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Strong Bounds for Linear Programs with Cardinality
Limited Violation (CLV) Constraint Systems
Ronald L. Rardin
University of Arkansas
Mark Langer, M.D.
Indiana University
Ali TuncelPurdue University
Jean-Philippe RichardPurdue University
Workshop on Mixed Integer ProgrammingAugust 4-7, 2008
Models LPCLV have all constraints and the objective linear, but some constraints belong to systems where up to k of m may be violated
LP’s with CLV Constraints
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If some outer limit on violation is available, the CLV system is easily modeled
But a difficult MILP often results, especially when there is a great difference between and
MIP Formulation of CLV Constraints
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LPCLV models arise in Value at Risk portfolio optimization, where decision variables xj represent the investment from budget P in asset j , and CLV constraints reflect risk under various scenarios i, at most k of which may exceed b
Application: Portfolio Optimization
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Application: Radiation Therapy (Fluence) Planning
• Radiation is delivered by an accelerator that rotates 360 degrees around the patient
• Full beam area is large, around 10 cm square. However, Intensity Modulated Radiation Therapy (IMRT) can deliver shaped intensity maps composed of 3-10mm rectangular beamlets with independent fluence
Application: Radiation Therapy (Fluence) Planning• The goal of intensity/fluence/timeon planning
optimization is to assign optimal beamlet intensities xj so that prescribed doses are delivered to different types of tissues: Primary Target: Tumor region. Maximize dose within
homogeneity requirements Secondary Targets: Minimum dose limits to eliminate
risk of microscopic infection. Healthy/Normal Tissues: Maximum dose limits on all
or specified fractions of surrounding tissues to avoid complications.
Dose-volume limits are ones on fractions of normal tissue volume (e.g. no more than 20% of points in the tissue can receive a dose exceeding 60)
Tissues are modeled as collections of imbedded points i
Dose at any point is approximately linear in beamlet fluence xj
Leads to a CLV system for each dose-volume constrained tissue with other constraints linear
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Dose-Volume Tissues
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Computational Challenge – Hard Cases
The difference between b and b values influences the initial optimality gap.
Rounding Heuristic
An LP-rounding heuristic often gives feasible integer solutions
1. Solve the LP relaxation
2. Sort the points in each dose-volume tissue by LP relaxation dose
3. Force all but the highest-dose k to satisfy b
4. Solve the modified LP relaxation Results are good if the LP is a fair to good
approximation of the MIP
Computational Challenge
LP relaxation and rounding heuristic
Compared to CPLEX 11.0
* feasible solution from later methods
Disjunctive Cutting Planes
LPCLVs As Disjunctive Programs
m=5, K=2
2 8
5 4
8 2
3 6
7 3
A
x1
x2
b = 20
Disjunctive programming studies optimization problems over disjunctions (unions) of polyhedra (LP feasible sets)
Easy to see how LPCLVs can be viewed this way as a union over all the LPs with different sets of (m-k) of the tighter b RHSs explicitly enforced (here all 3 of 5)
Valid Inequalities for Disjunctive Programming
Balas proved that all needed valid inequalities for disjunctions of polyhedra can be constructed by rolling up each of the member linear systems with nonnegative multipliers, and picking the lowest of the rolled coefficients on the LHS and the largest on the RHS
Valid Inequalities for LPCLV – Main Disjunctive Inequality
m=5, K=22 8
5 4
8 2
3 6
7 3
A
x1
x2
b = 20
8 8
7 6
5 4
3 3
2 2
A
Average of lowest 3 values
1 2
103 20
3x x Main Disjunctive Inequality
n.)disjunctio theofelement any in satisfied sconstraint CLV the
of )( on the )/(1 smultiplier Balas using tods(Correspon . that of
lowest )( theof average theis each where, side-hand-right and
tscoefficienover systems CLV of afor validis inequality The
Rardin) and Walters-Preciado Sen, and (Sherali.
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Valid Inequalities for LPCLV – A Family of Disjunctive Inequalities
Lemma. For any subset S of i=1,…,m with K or more elements, the implied LPCLV(A[S],b,K) is a relaxation of the full LPCLV(A,b,K) where A[S] is the row sub- matrix of A for rows in SProposition. For any subset S of i=1,…,m with more than K elements, the inequality
is valid for full system LPCLV(A,b,K), where eachis the average of the (|S|-K) smallest coefficients in rows in S.
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jSj
Sj
Valid Inequalities for LPCLV – Disjunctive Support Inequalities
m=5, K=2
2 8
5 4
8 2
3 6
7 3
A
x1
x2
b = 20
8 4
7 3
5 2
A
Average of the lowest value
1 25 2 20x x Disjunctive Support Inequality
Proposition. Let subset Sk of i=1,…,m be the K+1 rows with largest coefficients on xk . Then the inequality
is valid for full system LPCLV(A,b,K), where each is smallest j coefficient for rows in Sk . Furthermore the inequality supports the convex hull of solutions to the full LPCLV(A,b,K) (at xk = b / , other xj = 0)
bxj
jSjk
kSj
kSk
Results for Disjunctive Cuts
LP relaxation and rounding heuristic (except *)
LP relaxation with disjunctive cuts and rounding heuristic
* feasible solution from later methods
Single Constraint Enumeration
SCE Concept
Consider relaxations enforcing only one row of the CLV system at a time
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SCE Concept
Proposition. Let zi denote the optimal solution value of SCEi. Then the k+1st smallest such zi is a valid upper bound on the optimal solution value of the full LPCLV.
(Proof idea: There can be at most k of the single constraints violated in LPCLV, but thereafter all constraints must be satisfied at the b value.)Row
i zi
Row i zi
Row i zi
3 43 8 65 1 78
6 52 4 67 5 81
9 56 2 72 7 90
Example with m=9, k=3
Direct SCE (i) solves all one-row problems, (ii) sorts solution values, and (iii) picks the k+1st smallest
Do better by using one-row problems solved to get adaptive upper/lower bounds for unsolved SCEi
……
…
Each Box Shows SCE Value Upper – Lower Range with Bar for True Optimal Value
Solved Case
K+1st Lowest Upper Bound Is SCE Upper Bound
K+1st Lowest Lower Bound Is SCE Lower Bound
Bounds Improved by Solving Other SCEs
Adaptive Single Constraint Enumeration (ASCE)
ASCE – Global Bounds The K+1st lowest of the upper
bounds is a running upper bound on the ultimate SCE bound obtained (max of theseupper bounds max of their z’s >= max of ultimate K+1 z’s)
The K+1st lowest of the lower bounds is a running lower bound on the ultimate SCE bound obtained (max of these lower bounds <= max of lower bounds for ultimate K+1 <= max of their z’s)
ASCE – Surrogate Relaxation Upper Bounds Each SCEi solved produces dual multipliers that can be used
to quickly update running upper bounds on solution values for other SCEq
Construct a surrogate constraint valid for each SCEq using the dual solution for SCEi without any multipliers on the CLV constraints to roll up all the others Gx<=h
For each q not i in turn, solve a simple relaxation of SCEq having only such surrogate constraints and the single CLV inequality for q
These yield upper bounds on the solution values of the corresponding SCEq , and the running best upper bounds on their values can be updated
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ASCE – Common Feasible for Lower Bounds
If an SCEi solution vector is also feasible for constraint q of the CLV system, then i’s solution value is a lower bound on that SCEq and the running lower bound on that solution value can be updated
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ASCE - Pruning
If the running upper bound on the value of SCEq is already < the best known LPCLV feasible solution value, then q is certain to be among the lowest K+1 that establish the global bound. No further processing of SCEq is needed.
If at least K+1 of the running upper bounds are no greater that the running lower bound for SCEq then it cannot be in the lowest K+1. No further exploration of q is required
Results with SCE and ASCE
LP relaxation with disjunctive cuts and rounding heuristic
LP relaxation with disjunctive cuts and SCE/ASCE and rounding heuristic
Conclusions
Linear Programming Relaxations of MIP formulations of LPCLV problems can be arbitrarily weak
Commercial MIP solver (CPLEX 11) is not effective in closing the initial large optimality gaps
Efficient valid inequalities based on disjunctive programming theory can be generated to: Strengthen LP relaxations and significantly reduce initial
optimality gaps when problems are dense Help find better feasible solutions within the rounding
heuristic framework Improve run times
Conclusions
SCE procedure can be implemented as a stand-alone procedure or in branch and bound Significantly reduces upper bound values. In some cases, finds better feasible solutions
Adaptive SCE accomplishes the SCE computation in significantly reduced time
The combined algorithm utilizing disjunctive inequalities and the ASCE procedure within the rounding heuristic framework produces lower optimality gaps and better feasible solution values