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

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Presentation on theme: "Robust Optimization Concepts and Examples Yuriy Zinchenko Shane G. Henderson ORIE, Cornell University."— Presentation transcript:

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

2 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

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

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

5 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)

6 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

7 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?

8 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

9 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

10 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 )

11 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)

12 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)

13 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 )

14 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)

15 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!

16 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

17 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 

18 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)

19 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

20 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

21 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.)

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

23 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)

24 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

25 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

26 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

27 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

28 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

29 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)

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

31 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

32 Zinchenko and Henderson DVH – Treatment 1 det stoch

33 Zinchenko and Henderson DVH – Treatment 2 det stoch

34 Zinchenko and Henderson DVH – Treatment 3 det stoch

35 Zinchenko and Henderson DVH – Treatment 4 det stoch

36 Zinchenko and Henderson DVH – Treatment 5 det stoch

37 Zinchenko and Henderson DVH – Treatment 6 det stoch

38 Zinchenko and Henderson DVH – Treatment 7 det stoch

39 Zinchenko and Henderson DVH – Treatment 8 det stoch

40 Zinchenko and Henderson DVH – Treatment 9 det stoch

41 Zinchenko and Henderson DVH – Treatment 10 det stoch

42 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

43 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) ) …

44 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 

45 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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