Parallel Computing and Vehicle Routing Teodor Gabriel Crainic · 2018-06-05 · Parallel Computing...

Parallel Computing and Vehicle Routing

Teodor Gabriel CrainicTeodorGabriel.Crainic@CIRRELT.net

VeRoLog PhD School, Cagliari, June 1-2, 2018

Parallel Optimization

Several processes work simultaneously

on several processors

with the common goal of

solving a given problem instance

Solving Problems with O.R.

Difficult decision problems in planning and managing

complex systems

Transport, Logistics, Telecom, Production, Health, …

Operations Research methodology to address/solve them

Modelling: Mathematical formulations

Resolution: Solution algorithms/methods

Optimization and Solution Methods

Models with “nice” mathematical properties

Exact mathematical programming methods

e.g., Linear and convex programming

Hard optimization problems with “nice” mathematical

properties and small dimensions

Exact implicit enumeration methods

e.g., Branch-&-Bound

& cut & price & …

Polyhedral methods (cutting planes)

Larger instances? Decomposition + parallel computing

Optimization and Solution Methods (2)

All others?

Larger dimensions

Hard optimization problems, e.g., combinatorial

Optimization problems without “nice” mathematical

properties

Non-optimization problems

Heuristics, meta-heuristics, matheuristics

Decomposition + parallel computing

Parallel Computing

Parallel meta-heuristics

Cooperative search

A rapid perspective on Parallel Branch & …

We focus on algorithmic design not on implementation

(nor particular computing architectures)

The world is (massively) parallel ☺

Problems are large and complex (in space and time)

Parallel computing to better represent, simulate, “solve”,

understand them

Simulating actual systems (astrophysics, biology,

health, genomics, material engineering, …) and

virtual ones (games, movies, virtual reality, …)

Solving larger / more complex problems (VRP …)

Concurrency – performing several things

simultaneously, e.g., collaborative search

Better use of resources

Parallel Computing – Why ?

Several processes work simultaneously on several

processors with the common goal of solving a given

problem instance

Solving (for us) = Finding the (an) optimal or

a (good) feasible solution

Parallelism

Decompose the computational load (of solving)

Distribute the resulting tasks to available processors

Solve !

Extract (reconstruct) the solution

Parallel Computing – How ?

Solve more rapidly

Mathematical problems, exact solution algorithms

Aim for speedup =

Best sequential time / p-processor parallel time

The same algorithm, the same solution

Better representation

Simulation

Finer tasks, more complex interactions

Broader & more robust search

Heuristics

Parallel Computing Goals

Parallel Search – Sources of Parallelism

How to divide the task of solving the problem instance

into more or less independent tasks to be addressed more

or less simultaneously?

What is this “complete solving” task?

How to define tasks?

How to distribute the set of tasks (global search)?

Functional

The “algorithm”

Decompose computing-intensive tasks

Work on the same or dedicated parts of the data

Rarely changes the sequential algorithm

Search space separation

Decompose the problem domain (solution/search

space) or

Decompose the problem structure

Work on each part with particular solution method

Require “control” of the overall search

Parallel Computing Decomposition Types

Task granularity

Fine or coarse grained

Inter-task (processor) communications

Synchronous = All finish current task before

exchanging

Asynchronous = Individuals proceed when ready

Parallel Computing Decomposition

Performing the search & communications

How tasks (including the complete algorithm),

information (e.g., status), data (e.g., new solutions or

components), commands are initiated, exchanged,

terminatedSometimes together, sometimes separated

How the data is kept

Centralized (“master-slave)

A unique decision maker

Decentralized (“cooperation”, “collegial”)

Multi-level, hierarchical, …

Parallel Computing Control

Meta-heuristics

Master strategies (heuristics) to guide and modify other

heuristics to produce solutions beyond those normally

identified by local search heuristics (Glover 1986)

Neighbourhoods & populations

Avoiding getting trapped into

Local optima

Sequences of visited solutions (cycling)

Trying to avoid overlooking promising regions

Memories and learning capabilities

Local search is often one of the guided heuristics

Meta-heuristics (2)

Neighborhood Search Methods

Tabu Search, Adaptive Large Neighbourhood Search

(ALNS), Variable Neighbourhood Search (VNS +),

GRASP, Guided Local Search, Simulated Annealing,

and so on

Population-based methods

Evolutionary (genetic) methods, Scatter Search, Path

Relinking

Swarm intelligence

Ant colonies, bee swarms, schools of fish, …

Hybrids, Matheuristics

Meta-heuristics (3)

Different but with a few fundamental common design

elements and characteristics

Guiding strategies

Selecting neighbourhoods, individual-selection rules,

search phases (intensify/diversity, population

management), …

Exploring/Selecting in the neighbourhood (solution

transformation, individual matting, …)

Learning, explicit (memories), implicit (population), noneCommon strategies may be designed

Parallel Meta-heuristics Goals

Accelerate the search

For comparable solution quality (at least)

Broaden the search for better solutions

For comparable wall-clock time

Build a more robust search method for better solutions

The method performs well on different types of

problem instances

Looks “deeper” and “broader” for solutions with

good values for specific attributes

Sources of Parallelism in Parallel MH Search

The “complete solving” task

Finding a good, the best possible, solution

Significantly less predefined than exact solution

methods (e.g., Branch-and-Bound)

May change with the “decomposition” definition

A new class of meta-heuristics

How to define tasks?

Functional or search space separation

Functional Decomposition

Several tasks work in parallel on the “same” data

The “inner loop” of meta-heuristics, e.g.,

neighbourhood/fitness evaluation

Not much in meta-heuristics …,

But may become important for local search, hierarchical

schemes, etc.

Search Space Separation

A very rich source

Search Space (Domain) decomposition (data parallelism)

Explicit separation,

Same meta-heuristic on each component

Construct complete solution

Multi-search (threads, walks, …)

Implicit separation

Independent solvers (meta-heuristics, exact, …) work

on complete space or partial problems

Construct complete solution (latter case)

Combining Decompositions & Strategies

Combining strategies & methods (including exact ones)

“Hybrids”: not much information in this name …

Hierarchical designs

Cooperation

Integrative Cooperative Search

Partition or cover (region overlap)

Building complete solutions out of partial ones

Control/guidance of the overall search given the result of

concurrent explorations (if)

What & how & when information is exchanged (if any)

How to gather global information (if any)

How to create new information (if any)

What to do with received information

What search strategies

Parallel Search – Main Design Issues

A 3-Dimension Taxonomy of Algorithmic Design

Search control cardinality

Search control and

Communications

Search differentiation

KS C KC

Control of

- Work (who does what)

- Communications

(who talks to whom, why, how,

when, and about what)

- Knowledge

(who knows what, e.g., “are we

done, yet?”

what is created new?)

How many are controlling?

A 3- Dimension Taxonomy

Search control cardinalitySearch control and

Communications

KS C KC

Same/Different

Starting points/populations

Strategies

Rigid/Knowledge Synchronous

Asynchronous “Collegial”

Knowledge-creating Collegial

1-Control, RS/KS, SPSS: Functional Decomposition

Search control

cardinality

Search control and

communications

Low-level parallelism

Accelerates intensive tasks

Does not change the algorithm

(may change behavior)

1C/RS & KS – Master-Slave Model

1C/RS & KS: Functional Decomposition

Neighbourhood evaluation

Local Search

Education (individual improvement) for populations

Fitness evaluation for populations

Swarms

Master updates pheromone/global matrix

Slaves perform an individual’s work (e.g.,

construction heuristic)

Does not change the sequential behavior & solution

1C/RS & KS: Functional Decomposition (2)

1C/KS: Master-Slave Probing / Look Ahead

Allow slaves to execute a few iterations to select the most

promising trajectory

The parallel search method is different from the sequential one different solutions

Good speedups when evaluation is costly

QAP, VRP with ejection chains, fitness evaluation

Probing: better in quality, but, more complex to control,

outperformed by pC methods

Low-level decomposition interest: Large (very)

neighbourhoods/populations, Local Search in hierarchical

setups, swarms, using Graphic Processing Units

1C&pC/KS Search Space (Domain) Decomposition

Search control

cardinality

Search control and

communications

“Domain” decomposition

Perform the search on parts of

search space (“partition”)

Reconstruct global solution

Two types of control

1C/KS Explicit Search Space Decomposition

pC/KS Explicit Search Space Decomposition

Explicit Search Space Decomposition

How to separate? Eliminate or freeze variables?

Partition or cover?

Global solution reconstruction?

Moves restricted to own zone or not?

Modify decomposition and repeat?

To avoid missing subspaces in between partitions

1C = master-slave (real-time ambulances’ routing)

pC = reconstruction through synchronous exchanges

(first proposal for VRP 1993; ants for VRP 2006)

Potential for very large instances (e.g., time-dependent

network design), hierarchical strategies, …

Multi-search Parallel Meta-heuristics

Algorithm-based search space separation

Several “independent” meta-heuristics work

simultaneously on the same problem and search space

Communications = algorithm design

None: Independent Search

All processes synchronize through exogenous (hard

coded) or collegial control

Processes communicate asynchronously

The most successful

Direct or indirect communications/exchanges

Knowledge creation

pC/RS: Independent (Multi) Search

KS C KC

Search control

cardinality

Search control and

communications

One of the first strategies

Parallel multi-start

Easy to implement

“Not bad” ….

pC/RS Independent Search

Perform meta-h

SPDS/MPSS/MPDS

Return best

Perform meta-h

SPDS/MPSS/MPDS

Return best

Perform meta-h

SPDS/MPSS/MPDS

Return best

Select

best of best

Cooperative Multi-Search Strategies

Search control

cardinality

Search control and

communications

pC/KS pC/C pC/KC

Cooperative Multi-search Strategies

The most successful

Solvers participate to the cooperative search

Information is shared (and created)

Meaningful & timely

The information-sharing (and guidance) mechanism

How the solvers interact (who exchanges, when, how)

What information is shared (if any)

How is the shared information used

Locally: How each solver acts on / transforms the received

information / creates new one, before making it available

Globally (if)

Cooperative Multi-search Strategies (2)

Exchanged information

Meaningful

Timely

Enhance individual solver performance

Create “global” view of system to guide the search

What is exchanged

Good solutions, e.g., overall best, elite, local best at

end of improvement sequence

Context information, e.g., memories

Global guidance (control)

How it is exchanged – Communication topology

Directly at agreed meeting points, diffusion, or

broadcasting (humm, danger here, see next!)

Indirectly, through independent data structures (and

process)

How it is exchanged – Communication mode

Synchronously

Asynchronously

General remarks on cooperation

Tends to favor regions where good solutions were found Need for diversification mechanisms

No (major) differences between neighbourhood- and

population-based methods

Often combined

Main principle of swarm methods

Knowledge synchronization

Solvers stop and exchange, at pre- or dynamically-

determined moments

Collegial

Asynchronous exchanges initiated by one of the

solvers

Simple information exchanges

Knowledge collegial

Extract new information from exchanged data

Global guidance (control)

Synchronous Cooperation

Synchronous strategies show poor performance

compared to asynchronous and independent search

Less reactive to environment

Larger computation overheads

Premature convergence of the dynamic process

Global broadcasting: May yield a global random search

Broadcast-based Communications

Search

kSearch

Search

lSearch

Search

New best

solution !!

Better

solution

improving !

Synchronous Cooperation (2)

Need to control tightly the information exchanges

What, when, to whom, handling of import data, …

Should not be overwhelming

Asynchronous Cooperation

Most successful

Global solution (decision) emerges out of the collective

behaviour/decisions/solutions & exchanges of the

individual solvers New meta-heuristic class

More “agile”, adaptable to the search evolution

Exchange appropriate information in a timely manner

Aim for quality, meaningfulness, parsimony

What information is exchanged?

Good solutions – local best(s) (& diverse)

Context information (local “learning”)

Asynchronous Cooperation (2)

When is information exchanged?

Send on reaching new local optima or important

information (e.g., strategic variable) discovered

Require at decision moments, e.g., diversification or

generation renewal

Send & require jointly?

Reduces communication overhead

Maybe not, to make available significant new

information (improved solutions/individuals) sooner

Asynchronous Cooperation (3)

What does one do with imported data?

Integrate into choices and transform

What does one do with the exchanged data?

New information & guidance may be derived

Asynchronous Cooperation Exchanges

A logical communication graph to be mapped on the

computer architecture (for synchronous as well)

Direct, solver(s)-to-solver(s)

Most population-based methods (migration)

Swarms

Multi-level

Indirect, through data warehouse / blackboard / pool /

central (central / adaptive) memory

Solvers work on

Full problem

Partial problem + integration49

Controlled Direct Exchange Mechanisms

Search

kSearch

Search

lSearch

Search

Direct Exchange Mechanisms: Diffusion

Search

lSearch

Search

One-to-one or one-to-many mainly in genetic-based

population-based methods & some swarms

Island methods (ants: part of colony = island)

Migration initiated within one population

Send new best individual

Request when stagnant population

Individual accepted when different and better than

worst in receiving population

Complete communication graphs not necessary

Swarms: diffusion (often)

Direct Exchange Mechanisms

The Multi-level Paradigm

Search Phase

Coarsening Phase Refinement Phase

Multi-level Search

Search

Refine solution:

Interpolation (and search)

Coarsen the problem instance

Search

Multi-level Method Paradigm

Coarsen the solution space into hierarchical sub-spaces

Coarsening / aggregation operators

Find (search) a “good” solution at the top-most level

Search on the most coarsened problem

Project back (interpolate) the solution at the higher level

to guide the search at the current level

Projection / interpolation operators

(relatively “simple” search)

The search at level 0 yields the solution

Cooperative Multi-level Search

Search

Refine solution:

Interpolation (and search)

Elite solutions and memories

Coarsen the problem instance

Modify coarsening

Elite solutions and memories

Search

Cooperative Multi-level Search (2)

Several (re-)coarsening operators

Aggregation, variable fixing, partitioning, …

Coarsening: critical element

Several projection/interpolation operators

One/several searches

Use of local elite sets and memories to guide operations

at neighbouring levels

Excellent results: (hyper) graph partitioning, network

design, feature selection in bio-medical data, …

Cooperative Multi-level with Central Memory

Search

Memory

Elite SetNeeds to

be studied

Indirect, Memory-based Cooperation

Search

Memory, Pool

Reference Set

Elite Set

Data Warehouse

pC Memory-based Mechanisms

No direct exchanges among solvers

Data is deposited into memory according to internal logic of the solver

It is extracted on request according to guiding mechanism

Asynchronous operations and process independence easy to enforce

Easy to keep track of exchanged data, easy to manipulate it to build new data and search directions

Dynamic and adaptive structure

Same cooperation logic may be implemented without the common repository structure but it is not efficient

pC/C Indirect, Memory-based Cooperation

Solution set

(population)

+ Context information

+ Newly created

information

Solver set

SSearch

ControlGuidance

Memory

SOLUTIONS

SEND / REQUEST

(QUALITY, DIVERSITY)

EXTRACT & SEND ON REQUEST

Used with almost all classes of neighbourhood meta-

heuristics

Extraction from memory: random, biased by rank (and

diversity)

As in all cooperative strategies, solvers may belong to

different meta-heuristic or exact solution-method class

Different methods may be activated at different

phases within solvers

Context data (e.g., long-term central memories built on

short-term local ones) may be built

pC/C Indirect, Memory-based Cooperation

pC/KC Indirect, Memory-based Cooperation

There is a great richness in the information exchanged

Good and diverse solutions of various origins

Local context

Memory = Population of elite solutions dynamically

updated

One may analyze & learn & create new information,

new knowledge out of the exchanged one

Create new solutions (genetic, scatter, path relinking)

Build global context information

Build guiding information

Simple pC/KC Experiment for Network Design

pC/C with several Tabu Searches and Central Memory

A GA that takes its population from Memory and returns

improved individuals

Outperformed the initial pC/C

Best solution never from GA but improved by the Tabus

Worthwhile to involve different solvers and create new

solutions

Several

TABUsMemory

GENETIC

pC/KC Indirect, Memory-based Cooperation

Solution set

(population)

+ Newly created

information

Solver set

Search

Control

Guidance

Memory

SOLUTIONS

Local context

CREATE new

solutions +

global context

EXTRACT

guidance

SOLUTIONS

Guidance info

Adaptive Memory

Rochat & Taillard, 1995

In memory

Components of elite solutions (e.g., routes)

Measures of how good the solutions were

Possibly for other attribute-based measures

Memory

Extracts components out of received solutions

Computes a value for each component

Historic smoothing

Context data, e.g., frequency in good solutions

Adaptive Memory (2)

New information creation

Within solvers

(could be performed by different solvers)

Each solver

Selects components & constructs an initial solution

Biased random selection (in-memory solution value,

context data)

Improves the solution (meta-heuristic)

Sends best solution(s) + solution value(s) to memory

Adaptive Memory (3)

Very successful, widespread, VRP particularly

Several interesting ideas

Set-covering formulation to select components for

solvers (VRPTW; Schulze & Fahle, 1999)

Two-level hierarchical method for real-time vehicle

routing and dispatching (Gendreau et al, 1999)

First level: pC/KC/MPSS

Second level: master-slave with domain decomposition

Central Memory

Crainic, Toulouse, Gendreau, 1996

In memory

Complete solutions, received and created

Local context information received

Global context information created

Guidance information: solutions, solution

components, target variable or attribute values,

e.g., patterns of arcs in routes

Measures of performance for solutions, solvers,

guidance data, …

Central Memory (2)

New information creation

In memory

could be outsourced to a (group of) solver(s)

Memory

Memory = Elite population Generate new

solutions

Integrate local context information into global one

(includes performance measures)

Extract global attributes from population + contextGenerate guidance data

Central Memory (3)

Solvers

Request (good) solution(s) at appropriate times

e.g., diversification or parent selection

Receive information, solutions and guidance

Integrate it according to internal logic

Improve solutions

Heuristics (constructive, improving, post-optimizing),

neighbourhood (tabu search, VNS, etc.) or population (GA,

Path Relinking) meta-heuristics, exact methods

Send new best solution(s) to memory

Send context information

Simple pC/KC for VRPTW (Le Bouthillier & Crainic, 2005)

An Advanced pC/KS for VRPTW

Le Bouthillier, Crainic, Kropf, 2005.

Measures based on atomic element = arc

Independent of routes

Frequency & evolution of arc appearance in

good/average/bad elite solutions (memory)

Pattern = vector of arcs to consider

Build patterns (various lengths) of arcs based on

frequencies and search evolution and send to solvers to

Intensify / diversify the search, e.g., strongly

prefer / avoid the arcs in pattern when selecting arcs

pC/KS Structure (VRPTW)TABUROUTE

Ordered elite solutions

Extraction rules

Data creation

Postoptimization

heuristics

UNIFIED

(subchain)

(subchain)Construction

heuristics

Request

solution

Guidance

Request

parents

Guidance

Send new

Advanced pC/KSs for VRP

Jin, Crainic, Løkketangen, 2012, 2014

Cluster solutions based on arcs in common + history

& solution-quality measures = how well the region

was explored & how good solutions are

Intensification and diversification goals

Heuristic solvers to improve existing solutions + set

covering solvers generate new solutions from elite

solutions in memory

Gröer, Golden, Wasil, 2011

Set covering on routes in memory to generate new

and better ones + Meta-heuristic to improve solutions

Cooperation & Multi-Attribute Optimization

In most cooperative meta-heuristics, including pC/KC,

solvers work on the complete problem (& search space)

Increasingly, large number of interacting attributes

(characteristics)

Larger than in “classical” (“academic”) settings

Problem characterization

Objectives

Uncertainty

That one desires to (must ☺) address simultaneously

pC/KC with partial solvers and integrators

Initial complete problem

Complete solvers

Complete solution set

Partial problems

Partial solvers

Partial solution sets

DECOMPOSITION

GUIDANCE

SHARING CONTEXT DATA

Integration solvers

INTEGRATION

Decision-set decomposition to yield “simpler problems”,

e.g., MDPVRP → MDVRP & PVRP

Decision-Set decomposition: opportunistic along sets of

decision variables

Partial solvers in pC/KC groups

Integrators

Construct complete solutions out of partial ones

Complete-solution population (+ solvers)

Global Search Coordinator

Monitoring

Guidance

Crainic et al. (2006, 2009), Lahrichi et al. (2015)

Integrative Cooperative Search – Decomposition

Decision-Set decomposition

Group variables to yield known or “easier” problem

settings

MDPVRP: along depot and period decisions →

MDVRP & PVRP

Locate antennas and select values for 7 attributes:

location versus “local” attributes → location & 3 groups

of attributes

Fix (not eliminate) variables not in partial problem

setting

Partial Solution set Pi

+ Newly created information

Partial solver set Si

Partial Solver Group PSGi

Partial Solution set Pj

Partial solver set Sj

Partial Solver Group PSGj

Partial Solver Group

One or several solvers per partial problem

pC/KC organization

Solvers, central memory = elite population + context

Local Search Coordinator manages the central memory

+ interacts with the global search

Integrator set I

Integrative Cooperative Search – Integration

Integrators

Build complete solutions by mixing partial

solutions(from PSGs) with promising solution features

Aim for

Solution quality & Computational efficiency

Preserving/pass on critical features

Several different may work simultaneously

Simple transmission

Meta-heuristics, e.g., path relinking, Unified Hybrid

Genetic Search

Optimization models, e.g., set covering, selection with

critical-feature preservation82

Integrator set I

Complete Solution set P

Complete solver set S

Complete Solver Group CSG

Integrative Cooperative Search – Cooperation

Global Solver Group & Global Search Coordinator

The (global) central memory

Elite solution, context, guiding information

(“instructions”)

Solvers (possibly)

Integrator set I

Complete Solution set P

Complete solver set S

Complete Solver Group CSG

Integrative Cooperative Search – Cooperation

Global Solver Group & Global Search Coordinator

Context information

Out of solutions in central memory,

e.g., frequency of presence of (customer, depot,

period pattern) in MDPVRP solutions

Combining contexts of PSGs

Monitor PSG evolution for, e.g., loss of diversity,

search/population stagnation, unexplored zones, …

Build and send guiding information for PSGs to orient

the search toward promising features

New solutions or components, modify values of fixed

variables, change decomposition or solvers86

Perspectives

Low-level parallelism (1C) →

Search space (domain) decomposition →

Independent Search →

Cooperative multi-search (& hybridization)

Synchronous cooperation →

Asynchronous cooperation →

Asynchronous cooperation with knowledge creation

Each fulfills a particular type of task

All may be needed at some time

A meta-heuristic class/paradigm on its own

Perspectives (2)

Still many issues, gaps in knowledge, challenges

“Best” local search acceleration still best when

integrated in meta-heuristic ?

Usage of Graphic Processing units, behaviour of local

search, integration

“New” computing platforms

Search-space (& low-level) decomposition for

hierarchical methods

Dynamic separation

Integrating with cooperation

Perspectives (3)

Context data integration/generation/utilization

Solver relative performance

Strategic variable/characteristic identification

Learning & guidance

Creating new meaningful information out of shared data

Cooperation of various meta-heuristic and exact solvers

Understanding & modelling cooperation

New application fields, e.g., stochastic programing

Parallel Computing and Vehicle Routing Teodor Gabriel Crainic · 2018-06-05 · Parallel Computing...

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