OPERATIONS RESEARCH AND HIGH
ENERGY PHYSICS
Alberto De Min
Politecnico di Milano and INFN
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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INTRODUCTION
Operations Research (OR) is the discipline of applying
advanced analytical methods to help make better
decisions
Applications concern an uncountable number of fields: from
economics to finance, from logistics to supply chain
management, from social sciences to politics, from
engineering to physics
The aim of this talk is to illustrate possible applications to
high energy physics and review the existing
mathematical tools, restricting the scope to linear and
mixed-integer programming applications
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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LAYOUT
Basics of operations research
Two practical examples: Resource optimization, Set covering
Linear programming:
– Simplex method
– Interior-point method
Mixed-integer programming
Applications to high energy physics
Two pattern recognition examples:
– Vertex tracking
– Muon tracking
Other detector issues (calibration, alignment, design, logistics)
Mathematical libraries
Commercial software packages
Open source packages
Conclusions
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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EXAMPLE 1 (resource optimisation)
A firm produces tables and chairs in 3 phases: cutting, assemblying
and finishing. The firm wants to maximise total profit.
Cutting
Assemblying Finishing
Profit per item
Chair: time required
6/5h
1h
3/2h
$80
Table: time required
1h
1/2h
2h
$55
50h
120h
Available manpower (h/d) 72h
Objective function ->
Maximize z = 80 x + 55 y subject to:
Constraint 1 ->
6/5 x+ y  72
Constraint 2 ->
x + ½ y  50
Constraint 3 ->
3/2 x + 2y  120
Positiveness of variables ->
x,y 0
Solution: x = 35.0 chairs and y = 30.0 tables should be
produced per day
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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EXAMPLE 2 (set covering)
An airline needs to cover its flights with crew (based in city A).
1) Build all possible flight sequences (rotations) leaving and returning to A
2) Choose the subset of rotations (j) covering all flights (i) with minimum
A B
B C
C A
cost (Cj)
A
BB
A
A
C C
A
Objective function ->
Minimize z = j Cj xj subject to:
Set covering
(or set partitioning) ->
 flight i j Rij xj  1 ( or =1) where
Integrality of variables ->
xj ={0,1}
xj=1 (0) if rotation j is (not) selected
Rij = 1 (0) if rotation j does (not) include flight i
1) Solution is a set of integer (binary) values for xj
2) xj = 0.5 cannot be approximated to 0 (no covering) or 1 (no optimum)
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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LINEAR PROGRAMMING
1. Solution is a point in n-dim space
2. Linear functions are (hyper)planes
in that space
3. Linear constraints  () define the
half-space above (below) a plane
4. The space of feasible solutions is
all points inside the convex
polyedrum defined by the
constraint-planes (a «politope » or
« simplex »)
5. The objective function is an
infinity of parallel planes of which
we need the uppermost
intersecting the politope
6. The optimal solution is the point
of such intersection
A. De Min
Max z =
A01 x1 + A02 x2 + … subject to:
A11 x1 + A12 x2 + …  B1
A21 x1 + A22 x2 + …  B2
A31 x1 + A32 x2 + …  B3
x1, x2  0
CHEP06 Mumbai, India 13-17 February 2006
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THE SIMPLEX METHOD
1. The optimal solution cannot be an
internal point (as it would always be
possible to find a « more optimal »
point nearby!)
2. The optimal solution needs to be on
a corner (or an edge if degenerate)
3. Only the (few) corners need to be
analysed to reach optimality
4. The objective function determines
the direction to a better corner
5. A complex minimization task has
been reduced to an easy
combinatorial problem
6. The discovery of the simplex
method in 1947 gave Dantzig the
Nobel Price
7. This is an « external point method »
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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INTERIOR POINT METHODS
1. Take the politope of feasible solutions
2. Build an ellipsoid contained in the
politope
3. Find the tangence point A with the
objective plane
4. Build a new ellipsoid centered in A
contained in the politope
5. Repeat until size of ellipsoid is small
enough
6. Easy when matrix of coefficients is
sparse (many zeros!)
7. It works also for non-linear cases
8. For some problems faster than the
simplex
9. Developed in the last few decades
10.This is an « internal point method »
A. De Min
Objective function
A
CHEP06 Mumbai, India 13-17 February 2006
B
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(MIXED) INTEGER PROGRAMMING
Suppose some variables should be integer but non-integer values are found at optimum
Store corresponding value of objective function (OF) as a lower bound (LB) on the
solution
Take one such variables, say Xi=1.3, and branch into two separate problems (nodes or
trees) imposing the additional constraints Xi= 1 or Xi=2
Xi= 1.3
X= 1
i
Solve the corresponding LP problem for each node:
1.If the solution is infeasible or OF is larger than UB -> prune the node
2.If in the solution all required variables are integer -> :
Xi= 2
– If no upper bound (UB) exists on the solution -> store the corresponding OF as an initial
upper bound (UB)
– If the OF is smaller than an already existing UB -> update the upper bound
– If the OF is larger than an already existing UB -> prune the node
3.If in the solution some required variables are still non-integer -> branch further the
problem
Continue until all required variables are integer. This method is called «branch &
bound », alternative methods are: « branch & cut », « branch & price », « implicit
enumeration », etc.
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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HEP APPLICATIONS: EXAMPLE 1
Global VDET pattern recognition for ALEPH, P.Rensing, CHEP95
Extrapolating tracks from TPC to the silicon vertex detector (VDET), featuring two
double-sided lawyers (4 readout layers), presents ambiguities in hit assignment
process. How to assign VDET hits to tracks optimally?
Minimize z = ijklm Cijklm xijklm, with xijklm=0,1
subject to:
TPC
 track i
jklmxijklm=1
 hit j in layer 1
iklmxijklm=1,2
 hit k in layer 2
ijlmxijklm=1,2
 hit l in layer 3
ijkmxijklm=1,2
(1 configuration per track)
(1 or 2 tracks per
hit, depending on
hit signal amplitude)
 hit m in layer 4 ijklxijklm=1,2
1
2
VDET
3
4
Cijklm = 2 + 11NNon associated + 4N1 dimension
(i=0 means no assigned track; j,k,l,m=0 means no hit in VDET layer 1,2,3,4)
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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HEP APPLICATIONS: EXAMPLE 2
Linear pattern recognition algorithm for the CMS barrel muon detector,
A. De Min, CMS Note, Feb 2006, being submitted to CMS
In CMS  tracks are straight lines in a noisy environment.
Linear programming can be used to remove outliers:
1. Pi =(Xi,Zi) should be associated to straight line x=mz+q if |Xi-mZi+q|  i
(i is the « resolution » of Pi)
mZi+q-Xi  i
2. This is not a linear constraint! But it can be linearized:
-mZi-q+Xi i
3. Best (linear) fit is min  ( /  ) subject to: mZi+q-Xi i
i
i
i
-mZi-q+Xii
provided that only points with i  i are considered (pattern recognition)
4. Try combining pattern recognition and fit in a single problem by adding
the new integer (binary) variables i = {0,1} and new constraints
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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HEP APPLICATIONS: EXAMPLE 2
The problem can be formulated as follows (M is a big, fixed number):
Minimize residuals ->
Minimize f = i (i / i ) subject to,  i:
Straight line ->
mZi+q-Xi  i +iM
-mZi-q+Xi i +iM
Pattern recognition ->
i - i  iM
i - i  (1-i)M
New binary variable -> i = {0,1}
m,q  0
Hit
Zi
Xi
i
i
i
1
1.0
1.0
0.1
0.000
0
(i=1)  (i  i)  hit Pi out of the fit
2
1.0
2.1
0.1
0.1
1
The solution will provide a fast, unbiased
pattern recognition and a stable (linear)
fit in one go! It also provides the list of
included points (i=0) and their residuals i
3
2.3
1.6
0.1
0.072
0
4
3.0
2.2
0.1
0.1
1
5
3.9
2.5
0.1
0.000
0
6
5.0
3.1
0.1
0.031
0
(i=0)  (i  i)  hit Pi in the fit
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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OTHER HEP APPLICATIONS
Apart from pattern recognition problems, LP may be useful in
several other applications related to HEP. A few examples:
1. Detector design (ex. to optimize cabling network in order to
minimize the material in front of the calorimeters)
2. Detector construction (ex. optimal assembly schedule, optimal use
of resources, supply chain management and spare parts procurement)
3. Detector alignment (ex. tracker subparts alignment with charged
tracks)
4. Detector calibration (ex. optimal shower clustering in test-beams
for calorimeter)
5. Data analysis (ex. optimal association of calorimeter showers to
charged tracks)
6. Logistics in HEP laboratories (ex. shift roster of minimum cost,
optimal test-beam period allocation)
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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AVAILABLE LIBRARIES
Two kinds of packages:
Modeling systems:
• Designed to help people formulate LP problems and analyze their solutions.
• In practice they are modeling languages.
• Output is in standard format to feed any LP solver.
• Not of interest here, more dedicated to consultants who need to quickly
apply LP techniques to solve very different problems.
• An old but still widely used standard format is the MPS format (originally
from IBM). Essentially a matrix description convention.
LP solvers:
• Algorithmic codes devoted to finding optimal solutions to specific linear
problems.
• A code takes as input a the LP numerical coefficients in standard format
and produces as output a compact listing of optimal solution values.
• LP solvers are available as commercial packages and as open source
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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AVAILABLE LIBRARIES
Most used commercial packages:
–
–
–
ILOG/CPLEX: Simplex,Interior,Integer (www.ilog.com)
Xpress-MP: Simplex,Interior,Integer (www.dashoptimization.com)
IBM/OSL: Simplex,Interior,Integer (www.optimize.com)
In general rather sophisticated and expensive, but special agreements
possible for research use. Some versions are parallelizable for GRID platforms.
Best open source packages:
–
–
–
COIN-OR (COmputational INfrastructure for Operations Research): Simplex,Integer.
Originally from IBM and now available under Common Public Licence (www.coin-or.org)
GLPK (GNU Linear Programming Kit): Simplex,Interior,Integer. Available under the
GNU General Public Licence (www.gnu.org/software/glpk/glpk.html)
Lp_solve: Simplex,Integer. Available, under the GNU Lesser Public Licence (www.coinor.org)
Not as performing as commercial codes, but free and available in source
code. See licence for permitted uses.
For additional information see also:
http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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CONCLUSIONS
A wide class of problems encountered in HEP can be formulated
similarly to problems related to other fields of human activities,
normally covered by Operations Research
It is advisable that the HEP community is kept informed of the
latest OR achievements for what concerns both mathematical
algoritms and software packages and has quick access to all available
tools
Limited to (mixed-integer) linear programming, this talk has
provided a brief overview of the existing tools and of their possible
applications to HEP
A. De Min
CHEP06 Mumbai, India 13-17 February 2006
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