lift-and-project cuts in cplex 12.5.1

Decision OptimizationDecision Optimization

Lift-and-Project Cuts in CPLEX 12.5.1Andrea TramontaniCPLEX Optimization, IBM

INFORMS Fall Conference 2013

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Literature review– The Cut Generating Linear Program (CGLP)– L&P cuts à la Balas-Perregaard– L&P cuts and Gomory Mixed Integer (GMI) cuts– L&P cuts à la Bonami

L&P cuts in CPLEX 12.5.1: Performance Impact

Outline



Lift and Project Cuts Beginnings

Invented by Balas in the ’70s:– E. Balas, Disjunctive programming. Ann. Discrete. Math. 5, 3–51, 1979.

Succesfully exploited in practice in the ’90s:– E. Balas, S. Ceria, and G. Cornuejols, A lift-and-project cutting plane algorithm for

mixed-integer programs. Math. Progr. 58, 295-324, 1993.– E. Balas, S. Ceria, and G. Cornuejols: Mixed 0-1 programming by lift-and-project in a

branch-and-cut framework. Man. Science 42, 1229-1246, 1996.



Lift and Project Cuts: The Separation Problem

Given:

The LP relaxation of a MILP, defined by the underlying polyhedron P:

A fractional solution x* in P (typically, a vertex of P) to cut off

A binary fractional variable xj*: xj in {0,1}, 0 < xj* < 1

Find a violated cut

α x ≥ β

using the simple split disjunction

xj ≤ 0 OR xj ≥ 1

(P) A x ≥ bx ≥ 0



The Cut Generating LP (CGLP) of Balas, Ceria and Cornuejols

1. Consider the polyhedra P0 and P1 induced by the disjunction:

2. Separate a cut valid for the disjunctive polyhedron

Q = conv (P0 ⋃ P1)

by solving the following CGLP:

3. Strengthen the cut by Monoidal Strengthening (Balas and Jeroslow, Eur. J. Oper. Res. 1980)

P0

(u) A x ≥ b(w) x ≥ 0(u0) - xj ≥ 0

P1

(v) A x ≥ b(z) x ≥ 0(v0) xj ≥ 1

α = u A + w – u0 ejβ = u b

u, w, u0 ≥ 0

α = v A + z + v0 ejβ = v b + v0

v, z, v0 ≥ 0

min α x* - β

P0

P1

x*

αx ≥ β



CGLP: The Role of Normalization

The CGLP is unbounded by construction (if a violated cut exists)

Can be normalized in many different ways: see, e.g.,– Balas, Ceria and Cornuejols (Math. Prog.1993, Man. Science 1996)– Ceria and Soares (Tech. Report, 1997)

The normalization is the real objective function and makes the difference: see, e.g., – Fischetti, Lodi and T. (Math. Prog. 2011)

Two popular normalization constraints are:– The weak normalization (actually, the one used in CPLEX):

u0 + v0 = 1 (1)

– The strong normalization (Balas, Math. Prog. 1997):

1 u + 1 v + 1 w + 1 z + u0 + v0 = 1 (2)



Lift and Project Cuts à la Balas-Perregaard (BP L&P cuts)

Balas and Perregaard (Math. Prog. 2003) adopt normalization (2) and:

1. Provide a precise correspondence between feasible CGLP bases and (possibly infeasible) bases of the tableau of the original problem

2. Characterize Gomory Mixed Integer (GMI) cuts from the tableau as CGLP bases

3. Develop an efficient approach for solving the CGLP by a sequence of pivoting in the tableau of the original space.

Different variants of the idea avaliable in commercial and open-source solvers:

FICO XPRESS, see Balas and Perregaard (Math Progr. 2003)

CBC (COIN-OR B&C), see Balas and Bonami (Math. Prog. Comp. 2009)

Warmstart CGLP with GMI basis

Pivots in the spaceA x – I s = b

Optimal CGLPsolution



L&P Cuts à la Balas-Perregaard vs. Tableau GMI cuts

GMI cuts vs. Balas-Perregaard L&P cuts in Cgl 0.57.1. Test set: 52 MIPLIB3.0 problems.

Results taken from Pierre Bonami‘s talk in Bologna (Italy), 2012

%gap closed: 100 x (zcut - zinitial)/(zopt - zinitial), arithmetic means

Sep. Time: geometric means

Sep. Time

%gap closed # cuts # active density k

GMI - 1 round 0.01 28.79 20.60 9.08 95.95 4.53 x 1e+4

L&P - 1 round 0.07 30.11 25.33 11.90 79.44 5.19 x 1e+4

GMI - 10 rounds 0.06 49.33 212.52 20.62 216.12 5.64 x 1e+6

L&P - 10 rounds 2.31 60.77 310.49 38.87 170.40 2.37 x 1e+6

L&P closure 1.08 60.80 309.49 60.98 86.75 6.02 x 1e+5



L&P Cuts à la Balas-Perregaard: why better than tableau GMI cuts?

No dominance relationships between tableau GMI cuts and BP L&P cuts

But just a matter of normalization:

Empirical observation from Fischetti, Lodi and T. (Math. Prog. 2011):

Normalization (2) is much better than (1) because it:– Enforces the separation of cuts with sparse dual and primal support– Tends to produce low-rank inequalities – Breaks the correlation between optimization and separation

Tableau GMI cuts are Optimal CGLP cuts with u0 + v0 = 1



Why normalization (2) better than (1)?

Normalization (2):

Enforces the separation of sparse cuts:

Solving CGLP is equivalent to fix the violation as a constraint and use normalization (2) as the objective function: i.e., minimize the L-1 norm of the multipliers vector.

α = u A + w – u0 ejβ = u b

u, w, u0 ≥ 0

α = v A + z + v0 ejβ = v b + v0

v, z, v0 ≥ 0α x* - β= -1

min 1 u + 1 v + 1 w + 1 z + u0 + v0



1 u + 1 v + 1 w + 1 z + u0 + v0 = 1

α = u A + w – u0 ejβ = u b

u, w, u0 ≥ 0

α = v A + z + v0 ejβ = v b + v0

v, z, v0 ≥ 0

Why normalization (2) better than (1)?

Normalization (2):

Introduce a natural scaling that penalizes the separation of high rank inequalities

1. Each constraint is penalized in the normalization, but the penalty is constant and therefore affected by the scaling of the constraint.

The cut obtained using ai x >= bi with ui could be obtained using K ai x >= K bi with 1/K uithe smaler the norm, the larger the penalization

2. Cuts are obtained by aggregating constraints with multipliers that sum to 1:Each cut is automatically scaled down w.r.t. the constraints used to generate itIt will be penalized in the subquent separation

minα x* - β



Normalization (2) vs. Normalization (1): four pictures on one MIPLIB3.0 model

Average density of (u,v)

Average cut density

Dual bound evolution

Cut rank



L&P Cuts à la Balas-Perregaard vs. Tableau GMI cuts (Cont.d)

GMI cuts vs. Balas-Perregaard L&P cuts in Cgl 0.57.1. Test set: 52 MIPLIB3.0 problems.



Sep. Time, # cuts, # active, density: geometric means

k: condition number of the final optimal basis

Sep. Time


GMI - 1 round 0.01 28.79 20.60 9.08 95.95 4.53 x 1e+4

L&P - 1 round 0.07 30.11 25.33 11.90 79.44 5.19 x 1e+4

GMI - 10 rounds 0.06 49.33 212.52 20.62 216.12 5.64 x 1e+6

L&P - 10 rounds 2.31 60.77 310.49 38.87 170.40 2.37 x 1e+6

L&P closure 1.08 60.80 309.49 60.98 86.75 6.02 x 1e+5



Lift and Project Cuts: Our Idea

One more trivial empirical observation:

Separating rank-1 cuts:– Tends to produce cleaner and sparser cuts– Enforces the separation of low-rank inequalities (not a big surprise)– Breaks the correlation between optimization and separation

Sounds like normalization (2)

Separate some more rank-1 L&P cuts with u0 + v0 = 1

Keep tableau GMI cuts as they are (if your filter likes them enough), and



Lift and Project Cuts à la Bonami (this is what we did!)

Bonami (Math. Prog. Comp., 2012):

1. Adopts normalization u0 + v0 = 1

2. Applies some algebra to get rid of u0 and v0 in the CGLP

3. Takes the dual of the simplified CGLP and gets the so called Membership LP (MLP):

4. Provides promising computational results on MIPLIB 3.0 and MIPLIB 2003 models

The (strengthened) L&P closure:– Can be computed in practice– Provides a good approximation of the split closure

(MLP) max yj - δ*

(v) b δ* ≤ A y ≤ b δ* + s* (u)(z) 0 ≤ y ≤ x* (w)

s* := A x* - bδ* := xj*



Balas-Perregaard L&P cuts, Tableau GMI cuts, Rank-1 L&P cuts

GMI cuts vs. BP L&P cuts in Cgl 0.57.1 vs. Rank-1 L&P closure. Test set: 52 MIPLIB3.0 problems.



Sep. Time, # cuts. # active, density: geometric means

k: condition number of the final optimal basis

Sep. Time


GMI - 1 round 0.01 28.79 20.60 9.08 95.95 4.53 x 1e+4

L&P - 1 round 0.07 30.11 25.33 11.90 79.44 5.19 x 1e+4

GMI - 10 rounds 0.06 49.33 212.52 20.62 216.12 5.64 x 1e+6

L&P - 10 rounds 2.31 60.77 310.49 38.87 170.40 2.37 x 1e+6

L&P closure 1.08 60.80 309.49 60.98 86.75 6.02 x 1e+5



Lift and Project Cuts in CPLEX 12.5.1

Bonami MLP to separate rank-1 cuts only, plus some tricks:

Heuristic reduction of MLP to– Speed up the separation– Get rid of constraints you don‘t want to use to produce the cut

Clever selection of the disjunctions to use

Not explored but worthy to mention:

Dash and Goycoolea (Math. Prog. Comp., 2010)

Fischetti and Salvagnin (Math. Prog. Comp., 2011)



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L&P Cuts in CPLEX 12.5.1: Performance Impact All ModelsAll Models

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Date: 23 May 2013Testset: 3243 models Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threadsTimelimit: 10,000 sec



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L&P Cuts in CPLEX 12.5.1: Performance Impact

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Date: 23 May 2013Testset: 3243 models Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threadsTimelimit: 10,000 sec

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