Sparse Cutting-Planes

Marco MolinaroTU Delft

Santanu Dey, Andres IroumeGeorgia Tech

Analyze sparse inequalities under more “realistic” geometric models that incorporate characteristics from IP

IP AND CUTTING-PLANES

Max

integer for

Interested in general Mixed-integer Programs

Cutting-plane: linear inequality valid for all solutions

Extremely important for solving IPs in practice

Cutting-planes as better approximation to the integer hull

CUTTING-PLANES

IN THEORY• Can use any cutting-plane• Putting all gives exactly the integer hull• Many families of cuts, large literature, since 60’s

IN PRACTICE• Only want to use sparse inequalities

• Solvers use sparsity to filter out cuts– LP solvers are much faster if inequalities are sparse

(sparse lin. alg.)

• Very limited investigation [Andersen-Weismantel 10]

• Do not give integer hull

CUTTING-PLANES

𝑎. 𝑥≤𝑏at most non-zero entries

1-sparse

-sparse inequality:

IN PRACTICE• Only want to use sparse inequalities

• Solvers use sparsity to filter out cuts– LP solvers are much faster if inequalities are sparse

(sparse lin. alg.)

• Very limited investigation [Andersen-Weismantel 10]

• Do not give integer hull

SPARSE CUTTING-PLANES

𝑎. 𝑥≤𝑏at most non-zero entries

1-sparse

-sparse inequality:

GOAL: Understand sparse cutting-planes

• How strong are sparse cutting-planes?• How to generate sparse cuts?• ….

polytope in (e.g. integer hull) intersection of all -sparse inequalities

SPARSE APPROX OF POLYTOPES

𝑷𝑷 𝒌

Geometric abstraction: [Dey, M., Wang]

polytope in (e.g. integer hull) intersection of all -sparse inequalities

𝑷𝑷 𝒌

GOAL: How does behave?

SPARSE APPROX OF POLYTOPES

Geometric abstraction: [Dey, M., Wang]

[𝟎 ,𝟏 ]𝒏at most

Ex 1: = k-subset of

Ex 2: =

Ex 3: – convex hull of random 0/1 points (computational)

𝑑 (𝑃 ,𝑃𝑘 )

(density)

∝𝟏√𝒌

∝𝟏𝒌

good

bad

𝑛/2

√𝑛/2

RESULTS FROM [DMW]

min {√𝑛 ,−5 √𝒏√𝒌 √𝐥𝐨𝐠 (𝒏 .¿𝒗𝒆𝒓𝒕 (𝑷 )) ,−2√𝒏(𝒏𝒌−𝟏)}

𝑑 (𝑃 ,𝑃𝑘 )

• We proved in [DMW] matching upper and lower bounds for gap

• Sparse cuts are good if number of vertices is “small”; else can be bad

Strong lower bounds: Most random 0/1 polytopes with vertices and packing problems have gap even for

Sparse cuts are bad for IP ??

1. Does not allow any dense inequality in the approximation(in IP using a few dense cuts is ok)

2. What if we allow affine transformations to the polytope?(IP reformulation to improve strength of sparse cuts)

3. We require polytope to be approximated in every direction(in IP only care about objective function direction)

CAVEAT

Basic setting misses features of IP; may be too pessimistic

New: Add these features to basic setting to capture more closely features of IP [Dey, Iroume, M.]

Back to Ex 2: =

Adding only sparse ineq is bad: gap for density

But get exactly if we use sparse ineq + 1 dense inequality

1. SPARSE + FEW DENSE

Q: Can we show that sparse ineq+ a few dense ineqs always provide a good approximation?

A: No

1. SPARSE + FEW DENSE

Thm: There is a polytope such that adding all 50-sparse + dense cuts still leaves gap

𝑷 ′

Proof idea:• Construction of : take a polytope bad for

sparse cuts, replicate in every orthant

• There is point far from not cut by sparse cuts

• There are points far from not cut by sparse c.

• Intuition: dense cuts can’t cut all far points

• But: In high dim, one dense cut can cut multiple far points

• Use probabilistic argument to show one dense inequality can’t cut too many far points

~

𝑷𝟏

2. ROTATIONS

Quality of sparse closure is not invariant with respect to affine transformations

𝑷rotation

Q: Is there always some rotation that makes the polytope easy to approximate with sparse inequalities?

A: No

¿𝑹 (𝑷 )𝟏

2. ROTATIONS

Thm: Consider the polytope from previous result. Then every rotation is poorly approximated by -sparse cuts:

2. ROTATIONS

Thm: Consider the polytope from previous result. Then every rotation is poorly approximated by -sparse cuts:

Proof idea:1. From previous result, cannot approx with dense ineq

also cannot

2. Can approximate with dense ineq -sparse closure is essentially intersection of -dim polytopes

Can approximate a -dim (symm) polytope within using inequalities

3. Then must be very different from

Difficulty: Sparse closure changes a lot as we rotate the polytope

3. DIRECTIONAL APPROXIMATION

The basic model requires approximation in every direction

𝑷𝑑 (𝑃 ,𝑃𝑘 )= max

{𝑐:|𝑐|=1 }𝑔𝑎𝑝 (𝑐)

Q: Can we show that sparse cuts provide a good approximation in most directions?

A: No

𝑔𝑎𝑝 (𝑐 )=max𝑥∈𝑃 𝑘

𝑐𝑥−max𝑥∈ 𝑃

𝑐𝑥

𝑷 𝒌

3. DIRECTIONAL APPROXIMATION

Elements:1. where is random vector with iid Gaussian

coordinates

2. Explicit definition of and 3. Behavior of Gaussians: , ||, concentration4. Union bound

Thm: There is a polytope where is bad in most directions, namely

where is random unit vector.

CONCLUSION

Results: Bad examples, gap for sparsity , for all these settings

For positive results: Important to look at more structured IPs (see Santanu’s talk tomorrow)

Polytope sparse approximation: refined models that capture more closely features of IP [Dey, Iroume, M.]

1. Sparse + few dense inequalities2. Rotations3. Directional approximation

Sparse cuts can be quite weak for arbitrary IPs, no matter how you look at it

THANK YOU!