View
215
Download
0
Category
Preview:
Citation preview
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
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
Recommended