Investigating Mixed-Integer Hulls using a MIP-Solver Matthias Walter - - PowerPoint PPT Presentation

Investigating Mixed-Integer Hulls using a MIP-Solver

Matthias Walter Otto-von-Guericke Universit¨ at Magdeburg

Joint work with Volker Kaibel (OvGU)

Aussois Combinatorial Optimization Workshop 2015

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Outline

1

Polyhedral Combinatorics

1

Introduction

2

Usual Approach

3

Limitations 2

Vision

3

Facets

1

Polarity

2

Target Cuts 4

Affine Hull

5

Minimizing the 1-Norm of Basis Vectors

1

Problem

2

2 Vectors: Exact Approach

3

2 Vectors: Heuristic Approach

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 2 / 22

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Setup

Problems in Question

We consider mixed-integer programs with rational data: max c, x s.t. Ax ≤ b xi ∈ Z ∀i ∈ I ⊆ [n]

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Setup

Problems in Question

We consider mixed-integer programs with rational data: max c, x s.t. Ax ≤ b xi ∈ Z ∀i ∈ I ⊆ [n] Denote by R = {x ∈ Rn : Ax ≤ b} the relaxation polyhedron, by L = {x ∈ Rn : xi ∈ Z ∀i ∈ I} the integrality restrictions, and by P = conv.hull(R ∩ L) the mixed-integer hull.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 3 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Setup

Problems in Question

We consider mixed-integer programs with rational data: max c, x s.t. Ax ≤ b xi ∈ Z ∀i ∈ I ⊆ [n] Denote by R = {x ∈ Rn : Ax ≤ b} the relaxation polyhedron, by L = {x ∈ Rn : xi ∈ Z ∀i ∈ I} the integrality restrictions, and by P = conv.hull(R ∩ L) the mixed-integer hull.

Facts

P is a polyhedron again. For most (e.g., NP-hard) problems, P has many facets. MIP-solvers are really fast these days.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 3 / 22

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Polyhedral Combinatorics

Another Fact

The time a MIP-solver needs for solving depends on the strength of the relaxation, i.e., how well P is approximated by R.

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Polyhedral Combinatorics

Another Fact

The time a MIP-solver needs for solving depends on the strength of the relaxation, i.e., how well P is approximated by R.

Strengthening a Relaxation

Generic cutting planes: GMI, MIR, CG, Lift & Project, . . . Problem-specific inequalities: Problem-dependent

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 4 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Polyhedral Combinatorics

Another Fact

The time a MIP-solver needs for solving depends on the strength of the relaxation, i.e., how well P is approximated by R.

Strengthening a Relaxation

Generic cutting planes: GMI, MIR, CG, Lift & Project, . . . Problem-specific inequalities: Problem-dependent

Goals of Polyhedral Combinatorics

Given a MIP-model for a problem, find inequalities valid for P (but not for R), develop algorithms (exact or heuristics) to separate these inequalities if there are too many, determine the dimension of P, i.e., find valid equations, and prove if/when the inequalities define facets of P.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 4 / 22

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Usual Approach for IPs

Step 1: Find all feasible points

(a) By hand / handcrafted software (b) Some tool, e.g. PORTA’s vint functionality or azove

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Usual Approach for IPs

Step 1: Find all feasible points

(a) By hand / handcrafted software (b) Some tool, e.g. PORTA’s vint functionality or azove

Step 2: Compute Outer Description

There are quite a few tools (PORTA, Polymake, azove) and several algorithms: The Beneath-and-Beyond method The Double-Description method Lexicographic Reverse Search Pyramid decomposition (mixture of beneath-and-beyond and Fourier-Motzkin)

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 5 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Usual Approach for IPs

Step 1: Find all feasible points

(a) By hand / handcrafted software (b) Some tool, e.g. PORTA’s vint functionality or azove

Step 2: Compute Outer Description

There are quite a few tools (PORTA, Polymake, azove) and several algorithms: The Beneath-and-Beyond method The Double-Description method Lexicographic Reverse Search Pyramid decomposition (mixture of beneath-and-beyond and Fourier-Motzkin)

Step 3: Generalize Inequalities

There’s only one main tool here: The mathematician.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 5 / 22

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Limitations

Memory and Time

The dominant of the cut polytope (corresponds to MinCut problem) has among

• thers a facet per disjoint union of cycles joined together by any spanning tree!

Alevras (’99) enumerated the facets for this polyhedron for the complete graph on 8 nodes, including 2 billions of the above type.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 6 / 22

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Limitations

Memory and Time

The dominant of the cut polytope (corresponds to MinCut problem) has among

• thers a facet per disjoint union of cycles joined together by any spanning tree!

Alevras (’99) enumerated the facets for this polyhedron for the complete graph on 8 nodes, including 2 billions of the above type.

Continuous Variables

The mixed-integer case is usually harder for enumeration tools.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 6 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Limitations

Memory and Time

The dominant of the cut polytope (corresponds to MinCut problem) has among

• thers a facet per disjoint union of cycles joined together by any spanning tree!

Alevras (’99) enumerated the facets for this polyhedron for the complete graph on 8 nodes, including 2 billions of the above type.

Continuous Variables

The mixed-integer case is usually harder for enumeration tools.

Symmetry

Often, facets have a lot of symmetry and it is only sometimes possible to exploit (parts of it) during the convex-hull computation.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 6 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Limitations

Memory and Time

The dominant of the cut polytope (corresponds to MinCut problem) has among

• thers a facet per disjoint union of cycles joined together by any spanning tree!

Alevras (’99) enumerated the facets for this polyhedron for the complete graph on 8 nodes, including 2 billions of the above type.

Continuous Variables

The mixed-integer case is usually harder for enumeration tools.

Symmetry

Often, facets have a lot of symmetry and it is only sometimes possible to exploit (parts of it) during the convex-hull computation.

Specific Objective Functions

Which are the facets useful when optimizing specific objective functions?

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 6 / 22

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Vision

Fact Reminder

PORTA & friends need “a moment” for a 50-dimensional polytope, while a MIP with 50 variables is usually solved within a second!

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Vision

Fact Reminder

PORTA & friends need “a moment” for a 50-dimensional polytope, while a MIP with 50 variables is usually solved within a second! Goal of this work:

Use MIP-solvers to determine facets! (and the affine hull)

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Assumptions

Definition (MIP-oracle Pair)

A MIP-oracle pair for a rational MIP R ∩ L with relaxation R and integrality restrictions L consists of an oracle which solves max {c, x : x ∈ R ∩ L} for any objective c ∈ Qn, and an oracle which can separate over R, i.e., decides for given ˆ x, whether ˆ x ∈ R holds, and if not, yields a separating inequality.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 8 / 22

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Assumptions

Definition (MIP-oracle Pair)

A MIP-oracle pair for a rational MIP R ∩ L with relaxation R and integrality restrictions L consists of an oracle which solves max {c, x : x ∈ R ∩ L} for any objective c ∈ Qn, and an oracle which can separate over R, i.e., decides for given ˆ x, whether ˆ x ∈ R holds, and if not, yields a separating inequality.

Remark

We can use an ordinary MIP solver and apply rational reconstruction techniques to the floating-point solution

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 8 / 22

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Assumptions

Definition (MIP-oracle Pair)

A MIP-oracle pair for a rational MIP R ∩ L with relaxation R and integrality restrictions L consists of an oracle which solves max {c, x : x ∈ R ∩ L} for any objective c ∈ Qn, and an oracle which can separate over R, i.e., decides for given ˆ x, whether ˆ x ∈ R holds, and if not, yields a separating inequality.

Remark

We can use an ordinary MIP solver and apply rational reconstruction techniques to the floating-point solution (and hope for correctness).

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 8 / 22

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Facets

Finding Facets

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Polarity - Basics

Definition

Let P ⊆ Rn be a polyhedron containing the origin O. Then the set {y ∈ Rn : y, x ≤ 1 ∀x ∈ P} is called the polar dual of P.

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Polarity - Basics

Definition

Let P ⊆ Rn be a polyhedron containing the origin O. Then the set {y ∈ Rn : y, x ≤ 1 ∀x ∈ P} is called the polar dual of P. Since the polar dual has a lineality space if P is not full-dimensional, let P∗ := {y ∈ lin.hull (P) : y, x ≤ 1 ∀x ∈ P} . denote the truncated polar dual of P.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 10 / 22

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Polarity - Basics

Definition

Let P ⊆ Rn be a polyhedron containing the origin O. Then the set {y ∈ Rn : y, x ≤ 1 ∀x ∈ P} is called the polar dual of P. Since the polar dual has a lineality space if P is not full-dimensional, let P∗ := {y ∈ lin.hull (P) : y, x ≤ 1 ∀x ∈ P} . denote the truncated polar dual of P.

Proposition (see Schrijver ’86)

(P∗)∗ = P x is a point in (vertex of) P if and only if x, y ≤ 1 defines an inequality for (facet of) P∗. P∗ is bounded if and only if P contains O in its relative interior.

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Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 11 / 22

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Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

2

If not, we obtain a point ˆ x ∈ R \ P. Let’s find a facet separating ˆ x!

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 11 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

2

If not, we obtain a point ˆ x ∈ R \ P. Let’s find a facet separating ˆ x!

4

By polarity, we want a vertex y ∗ of (P − o)∗ with ˆ x − o, y ∗ > 1. For this we solve max ˆ x − o, y s.t. x − o, y ≤ 1 ∀x ∈ P y ∈ lin.hull(P − o) using the simplex method.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 11 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

2

If not, we obtain a point ˆ x ∈ R \ P. Let’s find a facet separating ˆ x!

4

By polarity, we want a vertex y ∗ of (P − o)∗ with ˆ x − o, y ∗ > 1. For this we solve max ˆ x − o, y s.t. x − o, y ≤ 1 ∀x ∈ P y ∈ lin.hull(P − o) using the simplex method.

5

The separation problem for the first constraint is – again by polarity – the problem to maximize ˆ y over P (Many oracle calls!).

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 11 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

2

If not, we obtain a point ˆ x ∈ R \ P. Let’s find a facet separating ˆ x!

3

Restrict to facets that are tight at points x∗

1 , . . . , x∗ k .

4

By polarity, we want a vertex y ∗ of (P − o)∗ with ˆ x − o, y ∗ > 1. For this we solve max ˆ x − o, y s.t. x − o, y ≤ 1 ∀x ∈ P y ∈ lin.hull(P − o) x∗

i − o, y= 1

∀i = 1, . . . , k using the simplex method.

5

The separation problem for the first constraint is – again by polarity – the problem to maximize ˆ y over P (Many oracle calls!).

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 11 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Using Polarity to Obtain Facets

Suppose we know a point o in the relative interior of P.

1

Solve maxx∈R c, x and maxx∈P c, x (2 oracle calls!). If objective values coincide, R approximates P well in direction c.

2

If not, we obtain a point ˆ x ∈ R \ P. Let’s find a facet separating ˆ x!

3

Restrict to facets that are tight at points x∗

1 , . . . , x∗ k .

4

By polarity, we want a vertex y ∗ of (P − o)∗ with ˆ x − o, y ∗ > 1. For this we solve max ˆ x − o, y s.t. x − o, y ≤ 1 ∀x ∈ P y ∈ lin.hull(P − o) x∗

i − o, y= 1

∀i = 1, . . . , k using the simplex method.

5

The separation problem for the first constraint is – again by polarity – the problem to maximize ˆ y over P (Many oracle calls!). Not new: Target Cuts invented by Buchheim, Liers & Oswald, ’08

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Choice of the Interior Point

Proposition

The objective value of a facet a, x ≤ β of P equals a, ˆ x − a, o β − a, o . ˆ x x∗

• maximize

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Choice of the Interior Point

Proposition

The objective value of a facet a, x ≤ β of P equals a, ˆ x − a, o β − a, o . ˆ x x∗

• maximize

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Choice of the Interior Point

Proposition

The objective value of a facet a, x ≤ β of P equals a, ˆ x − a, o β − a, o . ˆ x x∗

• maximize

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Choice of the Interior Point

Proposition

The objective value of a facet a, x ≤ β of P equals a, ˆ x − a, o β − a, o . ˆ x x∗

• ′

maximize

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Affine Hull

Computing Affine Hull & Dimension

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Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn.

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Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn. Output: Dimension d of P (d + 1)-many affinely independent points in P (n − d)-many irredundant equations Cx = d valid for P

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Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn. Output: Dimension d of P (d + 1)-many affinely independent points in P (n − d)-many irredundant equations Cx = d valid for P Ideas:

1

Maintain known equations Cx = d and points x1, x2, . . . , xℓ ∈ P.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 14 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn. Output: Dimension d of P (d + 1)-many affinely independent points in P (n − d)-many irredundant equations Cx = d valid for P Ideas:

1

Maintain known equations Cx = d and points x1, x2, . . . , xℓ ∈ P.

2

Repeatedly find a “useful” direction c ∈ Rn, and compute z+ := maxx∈P c, x and z− := minx∈P c, x (2 oracle calls!).

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 14 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn. Output: Dimension d of P (d + 1)-many affinely independent points in P (n − d)-many irredundant equations Cx = d valid for P Ideas:

1

Maintain known equations Cx = d and points x1, x2, . . . , xℓ ∈ P.

2

Repeatedly find a “useful” direction c ∈ Rn, and compute z+ := maxx∈P c, x and z− := minx∈P c, x (2 oracle calls!).

3

If we are lucky and z+ = z− holds, then c, x = z+ is a valid equation. To make progress, we want c to be linear independent of C’s rows.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 14 / 22

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Computing the Affine Hull

Input: MIP-oracle pair for ∅ = P = conv.hull(R ∩ L) ⊆ Rn. Output: Dimension d of P (d + 1)-many affinely independent points in P (n − d)-many irredundant equations Cx = d valid for P Ideas:

1

Maintain known equations Cx = d and points x1, x2, . . . , xℓ ∈ P.

2

Repeatedly find a “useful” direction c ∈ Rn, and compute z+ := maxx∈P c, x and z− := minx∈P c, x (2 oracle calls!).

3

If we are lucky and z+ = z− holds, then c, x = z+ is a valid equation. To make progress, we want c to be linear independent of C’s rows.

4

If we also choose c to be orthogonal to aff.hull(x1, . . . , xℓ) (that is, c, xi = c, x1 for all i = 2, . . . , ℓ), the result z+ > z− yields a point xℓ+1 affinely independent of x1, . . . , xℓ.

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Computing the Affine Hull (cont.)

Reminder: x1, . . . , xℓ ∈ P ⊆ Rn are affinely independent points found so far. Cx = d are the equations found so far. Algebra: Let D ∈ R(ℓ−1)×n be the matrix whose rows are xi − x1 for i = 2, . . . , ℓ. We can choose c ∈ (ker C ∩ ker D) \ O.

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Computing the Affine Hull (cont.)

Reminder: x1, . . . , xℓ ∈ P ⊆ Rn are affinely independent points found so far. Cx = d are the equations found so far. Algebra: Let D ∈ R(ℓ−1)×n be the matrix whose rows are xi − x1 for i = 2, . . . , ℓ. We can choose c ∈ (ker C ∩ ker D) \ O. Drawback: It works but we modeled linear independence by orthogonality!

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Computing the Affine Hull (cont.)

Reminder: x1, . . . , xℓ ∈ P ⊆ Rn are affinely independent points found so far. Cx = d are the equations found so far. Algebra: Let D ∈ R(ℓ−1)×n be the matrix whose rows are xi − x1 for i = 2, . . . , ℓ. We can choose c ∈ (ker C ∩ ker D) \ O. Drawback: It works but we modeled linear independence by orthogonality! Conclusion: Consider a basis c1, . . . , ck of ker D and use each ci as a direction (or skip ci if it linearly depends on C’s rows).

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Computing the Affine Hull (cont.)

Reminder: x1, . . . , xℓ ∈ P ⊆ Rn are affinely independent points found so far. Cx = d are the equations found so far. Algebra: Let D ∈ R(ℓ−1)×n be the matrix whose rows are xi − x1 for i = 2, . . . , ℓ. We can choose c ∈ (ker C ∩ ker D) \ O. Drawback: It works but we modeled linear independence by orthogonality! Conclusion: Consider a basis c1, . . . , ck of ker D and use each ci as a direction (or skip ci if it linearly depends on C’s rows). Open Question: Can one do better than 2n oracle calls?

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Minimize 1-Norm

Making Equations and Facets Pretty

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1-Norm Minimization

Motivation

Readability of produced equations & facets! a, x ≤ β with a ∈ Zn and β ∈ Z where (a⊺, β) is a primitive vector.

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1-Norm Minimization

Motivation

Readability of produced equations & facets! a, x ≤ β with a ∈ Zn and β ∈ Z where (a⊺, β) is a primitive vector. If dim(P) = n, facet representations are not unique and equation normals are an arbitrary basis of aff.hull (P)⊥.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 17 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

1-Norm Minimization

Motivation

Readability of produced equations & facets! a, x ≤ β with a ∈ Zn and β ∈ Z where (a⊺, β) is a primitive vector. If dim(P) = n, facet representations are not unique and equation normals are an arbitrary basis of aff.hull (P)⊥. Goal: Consider pairs of vectors and replace one by a nice linear combination.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 17 / 22

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1-Norm Minimization

Motivation

Readability of produced equations & facets! a, x ≤ β with a ∈ Zn and β ∈ Z where (a⊺, β) is a primitive vector. If dim(P) = n, facet representations are not unique and equation normals are an arbitrary basis of aff.hull (P)⊥. Goal: Consider pairs of vectors and replace one by a nice linear combination.

Problem (1-Norm Minimization in Dimension Two)

Input are two linearly independent vectors u, v ∈ Zn. Find λ, µ ∈ Q with λ = 0 and λu + µv ∈ Zn minimizing |λu + µv|1.

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2 Vectors, Step 1: Lattice Basis

Problem (1-Norm Minimization in Dimension Two)

Input are two linearly independent vectors u, v ∈ Zn. Find λ, µ ∈ Q with λ = 0 and λu + µv ∈ Zn minimizing |λu + µv|1.

Lemma

By computing the Hermite-Normal-Form of u⊺ v ⊺

• ∈ Z2×n, we can replace u, v

by a lattice basis u′, v ′ of lin.hull (u, v) ∩ Zn . . .

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2 Vectors, Step 1: Lattice Basis

Problem (1-Norm Minimization in Dimension Two)

Input are two linearly independent vectors u, v ∈ Zn. Find λ, µ ∈ Q with λ = 0 and λu + µv ∈ Zn minimizing |λu + µv|1.

Lemma

By computing the Hermite-Normal-Form of u⊺ v ⊺

• ∈ Z2×n, we can replace u, v

by a lattice basis u′, v ′ of lin.hull (u, v) ∩ Zn . . . . . . such that λu′ + µv ′ is a multiple of v if and only if λ = 0.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 18 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 1: Lattice Basis

Problem (1-Norm Minimization in Dimension Two)

Input are two linearly independent vectors u, v ∈ Zn. Find λ, µ ∈ Q with λ = 0 and λu + µv ∈ Zn minimizing |λu + µv|1.

Lemma

By computing the Hermite-Normal-Form of u⊺ v ⊺

• ∈ Z2×n, we can replace u, v

by a lattice basis u′, v ′ of lin.hull (u, v) ∩ Zn . . . . . . such that λu′ + µv ′ is a multiple of v if and only if λ = 0.

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 18 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm!

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ 10 u u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ 10 u 15 u + v u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ 10 u 15 u + v 15 2u − v 15 u − 3v u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ 10 u 15 u + v 15 2u − v 15 u − 3v 25 2u + v u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi. λ µ 10 u 15 u + v 15 2u − v 15 u − 3v 12 u − 2v 9 u − v 25 2u + v u =     1 −1 2 6         1 4 2     = v Objective: |λu + µv|1

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi.

Lemma

Let u, v ∈ Zn be lin. independent. Let −∞ = q0 < q1 < . . . < qk−1 < qk = ∞ be the sorted elements of Q :=

• − ui

vi : i ∈ [n], vi = 0

• ∪ {±∞}.

Then the sign-pattern of λu + µv is constant over all multiplier pairs λ, µ with with λ > 0 for which λ/µ is in any fixed interval [qj−1, qj].

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors, Step 2: Case Distinction

Problem (Restricted 1-Norm Minimization in Dimension Two)

Given a lattice basis u, v ∈ Zn, find λ, µ ∈ Z with λ > 0 minimizing |λu + µv|1. Let’s consider all sign-patterns of λu + µv in order to resolve the 1-norm! Observation: Sign of λui + µvi depends on µ/λ compared to −ui/vi.

Lemma

Let u, v ∈ Zn be lin. independent. Let −∞ = q0 < q1 < . . . < qk−1 < qk = ∞ be the sorted elements of Q :=

• − ui

vi : i ∈ [n], vi = 0

• ∪ {±∞}.

Then the sign-pattern of λu + µv is constant over all multiplier pairs λ, µ with with λ > 0 for which λ/µ is in any fixed interval [qj−1, qj]. Final Step: In such an interval the 1-norm is linear and integer programming in dimension 2 can be solved efficiently (see Eisenbrand & Laue, ’03).

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 19 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors: Heuristic Approach

Observation

Let u, v ∈ Zn be lin. independent. Let −∞ = q0 < q1 < . . . < qk−1 < qk = ∞ be the sorted elements of Q :=

• − ui

vi : i ∈ [n], vi = 0

• ∪ {±∞}.

Then the only multiplier pairs λ, µ ∈ Z for which λu + µv has zero entries not present in u nor v are those with λ/µ ∈ Q.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 20 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

2 Vectors: Heuristic Approach

Observation

Let u, v ∈ Zn be lin. independent. Let −∞ = q0 < q1 < . . . < qk−1 < qk = ∞ be the sorted elements of Q :=

• − ui

vi : i ∈ [n], vi = 0

• ∪ {±∞}.

Then the only multiplier pairs λ, µ ∈ Z for which λu + µv has zero entries not present in u nor v are those with λ/µ ∈ Q. Heuristic:

1

Let w := u.

2

For all i ∈ [n] with vi = 0 do:

1

Let w(i) := vi · u − ui · v.

2

Divide w(i) by the g.c.d. of its entries.

3

If |w(i)|1 < |w|1, replace w by w(i).

3

Return w.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 20 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Live Demo

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 21 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Questions?

Problems to be tackled by help of MIP-solvers: Compute all valid equations / the dimension d. Find facets useful when optimizing certain objective functions, together with d independent points proving that it is a facet. Compute the dimension of other faces, e.g. the optimal face. Compute the conflict graph for 0/1-variables.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 22 / 22

Polyhedral Comb. Vision Facets Affine Hull Min 1-Norm Final Slides

Questions?

Problems to be tackled by help of MIP-solvers: Compute all valid equations / the dimension d. Find facets useful when optimizing certain objective functions, together with d independent points proving that it is a facet. Compute the dimension of other faces, e.g. the optimal face. Compute the conflict graph for 0/1-variables. To be done: Finish basic code :-) Improve cutting plane procedure (stabilize, aging of cuts, . . . ). Carry out computational study. Find some nice facets for interesting polytopes. Investigate dimensions of non-facets.

Matthias Walter Investigating Mixed-Integer Hulls using a MIP-Solver Aussois 2015 22 / 22