Robust Optimization Concepts and Examples Yuriy Zinchenko Shane G. Henderson ORIE, Cornell...
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Transcript of Robust Optimization Concepts and Examples Yuriy Zinchenko Shane G. Henderson ORIE, Cornell...
Robust OptimizationConcepts and Examples
Yuriy ZinchenkoShane G. Henderson
ORIE, Cornell University
Zinchenko and Henderson 2005 2
Outline
• What can go wrong with LP?• A familiar blend problem• The general picture
– Robust linear programming– Software, resources, practicalities
• Radiation therapy for cancer treatment
Zinchenko and Henderson 2005 3
What can go wrong with LP?
Tough LP problem:
max x + ys/t 1 x 1
1 y 1 x, y 0?
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Blend Problem
blend to get output properties at minimum cost
$ $$ $$$
but properties change with time
for anyinput propertieswithin reason
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Blend constraints
• Typical constraint looks like Low ≤ 10 x1 + 12 x2 + 7 x3 ≤ High
• Changes to Low ≤ a1 x1 + a2 x2 + a3 x3 ≤ High
for any vector a that is “close” to (10, 12, 7)
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General robust LP
min cTx s/t A(1) x b1
A(2) x b2
A(3) x b3
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A more detailed view
Simple linear constrainta x 1
x 0with a “close” to 1, namely0 a 2
Want x to work for all such aHow do we deal with it?
Zinchenko and Henderson 2005 8
a x 1, x 0 for all 0 a 2
max a x 1, x 0 0 a 2, x 0
2 x 1 , x 0
x 1/2 , x 0
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A slightly more involved example:a x + b y 1
where (a, b) “close” to (1, 1), namely inEllipsoidal (spherical)
“uncertainty” set U(a, b) is in U if(a, b) = (a0, b0) + (a, b)
with (a0, b0) = (1, 1) and a2 + b2 1
Zinchenko and Henderson 2005 10
Ellipsoidal “uncertainty” set U(a, b) = (a0, b0) + (a, b)
(a0, b0) = (1, 1)
a2 + b2 1
Want (x, y) to satisfya x + b y 1, for all (a, b) from U
U
(a0, b0)
Zinchenko and Henderson 2005 11
a x + b y 1 for all (a, b) in U
max a x + b y 1 (a, b) in U
What can we say about a x + b y ?
a x + b y = (a0 + a) x + (b0 + b) y
= (a0 x + b0 y) + (a x + b y)
Zinchenko and Henderson 2005 12
For a moment, think of (x, y)as your objective function (fixed)
max a x + b y ( 1 ?) (a, b) in U
same as
(a0 x + b0 y) + max (a x + b y) ( 1 ?)
a2 + b2 1
U
(a0, b0)
(x, y)
Zinchenko and Henderson 2005 13
max (a x + b y) ( 1 - (a0 x + b0 y) ?)
a2 + b2 1
Here
a x + b y ||(x, y)|| = (x2 + y2)1/2
the “length” of (x, y)
U(a0, b0)
(x1, y1)
(x2, y2)
Zinchenko and Henderson 2005 14
a x + b y 1 for all (a, b) in U
max a x + b y 1(a, b) in U
(a0 x + b0 y) + max (a x + b y) 1a2 + b2 1
||(x, y)|| 1 - (a0 x + b0 y)
Zinchenko and Henderson 2005 15
Good newsCan handle constraints of this type
||(x, y)|| 1 - (1 x + 1 y)
easily (the so-called second-order conicprogramming (SOCP))
Not much harder than linear programming!
Zinchenko and Henderson 2005 16
General Robust LP formulation
Robust LP:
max cTxs/t A(i) x bi, i = 1,…,m
wherec, x Rn, A(i) R1 x n, A(i)=A(i)
0 + wi Pi
withwi R1 x ki, ||wi|| 1, i=1,…,m, Pi
Rki x n
Zinchenko and Henderson 2005 17
SOCP equivalent:
max cTxs/t || Pi
x || bi - A(i)0 x, i = 1,…,m
Probabilistic interpretation:think of A(i) taken from an -level set of your favorite probability distribution (e.g. multivariate normal)
the robust constraint will readsatisfy the constraint with a given probability
Zinchenko and Henderson 2005 18
Where’d the ellipse come from?
• Expert opinion• Statistics: Averages live in
ellipsoids• Doesn’t have to be an ellipse. Can
be some other shape (e.g., boxes)
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Commercial:• Mosek (http://www.mosek.com/)
“Free”:• SeDuMi (http://sedumi.mcmaster.ca/)
• SDPT3.x (http://www.math.nus.edu.sg/~mattohkc/sdpt3.html/)
Software
Zinchenko and Henderson 2005 20
Practicalities
• Realistic problem sizes– number of variables/constraints on the
order of 103 – 104
– depends (greatly) on the problem data structure/sparsity
• Possible to obtain a “good”, “inexpensive” approximation with LP
Zinchenko and Henderson 2005 21
Generality• Possible to extend this approach to
quite a few other convex programming problems
Resources• Lectures on Modern Convex Optimizatio
n: Analysis, Algorithms, and Engineering Applications by A. Ben-Tal, A. S. Nemirovskii
• Google for Robust Optimization (robust LP etc.)
Zinchenko and Henderson 2005 22
Joint work with Millie Chu (Cornell) and Michael B. Sharpe (Princess Margaret Hospital, Toronto)
Zinchenko and Henderson 2005 23
Cancer treatment
• About 1.3 million new cancer cases in the U.S. each year
• Nearly 60% receive radiation therapy (in conjunction with surgery, chemotherapy etc)
Zinchenko and Henderson 2005 24
External beam radiation therapy• Radiation
delivered by a linear accelerator
• Cancer cells more susceptible than normal cells
• Overlay beams from different angles
• Dose given in daily fractions for ~ 6 weeks
Zinchenko and Henderson 2005 25
Intensity Modulated Radiation Therapy
• Block parts of the radiation beam – discretize the whole beam into a grid of smaller “beamlets”
• Choose different intensities for each beamlet
Intensity Modulated Radiation Therapy
Collaborative Working Group, 2001
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Goal: Choose beam angles and beamlet intensities that deliver enough radiation to kill all tumor cells, while avoiding healthy organs & tissue as much as possible
Treatment Planning
Princess Margaret Hospital
- Take CT scan- Delineate target
region and healthy structures
- Discretize body as small cubes, or “voxels”
- Formulate & solve a mathematical program to find a “good” plan
Zinchenko and Henderson 2005 27
Robust Treatment Planning
Setup errors+ Patient motion+Structural changes during
treatment= uncertainty in geometry
• Don’t rescan patient much if at all• Use RO to “robustify”
mathematical program
Zinchenko and Henderson 2005 28
Model Formulation • Many different formulations exist – we use a penalty formulation
minimize: penalty objective
subject to:
Pr(Dose to voxel i in healthy structure k ≤ Uk) ≥ 0.95
Pr(Dose to voxel i in tumor ≥ L) ≥ 0.95
x = beamlet intensities ≥ 0
Zinchenko and Henderson 2005 29
Computational Results
• Prostate: tumor + 5 healthy regions• 5 equi-spaced beams, ~ 225
beamlets from each angle• Voxel size = 2 cm, ~ 400 total
voxels• Solver: Mosek, v. 3.0.1.18• Solve time = 6 seconds (LP), 45
minutes (SOCP)
Zinchenko and Henderson 2005 30
Dose-Volume HistogramsDose-Volume Histograms
deterministic solution’s plan
stochastic solution’s plan
% of structure receiving ≥ x Gy
DVH of expected dose
Zinchenko and Henderson 2005 31
Comparison
• Simulate 10 treatments (45 fractions each)
• For each of the 10 treatments, and for each solution (deterministic & stochastic),– calculated dose delivered to each voxel in
each fraction– summed over the 45 fractions to get total
dose delivered to each voxel– plotted DVH
Zinchenko and Henderson 2005 32
DVH – Treatment 1DVH – Treatment 1
det
stoch
Zinchenko and Henderson 2005 33
DVH – Treatment 2DVH – Treatment 2
det
stoch
Zinchenko and Henderson 2005 34
DVH – Treatment 3DVH – Treatment 3
det
stoch
Zinchenko and Henderson 2005 35
DVH – Treatment 4DVH – Treatment 4
det
stoch
Zinchenko and Henderson 2005 36
DVH – Treatment 5DVH – Treatment 5
det
stoch
Zinchenko and Henderson 2005 37
DVH – Treatment 6DVH – Treatment 6
det
stoch
Zinchenko and Henderson 2005 38
DVH – Treatment 7DVH – Treatment 7
det
stoch
Zinchenko and Henderson 2005 39
DVH – Treatment 8DVH – Treatment 8
det
stoch
Zinchenko and Henderson 2005 40
DVH – Treatment 9DVH – Treatment 9
det
stoch
Zinchenko and Henderson 2005 41
DVH – Treatment 10DVH – Treatment 10
det
stoch
Zinchenko and Henderson 2005 42
Conclusions
• LP “pushes you into a corner”• True situation never same as data• Robust LP: Find good solution that
is always feasible within reason• Efficient solution methods: can
solve real problems• Software available
Zinchenko and Henderson 2005 43
A Bit More DetailA Bit More Detail• Di(x) = Dose delivered to voxel i in N fractions, with intensities x, a random variable
Di(x) is the sum of N random variables (N = 45), assume iid,
apply CLT, so Di(x) is approximately normally distributed
• Take n sample shifts, s1,...,sn, with associated probabilities p = (p1,...,pn)T
• Let ai(∙)T = ai(s1)T ai(s2)T dose delivered to voxel i, shifted by sj,
from each beamlet with unit intensityai(sn)T
so that NpTai(∙)Tx = expected total dose delivered to voxel i, for N fractions.
• Let vi(x) = sample variance of dose delivered to voxel i
Di(x) ~ Normal ( NpTai(·)Tx, Nvi(x) )
…
Zinchenko and Henderson 2005 44
)(N
)(N
)(N
)(N)(
x
xapm
x
xapx
i
Ti
Tk
i
Ti
Ti
vv
DP
• Want constraints to be violated with low probability (say, δ = .05)
• Example: maximum dose constraint on voxel i in Hk:
Assuming Di(x) ~ Normal ( NpTai(∙)Tx, Nvi(x) ),
mk
1)(N
)(Nz
vi
Ti
Tk
x
xapm
N
)(N)(
1
zv
Ti
Tk
i
xapmx
N
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12
zR
Ti
TkT
i
xapmxa
• Second order cone program (SOCP)
Want P(Di(x) > mk) ≤ δ
Probabilistic ConstraintsProbabilistic Constraints
Zinchenko and Henderson 2005 45
Dose-Volume Constraints•Physicians like constraints of form:
“<= fraction fk of structure Hk gets >= dk”•0-1 var for each voxel: = 1 if dose is > dk.•MIP: Hard to solve!•Many voxels get near max allowed dose•Alternative: upper bound the “excess”
dose. For healthy structure Hk, we require:
•Linear constraints☺
kHi
kki gd )'N( Txa