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Flow-based extended formulations for feasible traffic light controls - - PowerPoint PPT Presentation

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Maximilian Merkert Flow-based extended formulations for feasible traffic light controls Aussois Combinatorial Optimization Workshop, January 8, 2019 joint work with Gennadiy Averkov, Do Duc Le Outline 1 Introduction 2 Flow-based Extended

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Maximilian Merkert

Flow-based extended formulations for feasible traffic light controls

Aussois Combinatorial Optimization Workshop, January 8, 2019 joint work with Gennadiy Averkov, Do Duc Le

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

1 Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

2 Introduction Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Centralized Optimization of Traffic-Light Controlled Intersections

  • All participants (cars and traffic lights)

are controlled centrally.

  • Our MINLP-model is time-discretized and

has binary variables due to trigger modeling for collision prevention on the intersection and enforcing feasible traffic light controls.

  • In this talk: focus on modeling of traffic

light regulations (independently from the rest of the model).

3 Introduction Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Traffic Light Regulations

Traffic light controls have to fulfill certain requirements in order to be reasonable or even legal, such as:

  • minimum length of green and red phases
  • minimum evacuation times
  • maximum length of green and red phases
  • minimum and maximum cycle times
  • pulse intervals
  • regulations on the switching order of multiple conflicting traffic lights

4 Introduction Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Traffic Light Regulations

Traffic light controls have to fulfill certain requirements in order to be reasonable or even legal, such as:

  • minimum length of green and red phases
  • minimum evacuation times
  • maximum length of green and red phases
  • minimum and maximum cycle times
  • pulse intervals
  • regulations on the switching order of multiple conflicting traffic lights

4 Introduction Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

5 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Extended Formulations

Definition (extended formulation)

An extension of a polytope P ⊆ Rn is a polyhedron Q ⊆ Rd together with an affine map p ∶ Rd → Rn with p(Q) = P. Any description of Q by linear equations and linear inequalities then (together with p) is called extended formulation of P. The size of the extended formulation is the number of its inequalities. P = {x ∈ Rn ∣ ∃y ∈ Q ∶ p(y) = x}

6 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: The Parity Polytope

The (even) parity polytope of dimension n is defined by Peven ∶= conv{x ∈ {0,1}n ∣ x has an even number of ones}

7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: The Parity Polytope

The (even) parity polytope of dimension n is defined by Peven ∶= conv{x ∈ {0,1}n ∣ x has an even number of ones} It is almost trivial to optimize a linear objective over Peven. However, Peven has exponentially many facets.

7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: The Parity Polytope

The (even) parity polytope of dimension n is defined by Peven ∶= conv{x ∈ {0,1}n ∣ x has an even number of ones} It is almost trivial to optimize a linear objective over Peven. However, Peven has exponentially many facets. There is a well-known linear size extended formulation for Peven, based on network flows: Extreme points of Peven ̂ = paths in the network

7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: The Parity Polytope

The (even) parity polytope of dimension n is defined by Peven ∶= conv{x ∈ {0,1}n ∣ x has an even number of ones} It is almost trivial to optimize a linear objective over Peven. However, Peven has exponentially many facets. There is a well-known linear size extended formulation for Peven, based on network flows: 1 1 1 1 Extreme points of Peven ̂ = paths in the network highlighted solution: x = (1,0,1,1,1)

7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: The Parity Polytope

The (even) parity polytope of dimension n is defined by Peven ∶= conv{x ∈ {0,1}n ∣ x has an even number of ones} It is almost trivial to optimize a linear objective over Peven. However, Peven has exponentially many facets. There is a well-known linear size extended formulation for Peven, based on network flows: x3 Extreme points of Peven ̂ = paths in the network

7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Finite Automata

Definition (finite automaton)

A deterministic finite automaton over the alphabet {0,1} is a quadruple (Q,δ,q0,F), where

  • Q is a nonempty finite set of states
  • δ ∶ Q × {0,1} → Q is the transition function
  • q0 ∈ Q is the initial state
  • F ⊆ Q is the set of accept states

The automaton recognizes a set L of 0 − 1 strings (a language) if for each string w = a0a1 ...ak ∈ L there exists a sequence of states q0 = r0,r1,...,rk in Q such that ri+1 = δ(ri,ai+1) and rk ∈ F.

8 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Finite Automata

Proposition [Fiorini, Pashkovich, 2013]

Let L denote a language over {0,1} and M = (Q,δ,q0,F) any deterministic finite automaton recognizing the language L. For each positive integer n, there exists an extended formulation of Pn(L) ∶= {x ∈ Rn ∣ x ∈ L} with size at most 2∣Q∣n.

9 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Example: Finite Automata & The Parity Polytope

even start

  • dd

1 1

10 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

11 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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The Min-up/min-down Polytope

P(L,l) ∶= conv{x ∈ {0,1}n ∣ sequences of 1s have length at least L, sequences of 0s have length at least l, except for the first and the last sequence}

  • corresponds to minimum red (xi = 0) and green (xi = 1) phases

12 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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The Min-up/min-down Polytope

P(L,l) ∶= conv{x ∈ {0,1}n ∣ sequences of 1s have length at least L, sequences of 0s have length at least l, except for the first and the last sequence}

  • corresponds to minimum red (xi = 0) and green (xi = 1) phases
  • modeling examples (for min. green):
  • xt − xt−1 ≤ xτ for τ ∈ {t + 1,...,t + L − 1} [Takriti, Krasenbrink, Wu, 2000]
  • xt−1 − xt ≤ 1

L ∑L j=1 xt−j [Sorgatz, 2016]

12 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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The Min-up/min-down Polytope

P(L,l) ∶= conv{x ∈ {0,1}n ∣ sequences of 1s have length at least L, sequences of 0s have length at least l, except for the first and the last sequence}

  • corresponds to minimum red (xi = 0) and green (xi = 1) phases
  • modeling examples (for min. green):
  • xt − xt−1 ≤ xτ for τ ∈ {t + 1,...,t + L − 1} [Takriti, Krasenbrink, Wu, 2000]
  • xt−1 − xt ≤ 1

L ∑L j=1 xt−j [Sorgatz, 2016]

  • complete description by [Lee, Leung, Margot, 2003]
  • linear size extended formulation by [Rajan, Takriti, 2005]

12 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Complete Description of the Min-up/min-down Polytope

For a nonnegative integer k, consider a nonempty set of 2k + 1 indices from the discrete interval [1;T]: Φ(1) < Ψ(1) < Φ(2) < Ψ(2) < ⋅⋅⋅ < Φ(k) < Ψ(k) < Φ(k + 1)

13 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Complete Description of the Min-up/min-down Polytope

For a nonnegative integer k, consider a nonempty set of 2k + 1 indices from the discrete interval [1;T]: Φ(1) < Ψ(1) < Φ(2) < Ψ(2) < ⋅⋅⋅ < Φ(k) < Ψ(k) < Φ(k + 1) such that Φ(k + 1) − Φ(1) ≤ L. Then we have the alternating up inequality −

k+1

∑

i=1

xΦ(i) +

k

∑

i=1

xΨ(i) ≤ 0 [Lee, Leung, Margot, 2003]

13 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Complete Description of the Min-up/min-down Polytope

For a nonnegative integer k, consider a nonempty set of 2k + 1 indices from the discrete interval [1;T]: Φ(1) < Ψ(1) < Φ(2) < Ψ(2) < ⋅⋅⋅ < Φ(k) < Ψ(k) < Φ(k + 1). Correspondingly, if Φ(k + 1) − Φ(1) ≤ l, we have the alternating down inequality

k+1

∑

i=1

xΦ(i) −

k

∑

i=1

xΨ(i) ≤ 1 [Lee, Leung, Margot, 2003] Exponentially many inequalities, separation is possible in polynomial time.

13 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Turn on/off Inequalities

Introduce new binary variables yt,yt ∈ {0,1} to indicate whether a 1-phase/0-phase is started at time period t: xt+1 = xt + yt − yt ∀t ∈ L,...,T − 1

t−1

∑

i=t−L

yi ≤ xt ∀t ∈ L + 1,...,T (Turn-On Inequality)

t−1

∑

i=t−l

yi ≤ 1 − xt ∀t ∈ l + 1,...,T (Turn-Off Inequality)

Proposition [Rajan, Takriti, 2005]

The turn on/off inequalities dominate the alternating up/down inequalities.

14 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Turn on/off inequalities - Computational Test

solution time [s] instance #Cars T[s] ∆t[s] previous formulation turn on/off ineq. p12_40 12 40 0.5 49.98 22.63 p24_40 24 40 0.5 814.95 124.95 p32_40 32 40 0.5 485.09 274.30 p12_50 12 50 0.5 770.62 219.62 p24_50 24 50 0.5 341.09 157.83 p32_50 32 50 0.5 3487.56 2179.76 p24_60 24 60 0.5 1122.62 223.38

Table: Comparison of solution times for different model formulations for minimum green and red phase restrictions.

15 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Flow-Based Extended Formulation for Min. Red/Green Phases

start g r 1 (0)l (1)L 1 (0)<l (1)<L

16 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Flow-Based Extended Formulation for Min. Red/Green Phases

L = 2,l = 3 extreme points of P(L,l) ̂ = paths in the network

17 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Flow-Based Extended Formulation for Min. Red/Green Phases

L = 2,l = 3 11 1 00 extreme points of P(L,l) ̂ = paths in the network highlighted solution: x = (0,1,1,1,0,0)

17 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Flow-Based Extended Formulation for Min. Red/Green Phases

L = 2,l = 3 x4 extreme points of P(L,l) ̂ = paths in the network Turn-on inequality ∑3

i=2 yi ≤ x4 can be interpreted as nonnegative flow on a horizontal arc.

17 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

18 Other Traffic Light Regulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Minimum & Maximum Red/Green Phases

length of green phase ∈ {2,3}, length of red phase ∈ {3,4}

19 Other Traffic Light Regulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Minimum Cycle Times

The cycle time c is the time between the beginning of a green period and its successive green period. g0 g1 gk r0 r1 rk 1 ... ... (0)l (0)l (0)l (1)L 1 where k = c − L − l; initial and final phase omitted for sake of simplicity

20 Other Traffic Light Regulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Switching Order of Multiple Traffic Lights

TL1 TL2 TL3 (0,1,0)L2 (0,0,1)L3 (1,0,0)L1 (1,0,0) (0,1,0) (0,0,1) initial and final phase omitted for sake of simplicity

21 Other Traffic Light Regulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Switching Order of Multiple Traffic Lights

TL1 TL2 TL3 (0,1,0)L2 (0,0,1)L3 (1,0,0)L1 (1,0,0) (0,1,0) (0,0,1) (0,0,0)k ⋅ (0,1,0)L2 Example extension: Switch from TL2 to TL3 possible, preceded by a fixed number k of all-red steps; initial and final phase omitted for sake of simplicity

21 Other Traffic Light Regulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Outline

1 Introduction 2 Flow-based Extended Formulations 3 Minimum Red and Green Phases 4 Other Traffic Light Regulations 5 Conclusion

22 Conclusion Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Conclusion

Summary:

  • Rediscovered complete description of the min-up/min-down polytope via

flow-based-extended formulations.

  • Using theoretically superior formulation has significant computational impact.
  • The concept extends to many other traffic light regulations.

23 Conclusion Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Conclusion

Summary:

  • Rediscovered complete description of the min-up/min-down polytope via

flow-based-extended formulations.

  • Using theoretically superior formulation has significant computational impact.
  • The concept extends to many other traffic light regulations.

Further Research:

  • Are there more applications of these extended formulations—in the context of traffic

light regulations or other MINLP substructures?

  • How to create cutting planes most efficiently from the different automata models?
  • Investigate cutting planes for the intersection of several flow-based extended

formulations.

23 Conclusion Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Conclusion

Summary:

  • Rediscovered complete description of the min-up/min-down polytope via

flow-based-extended formulations.

  • Using theoretically superior formulation has significant computational impact.
  • The concept extends to many other traffic light regulations.

Further Research:

  • Are there more applications of these extended formulations—in the context of traffic

light regulations or other MINLP substructures?

  • How to create cutting planes most efficiently from the different automata models?
  • Investigate cutting planes for the intersection of several flow-based extended

formulations. Thank you for your attention!

23 Conclusion Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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Conclusion

Summary:

  • Rediscovered complete description of the min-up/min-down polytope via

flow-based-extended formulations.

  • Using theoretically superior formulation has significant computational impact.
  • The concept extends to many other traffic light regulations.

Further Research:

  • Are there more applications of these extended formulations—in the context of traffic

light regulations or other MINLP substructures?

  • How to create cutting planes most efficiently from the different automata models?
  • Investigate cutting planes for the intersection of several flow-based extended

formulations. Thank you for your attention!

23 Conclusion Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls