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

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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?

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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

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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)

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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)

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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)

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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)

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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)

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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!

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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

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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

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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

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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

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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.)

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Joint work with Millie Chu (Cornell) and Michael B. Sharpe (Princess Margaret Hospital, Toronto)

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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)

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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

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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

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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

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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

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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)

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Dose-Volume HistogramsDose-Volume Histograms

deterministic solution’s plan

stochastic solution’s plan

% of structure receiving ≥ x Gy

DVH of expected dose

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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

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DVH – Treatment 1DVH – Treatment 1

det

stoch

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DVH – Treatment 2DVH – Treatment 2

det

stoch

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DVH – Treatment 3DVH – Treatment 3

det

stoch

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DVH – Treatment 4DVH – Treatment 4

det

stoch

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DVH – Treatment 5DVH – Treatment 5

det

stoch

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DVH – Treatment 6DVH – Treatment 6

det

stoch

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DVH – Treatment 7DVH – Treatment 7

det

stoch

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DVH – Treatment 8DVH – Treatment 8

det

stoch

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DVH – Treatment 9DVH – Treatment 9

det

stoch

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DVH – Treatment 10DVH – Treatment 10

det

stoch

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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

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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) )

…

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)(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

)(N)(

12

zR

Ti

TkT

i

xapmxa

• Second order cone program (SOCP)

Want P(Di(x) > mk) ≤ δ

Probabilistic ConstraintsProbabilistic Constraints

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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