Data-driven linear decision rule approach for distributionally robust optimization of on-line signal...

Data-driven Linear Decision Rule for Distributionally Robust Optimization

of On-line Signal Control

a Department of Industrial and Manufacturing Engineering, Penn State b Department of Civil and Environmental Engineering, Imperial College London

c Department of Civil and Environmental Engineering, Penn State

ISTTT 2015, Kobe, Japan

H. Liua, K. Hanb, V. Gayahc, T. Friesza, T. Yaoa

Outline

1. Linear Decision Rule for Responsive Signal Control

2. Distributionally Robust Optimization

3. Numerical Studies

Outline




Analytical/ closed-form

transformation

Decision Rule Approach for Responsive Signal Control

v  Real-time signal control: challenges

-  Computational burden -  Nonlinear and nonconvex objective -  Distributed vs. centralized control

v  Heuristic (genetic algorithm, fuzzy logic), inexact and sub-optimal

v  Two-stage Linear Decision Rule (LDR) approach for adaptive signal control:

ü  Historical and real-time data ü  Within-day and day-to-day variations ü  Distributionally Robust Optimization (DRO) to ensure

performance in the most adversarial scenario ü  Efficient on-line operation ü  Compatible with analytical computations and microsimulation

Real-time info

Signal control

parameters

Not optimal?

Decision rule

-- real-time information (flow, count, speed, queue)

-- Linear transformation

-- Projection onto feasible control set

-- Network performance measure (congestion, emission, fuel consumption)

Real-time Information

q

Signal Control

u

Linear Decision Rule

u = P![Aq+ b]

Network performance

measure

!(q,u)

Aq+ bu = P![Aq+ b]

Linear Decision Rule: Deterministic Formulation

q

Deterministic Formulation

Given real-time information q, find the best linear decision rule

Details of The Linear Decision Rule

Time

Location/ data type

past T observations

OBSERVATION

CONTROL

Details of The Linear Decision Rule, Continued

Ø Constraints on the signal control parameters

-  maximum/minimum green time -  fixed cycle -  all-red -  offset

Ø  : Set of feasible signal control parameters è

Ø  This is relevant to the mixed integer linear program

Proposition

If can be expressed using linear constraints, then the projection

is equivalent to a set of linear constraints with binary variables

Linear Decision Rule: Stochastic Extension

v  In reality, q is stochastic

v  Stochastic programming – exactly known probability distribution

v  Ambiguous information on the distribution with finite samples

v  Distributionally robust optimization

Ø  Worst-case scenario (‘max’),

Ø  among all candidate distributions

Ø  Subsumes stochastic optimization

Ø  Data-driven calibration of

Distributionally Robust Formulation Given stochastic info q, find the best linear decision rule in terms of A and b

“Replacing uncertain parameters with uncertain distributions”

Advantages of the Linear Decision Rule Approach

v  Finding the best responsive signal strategy è Finding A and b

v  Viable and efficient on-line operation

-  Off-line: Distributionally robust optimization (expensive) -  On-line: Linear transformation and projection (inexpensive)

v  Flexible sensor location, data type, and control resolution

v  User-defined feasible set for signal control parameters

v  Two solution procedures for the off-line problem:

-  Mixed integer linear program -  Metaheuristic search

Distributionally Robust Optimization

Outline




v  Kolmogorov-Smirnov (K-S) goodness-of-fit test (Massey, 1951; Bertsimas et al., 2013):

v  Sampled data:

v  Does a distribution well capture a finite set of sampled data?

v  Reject H0 at the level α if

Data-Driven Calibration of the Uncertainty Set

Set of candidate distributions

Uncertainty set

Change of Notation

= =

Off-Line Computation: Mixed Integer Linear Program

Off-Line Problem

Dual Formulation

Finite Formulation

Ø  Min-Max (bi-level)

Ø  Infinite-dimensional

Ø  Single-level

Ø  Infinite-dimensional

Ø  Finite-dimensional

Ø  Solvable

Ziliaskopoulos (2000); Lo (1999); Lin and Wang (2004); Han et al. (2015)

Duality

Finite sampling

Off-Line Computation: Metaheuristic Search

v  Limitations of the MILP approach

-  Linear signal constraints and objective -  Independence assumption: -  Computationally intractable

v  Heuristic method

-  Arbitrary feasible signal control set and objective -  Free of the independence assumption -  Large-scale and more realistic setting -  Zeroth-order information on the objective and constraints -  Compatible with analytical and simulation-based traffic models

v  Particle Swarm Optimization (Kennedy and Eberhart, 1995; Banks et al., 2007)

Metaheuristic Search: The K-S Test

v  Random Variable: , parameterized by

v  Uncertainty Set: , parameterized by

v  Fix , and consider K samples (historical data)

v  K-S test:

Uncertainty set

Metaheuristic Search: Evaluating the Objective Function

v  Random Variable (objective): , parameterized by

v  Lower and upper bounds of : , partitioned into W intervals

v  Fix (control),

g1 Lf Uf g2 gi-1 gi

. . . . . .

K-S test

Metaheuristic Search: Pseudo Code for Evaluating Objective

Input ; historical dataset Step1 For , conduct traffic simulation to calculate

Step 2 Calculate, for

Step 3 Evaluate the worst-case expected performance by solving the linear program

Evaluate the objective function for a given x

x = (A,b)

K-S test

Outline




Numerical Study: Test Network

Great Western Rd

Great Western Rd

Byre

s Rd

City Center

University of Glasgow

§  West end of Glasgow §  5 signalized intersections §  35 directed links

Network

Data §  Turn-by-turn flow count §  8-9 am, 7 June 2010 §  Daily variations are

generated synthetically using a variety of distributions (!!)

Benchmark §  Fixed signal timing

(optimized based on the averaged flow)

§  Approximation of SCOOT operation

Numerical Study: Calibration of the Uncertainty Set

0 20 40 60 80 100 120 140 160 180 200

0.05

0.15

0.25

0.35

Time

Inco

min

g f

low

on

lin

k 1

Average Sample path 1 Sample path 2 Sample path 3 Sample path 4 Sample path 5

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

Incoming flow (veh/s)C

um

ula

tive

pro

ba

bili

ty

KS upper (! = 0.05)

KS lower (! = 0.05)

KS upper (! = 0.30)

KS lower (! = 0.30)

KS upper (! = 0.75)

KS lower (! = 0.75)

True distribution

(a) (b)

(a): The average flow and 5 (out of 30) sampled flows (b): Constructed uncertainty set in terms of the CDF

Great Western Rd

Great Western Rd

Byre

s Rd

City Center


1 2 3 4

5

8

6

9

7

10 11

12

inflo

w

inflow inflow inflow

inflow

Great Western Rd

Mixed Integer Linear Program

West Glasgow

Numerical Study: Mixed Integer Linear Program

Numerical Study: Heuristic Search

Particle Swarm Optimization Great Western Rd

Great Western Rd

Byre

s Rd

City Center


§  Zeroth-order information on the objective and constraints

§  10 agents search in parallel (objective evaluation)

§  Flexible trade-off between solution quality and computational cost

§  Off-line computational time: 24h §  On-line computational time: negligible

Conclusion and On-going Research

Linear Decision

Rule Deterministic

Problem

Network Info

Daily variations

Stochastic Problem

Uncertainty In

Distribution Distributionally

Robust Optimization

K-S Test

Explicit reformulation 1.  Mixed integer linear program 2.  Heuristic search method

Conclusion and On-going Research

v  Integration with microsimulation (heuristic method)

v  Multi-objective responsive signal control (emission, fuel consumption)

v  Nonlinear decision rules (neural network)

On-going and future research

v  Explicit re-formulation: solve for A and b

v  Flexible real-time data input (space & time)

v  General signal constraint and objectives (heuristics)

v  Efficient on-line operation

v  Blend of statistical learning and traffic flow modeling

v  Tunable probabilistic guarantee

Thank you! Q&A

[email protected]

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