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 J. Witzig (ZIB): Cycle-Clustering 1 / 25
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 J. Witzig (ZIB): Cycle-Clustering 1 / 25
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 J. Witzig (ZIB): Cycle-Clustering 1 / 25
Problem Description ◮ Let P ∈ R n × n a transition matrix of a Markov process with stationary distribution π ∈ R n . ◮ A process is called reversible , if for all states i and j holds: π i p ij = π j p ji J. Witzig (ZIB): Cycle-Clustering 2 / 25
Problem Description ◮ Let P ∈ R n × n a transition matrix of a Markov process with stationary distribution π ∈ R n . ◮ A process is called reversible , if for all states i and j holds: π i p ij = π j p ji Goal Find clustering C 1 , . . . , C m , s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent . J. Witzig (ZIB): Cycle-Clustering 2 / 25
Problem Description ◮ Let P ∈ R n × n a transition matrix of a Markov process with stationary distribution π ∈ R n . ◮ A process is called reversible , if for all states i and j holds: π i p ij = π j p ji Goal Find clustering C 1 , . . . , C m , s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent . J. Witzig (ZIB): Cycle-Clustering 2 / 25
Problem Description ◮ Let P ∈ R n × n a transition matrix of a Markov process with stationary distribution π ∈ R n . ◮ A process is called reversible , if for all states i and j holds: π i p ij = π j p ji Goal Find clustering C 1 , . . . , C m , s.t. the process between two consecutive clusters is non-reversible and within each cluster coherent . J. Witzig (ZIB): Cycle-Clustering 2 / 25
Why is this interesting in practice? ◮ If a process is reversible the leading eigenvalues can be used to find a meaningful clustering. 1 Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov processes. Multiscale Modeling & Simulation, 14(4), 1319-1340. 2 Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB Rep, 15(35). J. Witzig (ZIB): Cycle-Clustering 3 / 25
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 circle 1 ⇒ only meaningful interpretation for dominant cycles, but not in general. 1 Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov processes. Multiscale Modeling & Simulation, 14(4), 1319-1340. 2 Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB Rep, 15(35). J. Witzig (ZIB): Cycle-Clustering 3 / 25
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 circle 1 ⇒ only meaningful interpretation for dominant cycles, but not in general. State-of-the-art methods for non-reversible processes are, e.g., ◮ Schur-decomposition 2 ◮ “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. 1 Conrad, N. D., Weber, M., Schütte, C. (2016). Finding dominant structures of nonreversible Markov processes. Multiscale Modeling & Simulation, 14(4), 1319-1340. 2 Weber, M., Fackeldey, K. (2015). G-PCCA: Spectral clustering for non-reversible Markov chains. ZIB Rep, 15(35). J. Witzig (ZIB): Cycle-Clustering 3 / 25
Our Goal 1. Find a clustering C 1 , . . . , C m , m ∈ N given. 2. Maximize the net-flow between consecutive clusters and the coherence within each cluster. J. Witzig (ZIB): Cycle-Clustering 4 / 25
Our Goal 1. Find a clustering C 1 , . . . , C m , 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. J. Witzig (ZIB): Cycle-Clustering 4 / 25
Our Goal 1. Find a clustering C 1 , . . . , C m , 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 optimality or to measure the solution quality . J. Witzig (ZIB): Cycle-Clustering 4 / 25
Cycle-Clustering MINLP � max net-flow + α · coherence of C t t ∈C � s . t . x it = 1 ∀ i ∈ S (1) t ∈C � x it ≥ 1 ∀ t ∈ C (2) i ∈S x it ∈ { 0 , 1 } ∀ i ∈ S , ∀ t ∈ C ◮ Set of clusters C ◮ Discrete set of states S J. Witzig (ZIB): Cycle-Clustering 5 / 25
Cycle-Clustering MINLP � max net-flow + α · coherence of C t t ∈C � s . t . x it = 1 ∀ i ∈ S (1) t ∈C � x it ≥ 1 ∀ t ∈ C (2) i ∈S x it ∈ { 0 , 1 } ∀ i ∈ S , ∀ t ∈ C ◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable x it = 1 ⇐ ⇒ state i is assigned to cluster t J. Witzig (ZIB): Cycle-Clustering 5 / 25
Cycle-Clustering MINLP � � max ǫ t + α · coherence of C t t ∈C t ∈C � s . t . x it = 1 ∀ i ∈ S (1) t ∈C � x it ≥ 1 ∀ t ∈ C (2) i ∈S � ǫ t = π i p ij ( x it x j φ ( t ) − x i φ ( t ) x jt ) ∀ t ∈ C (3) i � = j ∈S x it ∈ { 0 , 1 } ∀ i ∈ S , ∀ t ∈ C ǫ t ∈ R ≥ 0 ∀ t ∈ C ◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable x it = 1 ⇐ ⇒ state i is assigned to cluster t ◮ Cont. variable ǫ t representing the net-flow between C t and C φ ( t ) J. Witzig (ZIB): Cycle-Clustering 5 / 25
Cycle-Clustering MINLP � � � max ǫ t + α · π i p ij x it x jt t ∈C t ∈C i , j ∈S � s . t . x it = 1 ∀ i ∈ S (1) t ∈C � x it ≥ 1 ∀ t ∈ C (2) i ∈S � ǫ t = π i p ij ( x it x j φ ( t ) − x i φ ( t ) x jt ) ∀ t ∈ C (3) i � = j ∈S x it ∈ { 0 , 1 } ∀ i ∈ S , ∀ t ∈ C ǫ t ∈ R ≥ 0 ∀ t ∈ C ◮ Set of clusters C ◮ Discrete set of states S ◮ Binary variable x it = 1 ⇐ ⇒ state i is assigned to cluster t ◮ Cont. variable ǫ t representing the net-flow between C t and C φ ( t ) J. Witzig (ZIB): Cycle-Clustering 5 / 25
Linearizing the MINLP-Formulation ◮ automatic linearization by SCIP / textbook linearization ✗ J. Witzig (ZIB): Cycle-Clustering 6 / 25
Linearizing the MINLP-Formulation ◮ automatic linearization by SCIP / textbook linearization ✗ ◮ compact linearization for binary quadratic problems (L. Liberti, 2007) ✓ / ✗ J. Witzig (ZIB): Cycle-Clustering 6 / 25
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 ✓ J. Witzig (ZIB): Cycle-Clustering 6 / 25
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 , π i p ij + π j p ji > 0 } ◮ Binary variables y ij = 1 ⇔ states i and j are in the same cluster, ∀ i , j ∈ E : i < j ◮ Binary variables z ij = 1 ⇔ states i ∈ C t and j ∈ C φ ( t ) , ∀ i , j ∈ E J. Witzig (ZIB): Cycle-Clustering 6 / 25
MIP Model � � z ij ( π i p ij − π j p ji ) + α max y ij ( π i p ij + π j p ji ) ( i , j ) ∈ E ( i , j ) ∈ E : i < j � s . t . x it = 1 ∀ i ∈ S t ∈C � x it ≥ 1 ∀ t ∈ C i ∈S x it + x jt − y ij + z ij − x j φ ( t ) − x i φ − 1 ( t ) ≤ 1 ∀ t ∈ C , ( i , j ) ∈ E x it + x j φ ( t ) − z ij + y ij − x jt − x i φ ( t ) ≤ 1 ∀ t ∈ C , ( i , j ) ∈ E y ij + z ij + z ji ≤ 1 ∀ ( i , j ) ∈ E : i < j x it ∈ { 0 , 1 } ∀ i ∈ S , ∀ t ∈ C y ij , z ij , z ji ∈ { 0 , 1 } ∀ ( i , j ) ∈ E : i < j J. Witzig (ZIB): Cycle-Clustering 7 / 25
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