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Robust Optimization Concepts and Examples Yuriy Zinchenko Shane G. Henderson ORIE, Cornell University

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Zinchenko and Henderson 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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Zinchenko and Henderson What can go wrong with LP? Tough LP problem: maxx + y s/t1 x 1 1 y 1 x, y 0 ?

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Zinchenko and Henderson Blend Problem blend to get output properties at minimum cost $ $$$$$ but properties change with time for any input properties within reason

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Zinchenko and Henderson Blend constraints Typical constraint looks like Low ≤ 10 x x x 3 ≤ High Changes to Low ≤ a 1 x 1 + a 2 x 2 + a 3 x 3 ≤ High for any vector a that is “close” to (10, 12, 7)

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Zinchenko and Henderson General robust LP min c T x s/tA (1) x b 1 A (2) x b 2 A (3) x b 3

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Zinchenko and Henderson A more detailed view Simple linear constraint a x 1 x 0 with a “close” to 1, namely 0 a 2 Want x to work for all such a How do we deal with it?

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Zinchenko and Henderson a x 1, x 0for 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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Zinchenko and Henderson A slightly more involved example: a x + b y 1 where (a, b) “close” to (1, 1), namely in Ellipsoidal (spherical) “uncertainty” set U (a, b) is in U if (a, b) = (a 0, b 0 ) + ( a, b) with (a 0, b 0 ) = (1, 1) and a 2 + b 2 1

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Zinchenko and Henderson Ellipsoidal “uncertainty” set U (a, b) = (a 0, b 0 ) + ( a, b) (a 0, b 0 ) = (1, 1) a 2 + b 2 1 Want (x, y) to satisfy a x + b y 1, for all (a, b) from U U (a 0, b 0 )

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Zinchenko and Henderson a x + b y 1for 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 = (a 0 + a) x + (b 0 + b) y = (a 0 x + b 0 y) + ( a x + b y)

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Zinchenko and Henderson For a moment, think of (x, y) as your objective function (fixed) max a x + b y ( 1 ?) (a, b) in U same as (a 0 x + b 0 y) + max ( a x + b y) ( 1 ?) a 2 + b 2 1 U (a 0, b 0 ) (x, y)

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Zinchenko and Henderson max ( a x + b y) ( 1 - (a 0 x + b 0 y) ?) a 2 + b 2 1 Here a x + b y ||(x, y)|| = (x 2 + y 2 ) 1/2 the “length” of (x, y) U (a 0, b 0 ) (x 1, y 1 ) (x 2, y 2 )

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Zinchenko and Henderson a x + b y 1 for all (a, b) in U max a x + b y 1 (a, b) in U (a 0 x + b 0 y) + max ( a x + b y) 1 a 2 + b 2 1 ||(x, y)|| 1 - (a 0 x + b 0 y)

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Zinchenko and Henderson Good news Can handle constraints of this type ||(x, y)|| 1 - (1 x + 1 y) easily (the so-called second-order conic programming (SOCP)) Not much harder than linear programming!

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Zinchenko and Henderson General Robust LP formulation Robust LP: max c T x s/t A (i) x b i, i = 1,…,m where c, x R n, A (i) R 1 x n, A (i) =A (i) 0 + w i P i with w i R 1 x k i, ||w i || 1, i=1,…,m, P i R k i x n

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Zinchenko and Henderson SOCP equivalent: max c T x s/t || P i x || b i - 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 read satisfy the constraint with a given probability

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Zinchenko and Henderson 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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Zinchenko and Henderson Commercial: Mosek (http://www.mosek.com/)http://www.mosek.com/ “Free”: SeDuMi (http://sedumi.mcmaster.ca/)http://sedumi.mcmaster.ca/ SDPT3.x (http://www.math.nus.edu.sg/~mattohkc/sdpt3.html/)http://www.math.nus.edu.sg/~mattohkc/sdpt3.html/ Software

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Zinchenko and Henderson Practicalities Realistic problem sizes –number of variables/constraints on the order of 10 3 – 10 4 –depends (greatly) on the problem data structure/sparsity Possible to obtain a “good”, “inexpensive” approximation with LP

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Zinchenko and Henderson Generality Possible to extend this approach to quite a few other convex programming problems Resources Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications by A. Ben-Tal, A. S. NemirovskiiLectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications Google for Robust Optimization (robust LP etc.)

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

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Zinchenko and Henderson 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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Zinchenko and Henderson 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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Zinchenko and Henderson 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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Zinchenko and Henderson 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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Zinchenko and Henderson 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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Zinchenko and Henderson 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 ≤ U k )≥ 0.95 Pr(Dose to voxel i in tumor ≥ L)≥ 0.95 x = beamlet intensities ≥ 0

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Zinchenko and Henderson 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 Solve time = 6 seconds (LP), 45 minutes (SOCP)

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Zinchenko and Henderson Dose-Volume Histograms deterministic solution’s plan stochastic solution’s plan % of structure receiving ≥ x Gy DVH of expected dose

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Zinchenko and Henderson 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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Zinchenko and Henderson DVH – Treatment 1 det stoch

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Zinchenko and Henderson DVH – Treatment 2 det stoch

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Zinchenko and Henderson DVH – Treatment 3 det stoch

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Zinchenko and Henderson DVH – Treatment 4 det stoch

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Zinchenko and Henderson DVH – Treatment 5 det stoch

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Zinchenko and Henderson DVH – Treatment 6 det stoch

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Zinchenko and Henderson DVH – Treatment 7 det stoch

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Zinchenko and Henderson DVH – Treatment 8 det stoch

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Zinchenko and Henderson DVH – Treatment 9 det stoch

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Zinchenko and Henderson DVH – Treatment 10 det stoch

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Zinchenko and Henderson 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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Zinchenko and Henderson A Bit More Detail D i (x) = Dose delivered to voxel i in N fractions, with intensities x, a random variable D i (x) is the sum of N random variables (N = 45), assume iid, apply CLT, so D i (x) is approximately normally distributed Take n sample shifts, s 1,...,s n, with associated probabilities p = (p 1,...,p n ) T Let a i (∙) T =a i (s 1 ) T a i (s 2 ) T dose delivered to voxel i, shifted by s j, from each beamlet with unit intensity a i (s n ) T so that Np T a i (∙) T x = expected total dose delivered to voxel i, for N fractions. Let v i (x) = sample variance of dose delivered to voxel i D i (x) ~ Normal ( Np T a i (·) T x, Nv i (x) ) …

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Zinchenko and Henderson Want constraints to be violated with low probability (say, δ =.05) Example : maximum dose constraint on voxel i in H k : Assuming D i (x) ~ Normal ( Np T a i (∙) T x, Nv i (x) ), mkmk Second order cone program (SOCP) Want P(D i (x) > m k ) ≤ δ Probabilistic Constraints

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Zinchenko and Henderson Dose-Volume Constraints Physicians like constraints of form: “ = d k ” 0-1 var for each voxel: = 1 if dose is > d k. MIP: Hard to solve! Many voxels get near max allowed dose Alternative: upper bound the “excess” dose. For healthy structure H k, we require: Linear constraints ☺

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