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

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 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

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

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

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

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

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

Extended Formulations Definition (extended formulation) An extension of a polytope P ⊆ R n is a polyhedron Q ⊆ R d together with an affine map p ∶ R d → R n 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 ∈ R n ∣ ∃ y ∈ Q ∶ p ( y ) = x } 6 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

Example: The Parity Polytope The (even) parity polytope of dimension n is defined by P even ∶ = 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

Example: The Parity Polytope The (even) parity polytope of dimension n is defined by P even ∶ = conv { x ∈ { 0 , 1 } n ∣ x has an even number of ones } It is almost trivial to optimize a linear objective over P even . However, P even has exponentially many facets. 7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

Example: The Parity Polytope The (even) parity polytope of dimension n is defined by P even ∶ = conv { x ∈ { 0 , 1 } n ∣ x has an even number of ones } It is almost trivial to optimize a linear objective over P even . However, P even has exponentially many facets. There is a well-known linear size extended formulation for P even , based on network flows: Extreme points of P even ̂ = paths in the network 7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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

Example: The Parity Polytope The (even) parity polytope of dimension n is defined by P even ∶ = conv { x ∈ { 0 , 1 } n ∣ x has an even number of ones } It is almost trivial to optimize a linear objective over P even . However, P even has exponentially many facets. There is a well-known linear size extended formulation for P even , based on network flows: x 3 Extreme points of P even ̂ = paths in the network 7 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

Finite Automata Definition (finite automaton) A deterministic finite automaton over the alphabet { 0 , 1 } is a quadruple ( Q ,δ, q 0 , F ) , where ● Q is a nonempty finite set of states ● δ ∶ Q × { 0 , 1 } → Q is the transition function ● q 0 ∈ 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 = a 0 a 1 ... a k ∈ L there exists a sequence of states q 0 = r 0 , r 1 ,..., r k in Q such that r i + 1 = δ ( r i , a i + 1 ) and r k ∈ F . 8 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

Finite Automata Proposition [Fiorini, Pashkovich, 2013] Let L denote a language over { 0 , 1 } and M = ( Q ,δ, q 0 , F ) any deterministic finite automaton recognizing the language L . For each positive integer n , there exists an extended formulation of P n (L) ∶ = { x ∈ R n ∣ 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

Example: Finite Automata & The Parity Polytope 0 odd 1 1 even start 0 10 Flow-based Extended Formulations Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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

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 ( x i = 0) and green ( x i = 1) phases 12 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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 ( x i = 0) and green ( x i = 1) phases ● modeling examples (for min. green): ● x t − x t − 1 ≤ x τ for τ ∈ { t + 1 ,..., t + L − 1 } [Takriti, Krasenbrink, Wu, 2000] ● x t − 1 − x t ≤ 1 L ∑ L j = 1 x t − j [Sorgatz, 2016] 12 Minimum Red and Green Phases Maximilian Merkert // Flow-based extended formulations for feasible traffic light controls

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 ( x i = 0) and green ( x i = 1) phases ● modeling examples (for min. green): ● x t − x t − 1 ≤ x τ for τ ∈ { t + 1 ,..., t + L − 1 } [Takriti, Krasenbrink, Wu, 2000] ● x t − 1 − x t ≤ 1 L ∑ L j = 1 x t − 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

Complete Description of the Min-up/min-down Polytope For a nonnegative integer k , consider a nonempty set of 2 k + 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