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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!
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