Mixed-Integer Programming for Cycle Detection in Non-reversible - - PowerPoint PPT Presentation

Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes

Isabel Beckenbach Leon Eifler Ambros Gleixner Jakob Witzig

Mathematical Optimization, Zuse Institute Berlin

Konstantin Fackeldey Andreas Grever Marcus Weber

Computational Molecular Design, Zuse Institute Berlin

SCIP Workshop 2018 · Aachen

Motivation

◮ Many chemical and biological

processes can be interpreted as a Markov State Models

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Motivation

◮ Many chemical and biological

processes can be interpreted as a Markov State Models

◮ After simulation (e.g.

Monte-Carlo) a discrete set of states S = {1, . . . , n} and a transition matrix P describe the underlying Markov process

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Motivation

◮ Many chemical and biological

processes can be interpreted as a Markov State Models

◮ After simulation (e.g.

Monte-Carlo) a discrete set of states S = {1, . . . , n} and a transition matrix P describe the underlying Markov process

◮ A clustering of the states can be

used to analyze the process

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Problem Description

◮ Let P ∈ Rn×n a transition matrix of a Markov process with stationary distribution

π ∈ Rn.

◮ A process is called reversible, if for all states i and j holds:

πipij = πjpji

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Problem Description

◮ Let P ∈ Rn×n a transition matrix of a Markov process with stationary distribution

π ∈ Rn.

◮ A process is called reversible, if for all states i and j holds:

πipij = πjpji Goal Find clustering C1, . . . , Cm, s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent.

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Problem Description

◮ Let P ∈ Rn×n a transition matrix of a Markov process with stationary distribution

π ∈ Rn.

◮ A process is called reversible, if for all states i and j holds:

πipij = πjpji Goal Find clustering C1, . . . , Cm, s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent.

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Problem Description

◮ Let P ∈ Rn×n a transition matrix of a Markov process with stationary distribution

π ∈ Rn.

◮ A process is called reversible, if for all states i and j holds:

πipij = πjpji Goal Find clustering C1, . . . , Cm, s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent.

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Why is this interesting in practice?

◮ If a process is reversible the leading eigenvalues can be used to find a meaningful

clustering.

1Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov

• processes. Multiscale Modeling & Simulation, 14(4), 1319-1340.

2Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB

Rep, 15(35).

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Why is this interesting in practice?

◮ If a process is reversible the leading eigenvalues can be used to find a meaningful

clustering.

◮ If a process is non-reversible some of the leading eigenvalues are complex and close

to the unit circle1 ⇒ only meaningful interpretation for dominant cycles, but not in general.

1Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov

• processes. Multiscale Modeling & Simulation, 14(4), 1319-1340.

2Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB

Rep, 15(35).

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Why is this interesting in practice?

◮ If a process is reversible the leading eigenvalues can be used to find a meaningful

clustering.

◮ If a process is non-reversible some of the leading eigenvalues are complex and close

to the unit circle1 ⇒ only meaningful interpretation for dominant cycles, but not in general. State-of-the-art methods for non-reversible processes are, e.g.,

◮ Schur-decomposition2 ◮ “Fuzzy” clustering (i.e. assign fractions of states to clusters) and apply a rounding

heuristic but none of them can prove optimality or has some guarantee for the solution quality.

1Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov

• processes. Multiscale Modeling & Simulation, 14(4), 1319-1340.

2Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB

Rep, 15(35).

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Our Goal

• 1. Find a clustering C1, . . . , Cm, m ∈ N

given.

• 2. Maximize the net-flow between

consecutive clusters and the coherence within each cluster.

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Our Goal

• 1. Find a clustering C1, . . . , Cm, m ∈ N

given.

• 2. Maximize the net-flow between

consecutive clusters and the coherence within each cluster.

• 3. Should be applicable for all

non-reversible processes with stationary distribution.

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Our Goal

• 1. Find a clustering C1, . . . , Cm, m ∈ N

given.

• 2. Maximize the net-flow between

consecutive clusters and the coherence within each cluster.

• 3. Should be applicable for all

non-reversible processes with stationary distribution.

• 4. We want to be able to either prove
• ptimality or to measure the solution

quality.

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Cycle-Clustering MINLP

max net-flow + α ·

• t∈C

coherence of Ct s.t.

• t∈C

xit = 1 ∀i ∈ S (1)

• i∈S

xit ≥ 1 ∀t ∈ C (2) xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C

◮ Set of clusters C ◮ Discrete set of states S

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Cycle-Clustering MINLP

max net-flow + α ·

• t∈C

coherence of Ct s.t.

• t∈C

xit = 1 ∀i ∈ S (1)

• i∈S

xit ≥ 1 ∀t ∈ C (2) xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C

◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable xit = 1 ⇐

⇒ state i is assigned to cluster t

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Cycle-Clustering MINLP

max

• t∈C

ǫt + α ·

• t∈C

coherence of Ct s.t.

• t∈C

xit = 1 ∀i ∈ S (1)

• i∈S

xit ≥ 1 ∀t ∈ C (2) ǫt =

• i=j∈S

πipij(xitxjφ(t) − xiφ(t)xjt) ∀t ∈ C (3) xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C ǫt ∈ R≥0 ∀t ∈ C

◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable xit = 1 ⇐

⇒ state i is assigned to cluster t

◮ Cont. variable ǫt representing the net-flow between Ct and Cφ(t)

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Cycle-Clustering MINLP

max

• t∈C

ǫt + α ·

• t∈C
• i,j∈S

πipijxitxjt s.t.

• t∈C

xit = 1 ∀i ∈ S (1)

• i∈S

xit ≥ 1 ∀t ∈ C (2) ǫt =

• i=j∈S

πipij(xitxjφ(t) − xiφ(t)xjt) ∀t ∈ C (3) xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C ǫt ∈ R≥0 ∀t ∈ C

◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable xit = 1 ⇐

⇒ state i is assigned to cluster t

◮ Cont. variable ǫt representing the net-flow between Ct and Cφ(t)

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Linearizing the MINLP-Formulation

◮ automatic linearization by SCIP / textbook linearization

✗

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Linearizing the MINLP-Formulation

◮ automatic linearization by SCIP / textbook linearization

✗

◮ compact linearization for binary quadratic problems (L. Liberti, 2007)

✓/✗

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Linearizing the MINLP-Formulation

◮ automatic linearization by SCIP / textbook linearization

✗

◮ compact linearization for binary quadratic problems (L. Liberti, 2007)

✓/✗

◮ problem specific linearization which uses the cyclic structure

✓

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Linearizing the MINLP-Formulation

◮ automatic linearization by SCIP / textbook linearization

✗

◮ compact linearization for binary quadratic problems (L. Liberti, 2007)

✓/✗

◮ problem specific linearization which uses the cyclic structure

✓

◮ Set of relevant transitions: E := {(i, j) ∈ S × S : i = j, πipij + πjpji > 0} ◮ Binary variables yij = 1 ⇔ states i and j are in the same cluster, ∀i, j ∈ E : i < j ◮ Binary variables zij = 1 ⇔ states i ∈ Ct and j ∈ Cφ(t), ∀i, j ∈ E

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MIP Model

max

• (i,j)∈E

zij(πipij − πjpji) + α

• (i,j)∈E : i<j

yij(πipij + πjpji) s.t.

• t∈C

xit = 1 ∀i ∈ S

• i∈S

xit ≥ 1 ∀t ∈ C xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E yij + zij + zji ≤ 1 ∀(i, j) ∈ E : i < j xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C yij, zij, zji ∈ {0, 1} ∀(i, j) ∈ E : i < j

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MIP Model

max

• (i,j)∈E

zij(πipij − πjpji) + α

• (i,j)∈E : i<j

yij(πipij + πjpji) s.t.

• t∈C

xit = 1 ∀i ∈ S

• i∈S

xit ≥ 1 ∀t ∈ C xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E yij + zij + zji ≤ 1 ∀(i, j) ∈ E : i < j xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C yij, zij, zji ∈ {0, 1} ∀(i, j) ∈ E : i < j

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MIP Model

max

• (i,j)∈E

zij(πipij − πjpji) + α

• (i,j)∈E : i<j

yij(πipij + πjpji) s.t.

• t∈C

xit = 1 ∀i ∈ S

• i∈S

xit ≥ 1 ∀t ∈ C xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E yij + zij + zji ≤ 1 ∀(i, j) ∈ E : i < j xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C yij, zij, zji ∈ {0, 1} ∀(i, j) ∈ E : i < j

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MIP Model

max

• (i,j)∈E

zij(πipij − πjpji) + α

• (i,j)∈E : i<j

yij(πipij + πjpji) s.t.

• t∈C

xit = 1 ∀i ∈ S

• i∈S

xit ≥ 1 ∀t ∈ C xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 ∀t ∈ C, (i, j) ∈ E yij + zij + zji ≤ 1 ∀(i, j) ∈ E : i < j xit ∈ {0, 1} ∀i ∈ S, ∀t ∈ C yij, zij, zji ∈ {0, 1} ∀(i, j) ∈ E : i < j

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Facets of the Linearization

Theorem

Let (i, j) ∈ E, t ∈ C, and m ≥ 4. Then the inequalities xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 and xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 are facet-defining for the cycle-clustering polytope (CCP).

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Facets of the Linearization

Theorem

Let (i, j) ∈ E, t ∈ C, and m ≥ 4. Then the inequalities xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 and xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 are facet-defining for the cycle-clustering polytope (CCP). very technical proof

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Facets of the Linearization

Theorem

Let (i, j) ∈ E, t ∈ C, and m ≥ 4. Then the inequalities xit + xjt − yij + zij − xjφ(t) − xiφ−1(t) ≤ 1 and xit + xjφ(t) − zij + yij − xjt − xiφ(t) ≤ 1 are facet-defining for the cycle-clustering polytope (CCP). very technical proof Proof idea

• 1. Let ˆ

ax + ˆ by + ˆ cz ≤ ˆ δ facet-defining with {(x, y, z) ∈ CCP | ax + by + cz = δ} ⊆ {(x, y, z) ∈ CCP | ˆ ax + ˆ by + ˆ cz = ˆ δ}

• 2. Compare coefficients and show that ax + by + cz ≤ δ is a multiple of

ˆ ax + ˆ by + ˆ cz ≤ ˆ δ Follows the proof technique of Chopra and Rao (1993).

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Valid Inequalities

• 1. various types of triangle inequalities

◮ special case: m = 3 ◮ general case: m ≥ 4 (motivated by S. Chopra and M. R. Rao. (1993); M. Grötschel

and Y. Wakabayashi (1990))

• 2. subtour and path inequalities

◮ motivated by TSP subtour elimination inequalities

• 3. partitioning inequalities

◮ modification of M. Grötschel and Y. Wakabayashi (1990)

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Valid Inequalities

• 1. various types of triangle inequalities

◮ special case: m = 3 ◮ general case: m ≥ 4 (motivated by S. Chopra and M. R. Rao. (1993); M. Grötschel

and Y. Wakabayashi (1990))

• 2. subtour and path inequalities

◮ motivated by TSP subtour elimination inequalities

• 3. partitioning inequalities

◮ modification of M. Grötschel and Y. Wakabayashi (1990)

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Simple Triangle Inequalities

i k j i j k

special case m = 3: zij + zjk − zki ≤ 1 ∀(i, j), (j, k), (k, i) ∈ E holds in general: yij + yjk − yik ≤ 1 ∀(i, j), (j, k), (k, i) ∈ E

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Triangle Inequalities Involving 2 Clusters

k j i

k is in the successor cluster of i

k j i

k is in the predecessor cluster of i

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Triangle Inequalities Involving 2 Clusters

k j i

k is in the successor cluster of i yij + zik − zjk ≤ 1

k j i

k is in the predecessor cluster of i yij + zki − zkj ≤ 1

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Facet-Defining Triangle Inequalities

A combination of

◮ yij + yjk − yik ≤ 1 ◮ yij + zik − zjk ≤ 1 ◮ yij + zki − zkj ≤ 1

yield a facet-defining inequality.

i j k k j i k j i

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Facet-Defining Triangle Inequalities

A combination of

◮ yij + yjk − yik ≤ 1 ◮ yij + zik − zjk ≤ 1 ◮ yij + zki − zkj ≤ 1

yield a facet-defining inequality.

i j k k j i k j i

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E. Then the triangle inequality yij + yjk − yik + 0.5 (zij + zji + zjk + zkj − zik − zki) ≤ 1 is facet-defining for the cycle clustering polytope.

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Triangle Inequalities Involving 2 Clusters (cont.)

i j k

i is in the successor cluster of j and k zij + zik − yjk ≤ 1

i j k

i is in the predecessor cluster of j and k zji + zki − yjk ≤ 1

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Facet-Defining Triangle Inequalities (cont.)

Theorem

Let i, j, k ∈ S with (i, j), (j, k), (i, k) ∈ E.

◮ If m > 4, then

zij + zik − yjk ≤ 1 and zji + zki − yjk ≤ 1 are facet-defining for the cycle clustering polytope.

◮ If m = 4, then

zij + zik − 2yjk − (zjk + zkj + zji + zki) ≤ 0 is facet-defining for the cycle clustering polytope.

i j k

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Valid Inequalities

• 1. various types of triangle inequalities

◮ special case: m = 3 ◮ general case: m ≥ 4 (motivated by S. Chopra and M. R. Rao. (1993); M. Grötschel

and Y. Wakabayashi (1990))

• 2. subtour and path inequalities

◮ motivated by TSP subtour elimination inequalities

• 3. partitioning inequalities

◮ modification of M. Grötschel and Y. Wakabayashi (1990)

• J. Witzig (ZIB): Cycle-Clustering

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Subtour Elimination Inequalities

◮ Let K = {(i1, i2), (i2, i3), . . . , (is−1, is), (is, i1)} ⊂ E with 1 < |K| < m ◮ U ⊂ K, |U| ≤ |K| − 1

i1 i2 i3 zi1i2 zi2i3 zi3i1 yi1i2 zi2i3 zi3i4 zi4i1

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Subtour Elimination Inequalities

◮ Let K = {(i1, i2), (i2, i3), . . . , (is−1, is), (is, i1)} ⊂ E with 1 < |K| < m ◮ U ⊂ K, |U| ≤ |K| − 1

i1 i2 i3 zi1i2 zi2i3 zi3i1

• (i,j)∈K

zij ≤ |K| − 1

yi1i2 zi2i3 zi3i4 zi4i1

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Subtour Elimination Inequalities

◮ Let K = {(i1, i2), (i2, i3), . . . , (is−1, is), (is, i1)} ⊂ E with 1 < |K| < m ◮ U ⊂ K, |U| ≤ |K| − 1

i1 i2 i3 zi1i2 zi2i3 zi3i1

• (i,j)∈K

zij ≤ |K| − 1

yi1i2 zi2i3 zi3i4 zi4i1

• (i,j)∈K

zij +

• (i,j)∈U

yij ≤ |K| − 1

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Subtour Elimination Inequalities

◮ Let K = {(i1, i2), (i2, i3), . . . , (is−1, is), (is, i1)} ⊂ E with 1 < |K| < m ◮ U ⊂ K, |U| ≤ |K| − 1

i1 i2 i3 zi1i2 zi2i3 zi3i1

• (i,j)∈K

zij ≤ |K| − 1

yi1i2 zi2i3 zi3i4 zi4i1

• (i,j)∈K

zij +

• (i,j)∈U

yij ≤ |K| − 1

◮ none of them is facet-defining ◮ but can be separated in polynomial time

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Valid Inequalities

• 1. various types of triangle inequalities

◮ special case: m = 3 ◮ general case: m ≥ 4 (motivated by S. Chopra and M. R. Rao. (1993); M. Grötschel

and Y. Wakabayashi (1990))

• 2. subtour and path inequalities

◮ motivated by TSP subtour elimination inequalities

• 3. partitioning inequalities

◮ modification of M. Grötschel and Y. Wakabayashi (1990)

• J. Witzig (ZIB): Cycle-Clustering

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Partition Inequalities

B A

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Partition Inequalities

Let A, B ⊂ S, A ∩ B = ∅. The partitioning inequality

• i∈A, j∈B

zij −

• i<j∈A

yij −

• i<j∈B

yij ≤ min{|A|, |B|} (4) is valid for the cycle clustering polytope.

B A

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Partition Inequalities

Let A, B ⊂ S, A ∩ B = ∅. The partitioning inequality

• i∈A, j∈B

zij −

• i<j∈A

yij −

• i<j∈B

yij ≤ min{|A|, |B|} (4) is valid for the cycle clustering polytope.

B A

Theorem

The partitioning inequalities of type (4) are facet-defining if m > 4 and ||A| − |B|| = 1.

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How do the inequalities perform in practice?

Our test instances:

• 1. Artificial potentials with

3, 4 or 6 minima

• 2. Repressilator (synthetic

genetic regulatory network)

• 3. Hindmarsh-Rose (simulate

membrane potential of cells in a human heart)

potential with 4 minima Repressilator (Elowitz and Leibler, 2000)

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Only one slide left before you see some numbers. . .

pA mA pB mB pC mC 5 10 15 pA mA pB mB pC mC pA mA pB mB pC mC ǫ1 ǫ2 ǫ3

Schur Clustering

pA mA pB mB pC mC 5 10 15 pA mA pB mB pC mC pA mA pB mB pC mC ǫ1 ǫ2 ǫ3

Cycle Clustering

(solutions of a repressilator instance)

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Computational Results: Impact of Inequalities

Inequ. time [s] nodes dual integral default 130.6

• 256.6
• 1213.3
• triangle

77.6 0.59 28.8 0.11 474.1 0.39 subtour 122.6 0.94 174.3 0.68 1036.4 0.85 partition 122.6 0.94 121.6 0.47 933.1 0.77 all 68.3 0.52 15.4 0.06 396.6 0.33

Test set of 176 instances with 50 – 250 states. Time limit 1h.

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Mixed Integer Programming is more than cutting planes. . .

Beside of three different type of cutting planes we designed primal heuristics to improve the primal side as well:

◮ greedy heuristic assigning states to clusters ◮ exchange heuristic that swaps states between clusters

(even if it gets worse in between)

◮ problem-tailored LP rounding heuristic

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Mixed Integer Programming is more than cutting planes. . .

Beside of three different type of cutting planes we designed primal heuristics to improve the primal side as well:

◮ greedy heuristic assigning states to clusters

−32% primal integral

◮ exchange heuristic that swaps states between clusters

−53% primal integral (even if it gets worse in between)

◮ problem-tailored LP rounding heuristic

−32% primal integral

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Now, put it all together

Solver solved time [s] primal int dual int SCIP 113 182.4 3274.8 1794.0 CC-SCIP 130 64.7 82.9 327.1 (out of competition) Gurobi 118 153.1 1236.6 1524.9

Test set of 176 instances with 50 – 250 states. Time limit 1h.

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Conclusion

Summary

◮ First MIP model for detecting cyclic clustering of non-reversible Markov State

Models

◮ Various (facet-defining) valid inequalities for the cycle-clustering polytope ◮ Very brief description of problem-tailored primal heuristics

References

◮ Model: Witzig, J., Beckenbach, I., Eifler, L., Fackeldey, K., Gleixner, A., Grever, A.,

& Weber, M. (2018). Mixed-Integer Programming for Cycle Detection in Nonreversible Markov Processes. Multiscale Modeling & Simulation, 16(1), 248-265.

◮ Methods: Master thesis of Leon Eifler, publication coming soon. . .

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The End Thank you for your attention. Questions?

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