, x3 ` x4 ď 5, x2 ` x4 ď 5u

U “ conv´

ei, i P t1, . . . , 4u(

¯

zRO “ 1, and XRO “ tx P X : x ě 1u

x1 ““

1 2 4 1‰T

, x2 ““

1 4 2 1‰T

P XPRO

0.5x1 ` 0.5x2 is Pareto dominated by“

1 3 3 2‰T

P XRO.

13 / 26

Optimizing over / Understanding XPRO

Secondary objective r: can we solve

maximize rTx

subject to x P XPRO?

Proposition

If XRO ‰ XPRO, then XPRO X ripXROq “ H.

Whether the solution to a nominal RO is PRO depends on algorithmused for solving LP

Simplex vs. interior point

13 / 26

Optimizing over XPRO by MILP

Proposition

For any r P Rn and any p P ripUq, let px‹, µ‹, η‹, z‹q be an optimal

solution of the following MILP

maximize rTx

subject to x P XRO

µ ď Mp1 ´ zq

b ´ Ax ď Mz

DATµ ´ d η ě Dp

µ ě 0, η ě 0, z binary.

Then, x‹ P argmaxxPXPRO

rTx.

14 / 26

Optimizing over XPRO by sampling

Algorithm 1 Sampling heuristic for solving the problem maxxPXPRO rTx

1: XPRO Ð H2: for i Ð 1, . . . , N do3: Sample a point p P ripUq.4: XPRO Ð XPRO Y argmaxxPXRO pTx.5: end for6: Solve max

xPXPRO rTx.

Theorem

x P XPRO if and only if D p P ripUq such that argmaxxPXRO pTx.

15 / 26

Generalizations — Robust MILP

Main results all readily extend!§ Testing, finding and optimizing over PRO solutions is as hard assolving the classical robust counterpart

16 / 26

Generalizations — Robust MILP

Main results all readily extend!§ Testing, finding and optimizing over PRO solutions is as hard assolving the classical robust counterpart

Only change: when XRO ‰ XPRO, there may existx P XPRO X ripXROq.

X “!

px1, x2, x3q P Z2` ˆ R :

1

2x1 `

1

5x2 ď 1, x3 ě ´1, x3 ď 0

)

U “ conv`

te1, e2, e3u˘

XRO “

x P X : x3 “ 0(

XPRO “

r0 2 0sT , r5 0 0sT , r1 2 0sT(

x “ r1 2 0sT P ri`

convpXROq˘

16 / 26

Generalizations – uncertainty in the constraints

minimize cTx

subject to Ax ě b, @A P UA,

with UA Ă Rmˆn a bounded polyhedral uncertainty set.

17 / 26

Generalizations – uncertainty in the constraints

minimize cTx

subject to Ax ě b, @A P UA,

with UA Ă Rmˆn a bounded polyhedral uncertainty set.

Vector of slacks spx,Aqdef“ Ax ´ b, @x P R

n, A P UA

RO solutions

XRO “

x P Rn : cTx ď zRO, spx,Aq ě 0, @A P UA

(

RO solution guarantees that no other solution exists yieldingnonnegative slacks for all A, and at a lower cost than zRO

17 / 26

PRO solutions under constraint uncertainty

minimize cTx

subject to Ax ě b, @A P UA,(3)

To compare two slack vectors: slack value vector v

§ quantifies the relative value of slack in each constraint

§ E.g., if (3) comes from epigraph formulation, choose v “ e1

Definition

A solution x is called a Pareto Robustly Optimal (PRO) solution forProblem (3) if x P XRO, and there is no other x P XRO such that

vT spx, Aq ě vT spx,Aq, @A P UA, and

vT spx, Aq ą vT spx, Aq, for some A P UA.

All previous results directly apply

18 / 26

Numerical experiments

19 / 26

Numerical experiments

Example (Portfolio)

n ` 1 assets, with returns ri

ri “ µi ` σi ζi, i “ 1, . . . , n, rn`1 “ µn`1

ζ unknown, U “ tζ P Rn : ´1 ď ζ ď 1, 1T ζ “ 0u

Objective: select weights x to maximize worst-case portfolio return

19 / 26

Numerical experiments

Example (Portfolio)

n ` 1 assets, with returns ri

ri “ µi ` σi ζi, i “ 1, . . . , n, rn`1 “ µn`1

ζ unknown, U “ tζ P Rn : ´1 ď ζ ď 1, 1T ζ “ 0u

Objective: select weights x to maximize worst-case portfolio return

Example (Inventory)

One warehouse, N retailer where uncertain demand is realized

Transportation, holding costs and profit margins differ for each retailer

Demand driven by market factors di “ d0i ` qTi z, i “ 1, . . . , N

Market factors z are uncertain

z P U “ tz P RN : ´b ¨ 1 ď z ď b ¨ 1, ´B ď 1

T z ď Bu

19 / 26

Numerical experiments

Example (Project management)

A PERT diagram given by directed, acyclic graph G “ pN , Eq

N are project events, E are project activities / tasks

S C

A

B

F

c

a

b

d

e g

f

S 4

3

2

5

6

7

8

F

20 / 26

Numerical experiments

Example (Project management)

A PERT diagram given by directed, acyclic graph G “ pN , Eq

N are project events, E are project activities / tasks

Task e P E has uncertain duration τe “ τ0e ` δe

δ P

δ P R|E|` : δ ď b ¨ 1, 1

T δe ď B(

Task e P E can be expedited by allocating a budgeted resource xe

τe “ τ0e ` δe ´ xe

1Tx ď C

Goal: find resource allocation x to minimize worst-case completiontime

20 / 26

Methodology

Generate 10,000 random problem instances

For every problem, find x P XRO using simplex method

Test whether x P XPRO

§ If not, find x P XPRO that Pareto dominates x

§ Record pT px´xqpT x

, where p is “nominal” scenario

§ Record maxpPUpT px´xq

pT x.

21 / 26

Results – finance and inventory examples#

ofoccurrences

#of

occurrences

nominal gain (%) maximum gain (%)

0000

0000

2020

2020

4040

4040

1010

1010

3030

3030

5050

5050

200200

200200

400400

400400

600600

600600

Figure: TOP: portfolio example. BOTTOM: inventory example. LEFT:performance gains in nominal scenario. RIGHT: maximal performance gains.

22 / 26

Results – project management example#

ofoccurrences

#of

occurrences

nominal performance gain (in %) maximum performance gain (in %)

0000

0000

2020

2020

4040

4040

1010

1010

3030

3030

5050

100100

100100

150150

200200

200200

300300

400400

(a) (a)

(b) (b)

Figure: TOP: configuration (a). BOTTOM: configuration (b). LEFT:performance gains in nominal scenario. RIGHT: maximal performance gains

23 / 26

Conclusions

Adapted the well known concept of Pareto efficiency in the context ofthe RO methodology

By focusing exclusively on worst-case outcomes, the classical ROparadigm can lead to inefficiencies and sub-optimal performance inpractice

Extended the RO framework via practical methods that verify Paretooptimality, and generate PRO solutions

Preserved complexity of the underlying robust problems

ñ PRO solutions have a significant upside, at no extra cost or downside

D. Iancu, N. Trichakis. Pareto Efficiency in Robust Optimization, 2012 -

Optimization Online.

24 / 26

References I

A. Ben-Tal and A. Nemirovski. Robust convex optimization. Mathematics of Operations Research, 23(4):769–805, 1998.

A. Ben-Tal and A. Nemirovski. Robust optimization - methodology and applications. Mathematical Programming, 92(3):453–480, 2002.

A. Ben-Tal, D. Bertsimas, and D. B. Brown. A soft robust model for optimization under ambiguity. Operations Research, 58:1220–1234, 2010.

D. Bertsimas and M. Sim. Robust discrete optimization and network flows. Mathematical Programming, 98(1-3):49–71, 2003.

D. Bertsimas and M. Sim. The price of robustness. Operations Research, 52(1):35–53, 2004.

D. Bertsimas, D. A. Iancu, and P. A. Parrilo. Optimality of affine policies in multistage robust optimization. Mathematics of

Operations Research, 35(2):363–394, 2010.

D. Bertsimas, V. F. Farias, and N. Trichakis. On the efficiency-fairness tradeoff. Management Science, 2012. To appear.

L. El-Ghaoui, F. Oustry, and H. Lebret. Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1):33–52, 1998.

M. Fischetti and M. Monaci. Light robustness. In R. Ahuja, R. Mohring, and C. Zaroliagis, editors, Robust and Online

Large-Scale Optimization, volume 5868 of Lecture Notes in Computer Science, pages 61–84. Springer Berlin / Heidelberg,2009.

V. Gabrel, C. Murat, and L. Wu. New models for the robust shortest path problem: Complexity, resolution and generalization.Annals of Operations Research, pages 1–24, 2011.

D. A. Iancu, M. Sharma, and M. Sviridenko. Supermodularity and affine policies in dynamic robust optimization. Submitted forpublication, 2012.

R. Kalai, C. Lamboray, and D. Vanderpooten. Lexicographic α-robustness: An alternative to min–max criteria. EuropeanJournal of Operational Research, 220(3):722 – 728, 2012.

W. Ogryczak. On the lexicographic minimax approach to location problems. European Journal of Operational Research, 100(3):566 – 585, 1997.

B. Roy. Robustness in operational research and decision aiding: A multi-faceted issue. European Journal of Operational

Research, 200(3):629 – 638, 2010.

25 / 26

References II

L. J. Savage. The Foundations of Statistics. Dover Publications, Inc., second edition, 1972.

A. L. Soyster. Technical Note–Convex Programming with Set-Inclusive Constraints and Applications to Inexact LinearProgramming. Operations Research, 21(5):1154–1157, 1973.

M.-h. Suh and T.-y. Lee. Robust optimization method for the economic term in chemical process design and planning. Industrial

& Engineering Chemistry Research, 40(25):5950–5959, 2001.

H. P. Young. Equity: In Theory and Practice. Princeton University Press, 1995.

26 / 26