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Robust Optimization Concepts and Examples

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

2 Zinchenko and Henderson 2005
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 + y s/t 1 x  y  1 x, y  0 ? Zinchenko and Henderson 2005

4 Zinchenko and Henderson 2005
Blend Problem but properties change with time $$ $$$ $ blend to get output properties at minimum cost for any input properties within reason Zinchenko and Henderson 2005

5 Zinchenko and Henderson 2005
Blend constraints Typical constraint looks like Low ≤ 10 x 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) Zinchenko and Henderson 2005

6 Zinchenko and Henderson 2005
General robust LP min cTx s/t A(1) x  b1 A(2) x  b2 A(3) x  b3 Zinchenko and Henderson 2005

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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? Zinchenko and Henderson 2005

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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 Zinchenko and Henderson 2005

9 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) = (a0, b0) + (Da, Db) with (a0, b0) = (1, 1) and Da2 + Db2  1 Zinchenko and Henderson 2005

10 Zinchenko and Henderson 2005
Ellipsoidal “uncertainty” set U (a, b) = (a0, b0) + (Da, Db) (a0, b0) = (1, 1) Da2 + Db2  1 Want (x, y) to satisfy a x + b y  1, for all (a, b) from U U (a0, b0) Zinchenko and Henderson 2005

11 What can we say about a x + b y ?
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 + Da) x + (b0 + Db) y = (a0 x + b0 y) + (Da x + Db y) Zinchenko and Henderson 2005

12 Zinchenko and Henderson 2005
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 (Da x + Db y) ( 1 ?) Da2 + Db2  1 (x, y) U (a0, b0) Zinchenko and Henderson 2005

13 Zinchenko and Henderson 2005
max (Da x + Db y) ( 1 - (a0 x + b0 y) ?) Da2 + Db2  1 Here Da x + Db y  ||(x, y)|| = (x2 + y2)1/2 the “length” of (x, y) (x1, y1) U (x2, y2) (a0, b0) Zinchenko and Henderson 2005

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a x + b y  for all (a, b) in U max a x + b y  1 (a, b) in U (a0 x + b0 y) + max (Da x + Db y)  Da2 + Db2  1 ||(x, y)||  1 - (a0 x + b0 y) Zinchenko and Henderson 2005

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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! Zinchenko and Henderson 2005

16 General Robust LP formulation
max cTx s/t A(i) x  bi, i = 1,…,m where c, x Î Rn, A(i) Î R1 x n, A(i)=A(i)0 + wi Pi with wi Î R1 x ki, ||wi||  1, i=1,…,m, Pi Î Rki x n Zinchenko and Henderson 2005

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SOCP equivalent: max cTx s/t || Pi x ||  bi - A(i)0 x, i = 1,…,m Probabilistic interpretation: think of A(i) taken from an a-level set of your favorite probability distribution (e.g. multivariate normal) the robust constraint will read satisfy the constraint with a given probability a 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) Zinchenko and Henderson 2005

19 Zinchenko and Henderson 2005
Software Commercial: Mosek (http://www.mosek.com/) “Free”: SeDuMi (http://sedumi.mcmaster.ca/) SDPT3.x (http://www.math.nus.edu.sg/~mattohkc/sdpt3.html/) Zinchenko and Henderson 2005

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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 Zinchenko and Henderson 2005

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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. Nemirovskii Google for Robust Optimization (robust LP etc.) Zinchenko and Henderson 2005

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In practice... Joint work with Millie Chu (Cornell) and Michael B. Sharpe (Princess Margaret Hospital, Toronto) Zinchenko and Henderson 2005

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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) 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 Zinchenko and Henderson 2005

26 Zinchenko and Henderson 2005
Treatment Planning 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 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 Princess Margaret Hospital

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 Zinchenko and Henderson 2005
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 Solve time = 6 seconds (LP), 45 minutes (SOCP) Zinchenko and Henderson 2005

30 Dose-Volume Histograms
% of structure receiving ≥ x Gy deterministic solution’s plan DVH of expected dose stochastic solution’s plan Zinchenko and Henderson 2005

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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 Zinchenko and Henderson 2005

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DVH – Treatment 1 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 2 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 3 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 4 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 5 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 6 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 7 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 8 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 9 det stoch Zinchenko and Henderson 2005

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DVH – Treatment 10 det stoch Zinchenko and Henderson 2005

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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 The End Zinchenko and Henderson 2005

43 Zinchenko and Henderson 2005
A 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 intensity ai(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 Probabilistic Constraints
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 Want P(Di(x) > mk) ≤ δ Second order cone program (SOCP) 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☺ Zinchenko and Henderson 2005


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