Reformulating the disjunctive cut generating linear programThiago Serra [email protected] @thserra Advisors: Egon Balas & Francois Margot
1.| Introduction
Lift-and-project resembles a Swiss armyknife to define and generate valid cuts.
But why using just one blade to cut in asingle way? Perhaps we could avoidgetting increasingly more parallel andshallow cuts by diversifying in the CGLP.
2.| Background
Consider a binary linear program
min{cTx : Ax ≥ b, x ∈ {0,1}n}with linear relaxation
P := {x : Ax ≥ b, 0 ≤ x ≤ 1}:= {x : Ax ≥ b}
having a fractional basic solution
x = arg min{cTx : x ∈ P}, 0 < xj < 1.
Feasible solutions are covered by
Dj :={x : Ax ≥ b, − xj ≥ 0}∪{x : Ax ≥ b, xj ≥ 1},
which convexifies as Pj := conv(Dj).From Farkas lemma, any valid αTx ≥ βfor Pj is found in the (α, β)-projection of
α −uT A +u0ej = 0α −vT A −v0ej = 0β −uT b ≤ 0β −vT b −v0 ≤ 0
u, v , u0, v0 ≥ 0
(C)j
For Pj, Balas et al. [1993] show that allnon-dominated inequalities are found in
α −uT A +u0ej = 0α −vT A −v0ej = 0β −uT b = 0β −vT b −v0 = 0
u, v , u0, v0 ≥ 0
(C)j
and it has as dual the lift-and-projectdescription of x ∈ Pj as x = x0 + x1 for
Ax0 −bx00 ≥ 0
−x0j ≥ 0
Ax1 −bx10 ≥ 0
x1j −x1
0 ≥ 0x0
0 + x10 = 1
(D)j
with the nonnegativity of x00 , x
10 ≥ 0
relaxed [Ceria and Soares, 1997].
We look for αTx ≥ β separating x with aCut Generating Linear Program (CGLP)minimizing αTx − β and bounding (C)j :
min αT x − βs.t. (C)j
uTe + vTe + u0 + v0 = 1(CGLP)j
However, the cut may be strictlydominated [Fischetti et al., 2011]:
For a split on j = 1, (CGLP)j yields:(c1) if k ≤ 8; (c2) is k = 8; (c3) if k ≥ 8
3.| Reverse polar CGLP
For some p ∈ Pj, we propose thefollowing formulation:
min αTp − βs.t. (C)j
β − αT x = 1(RP-CGLP)pj
Using (C)j instead, (RP-CGLP)pj is an
extended formulation of the reverse polar(Pj−x
)−:={
y : yT (x − x) ≥ 1 ∀x ∈ Pj}
.
Following the reverse polar notation, wedenote cuts in the form yT (x − x) ≥ 1.
¶ Cuts from (RP-CGLP)pj define
supporting hyperplanes of Pj
x
p
Among parallel inequalities, gettingcloser to p reduces ‖y‖ since distance to x× cut norm = 1, thus reducing yT (p − x).
Ü No cutting plane is strictly dominated
How facets of Pj are combined in cuttingplanes from (RP-CGLP)p
j ?
Let the facet-defining inequalities of Pj
define F ={
(γ i)T (x − x) ≥ δi}
i∈F with Ffinite for A,b rational. W.l.o.g., let uspartition F = F + ∪ F 0 ∪ F−, where:
δi =
1, i ∈ F +
0, i ∈ F 0
−1, i ∈ F−
A cut yT (x − x) ≥ 1 from (RP-CGLP)pj
has a nonnegative combination {λi}i∈F :
∑i∈F
λiγi =y∑
i∈F+
λi −∑i∈F−
λi =1
· For a cut yT (x − x) ≥ 1 from(RP-CGLP)p
j , either λi = 0 or(γ i)T (p − x)− δi = 0 ∀i ∈ F 0 ∪ F−
If neither holds, we could find aface-defining cut with strictly betterobjective value by reducing such λi andaccordingly other multipliers λj, j ∈ F +.
¸ If p ∈ int(Pj), a cut from (RP-CGLP)pj
is a combination of facet-defininginequalities separating x
If p ∈ int(Pj), then λi = 0 ∀i ∈ F 0 ∪ F−.
¹ For a cut yT (x − x) ≥ 1 from(RP-CGLP)p
j , it holds thatyT (p − x) = (γ i)T (p − x) ∀i ∈ F + : λi > 0
If facet i has larger slack than facet j in aproper combination, there is aface-defining cut with strictly bettervalue where λj ← λj + λi and λi = 0.
Î A cut yT (x − x) ≥ 1 from (RP-CGLP)pj
is a combination of inequalities in Fthat first intersect the ray from x to p
The objective becomes minαT (p − x)− 1by replacing β. Points defining the sameray with x yield the same cuts. For thoseclosest to x , (γ i)T (p − x)− δi = 0 ∀i ∈ F +.
4.| Equivalent formulations
A similar reformulation is proposedby Balas and Perregaard [2002]:
min αT x − βs.t. (C)j
αT (p − x) = 1(BP-CGLP)pj
Ü They prove ¶ and º for (BP-CGLP)pj
Inspired by Cadoux and Lemarechal[2013], we denote as the polar CGLP thefollowing bounded CGLP formulation:
min αT x − βs.t. (C)j
αTp − β = 1(P-CGLP)pj
Ü Property ¶ follows similarly. Withobjective restated as minαT (x − p)− 1,getting closer to p increases ‖α‖.
Buchheim et al. [2008] enumeratesextreme points to define a GeneralizedIntersection Cut (GIC) equivalent [Balasand Kis, 2016] of formulation (P-CGLP)p
j ,but replacing Pj with P∩{0,1}n.
» (RP-CGLP)pj , (BP-CGLP)p
j , and(P-CGLP)p
j yield the same cuts
Let us denote cuts separating x asµTx ≥ ν, ‖ν‖ = 1. For each CGLP, there isa scale factor θ > 0 : (α, β) = θ(µ, ν).
(RP-CGLP)pj : θ = 1
ν−µT x
minαTp − β = min θ(µTp − ν) = min µT p−νν−µT x
(P-CGLP)pj : θ = 1
µT p−ν
minαT x −β = min θ(µT x − ν) = max ν−µT xµT p−ν
(BP-CGLP)pj : θ = 1
(µ)T (p−x)
minαT x − β = . . . = min µT p−νν−µT x + 1
¼ CGLP dual yields first intersection
The lift-and-project primal of (BP-CGLP)pj
yields x0 + x1 as the intersection from º:
min ω
s.t. (D)jx0 + x1 + ω(x − p) = x
(BP-L&P)pj
5.| Finding p
½ Any proper convex combination ofpoints in the relative interior of eachterm of Dj is in the relative interior of Pj
¾ Any point in the relative interior of Pj
has a convex combination of points inthe relative interior of each term of Dj
Ü For any cut, all we need is a point inthe relative interior of each term of Dj
Using ½, we can get a cut satisfyingproperty ¸ if Pj is full-dimensional.
On each term of the disjunction, weidentify inequalities always satisfied atequality and get a point maximizing theminimum slack on the remaining ones.
Let p be a combination in proportion tothe minimum slacks of those points.
6.| Results
Similar performance in the first roundand better performance in the second
Gap closed (%) by adding split cuts onfractional variables with standard (S) andRP CGLP (R) and resolving on MIPLIB:
1 round 2 roundsS R S+S S+R R+S R+R
bm23 5.6 5.7 12.2 13.4 11.4 13.6lseu 4.2 4.2 14.8 31.2 17.0 31.2mod008 0.1 0.1 1.6 1.9 0.5 1.9p0033 2.6 2.2 5.6 11.0 6.6 11.0p0040 6.7 6.7 11.5 11.4 11.0 11.4p0201 0.0 0.0 16.9 16.5 15.6 0.0p0282 48.4 62.9 54.3 71.5 74.0 78.3p0291 50.5 46.9 84.9 84.9 91.0 91.3sentoy 8.2 7.8 15.2 16.3 14.3 17.0
Varied CGLPs on x and p ∈ int(Pj) havegood properties and yield the same cuts
The cut is a combination of facets from Pj
separating x and laying between p and x .
Ü Contraposes the concern by Cadouxand Lemarechal [2013] with reverse polarnormalization due to its unboundedness.
Ü Extends properties in Buchheim et al.[2008] to lift-and-project formulations,both bringing the supporting hyperplanemethod from Veinott [1967] to MIP.
RP CGLP is better for multiple root cuts
The feasible set is invariant to p, hencechanging it only affects cut evaluation.
7.| Next steps
Beyond finding just some p ∈ int(Pj):
Ü find p yielding one, facet-defining cut
Ü find such points yielding disjoint cuts
If Pj is not full-dimensional, we couldintegrate facial decomposition [Borweinand Wolkowicz, 1981] to the CGLP.
Generating p and solving the CGLP isexpensive. Learning what we need fromboth, we should aim for slim equivalents:
| References
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E. Balas, S. Ceria, and G. Cornuejols. A lift-and-project cutting plane algorithm for mixed0-1 programs. Math. Progr., 58:295–324, 1993.
J. Borwein and H. Wolkowicz. Regularizing theabstract convex program. Journal of MathematicalAnalysis and Applications, 83(2):495 – 530, 1981.
C. Buchheim, F. Liers, and M. Oswald. Local cutsrevisited. Ops. Res. Letters, 36:430–433, 2008.
F. Cadoux and C. Lemarechal. Reflections ongenerating (disjunctive) cuts. EURO J. Comput.Optim., 1(1-2):51–69, 2013.
S. Ceria and J. Soares. Disjunctive cuts for mixed0-1 programming: duality and lifting. GSB, 1997.
M. Fischetti, A. Lodi, and A. Tramontani. On theseparation of disjunctive cuts. MathematicalProgramming A, 128:205–230, 2011.
A.F. Jr. Veinott. The supporting hyperplanemethod for unimodal programming. OperationsResearch, 15(1):147–152, 1967.
