Strong Bounds for Linear Programs with Cardinality

Limited Violation (CLV) Constraint Systems

Ronald L. Rardin

University of Arkansas

rrardin@uark.edu

Mark Langer, M.D.

Indiana University

mlanger@iupui.edu

Ali TuncelPurdue University

atuncel@purdue.edu

Jean-Philippe RichardPurdue University

jprichar@purdue.edu

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

njx

x

mikmbxa

xc

j

n

jjij

n

jjj

,,1 0

)on sconstraintlinear other (

,,1 of )(on

max

1

1

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

b

miy

ky

mibybbxa

i

m

ii

i

n

jjij

,,1 }1,0{

,,1 )(

1

1

b

b

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

njx

Px

mikmbxa

xc

j

n

jj

n

jjij

n

jjj

,,1 0

budget) (portfolio

,,1 of )(on

return)mean (max max

1

1

1

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

njx

mikmbxa

xc

j

n

jjij

n

jjj

,,1 0

y)homogeneit and targets,secondary

ues,other tisson sconstraint(linear

,,1 of )(on

dose) (max tumor max

1

1

Dose-Volume Tissues

+

+

++

+

+++

++

+

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.

j

kmkmja

kmba

mkbx

ij

ij

jjj

Theorem

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.

bxj

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

njx

x

bxa

xc

j

i

n

jjij

n

jjj

,,1 0

)on sconstraintlinear other (

)(SCE

max

1

1

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

njx

huGxu

bxa

xcz

j

ii

n

jjqj

n

jjjq

,,1 0

max

1

1

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

?

,,1 0

max

1

1

1

bxa

njx

hGx

bxa

xcz

n

jjqj

j

n

jjij

n

jjjq

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