using hybrid optimization algorithms for very-large graph...

Using hybrid optimization algorithms for very-large graph problems

and for small real-time problems

Karla HoffmanGeorge Mason University

Joint work with:

Brian Smith, Tony Coudert, Rudy Sultana and James Costa (NCI, Inc)Paul Nicholas (Applied Physics Lab, JHU)

Ryan O;Neil (Grubhub, Inc.)

4/1/2018 INFORMS Optimization Society Meeting 1

The opinions expressed in this talk are those of the authors and do not necessarily represent

the views of the FCC or any of its staff.

INFORMS Optimization Society Meeting

What to do when state-of-the-art software does not solve the problem?

Three Important and Difficult Problems:

• The Federal Communications Commission Incentive Auction• Why the problem is important• Approach to solving the problems• Impact to government policy making

• Assigning Channel Assignments to Radios on the Battlefield• Similar to the FCC problem, but now there is a dynamic

aspect to the problem

• Getting near-optimal solutions to routing problems for the App Economy• The future of optimization

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The FCC: UNLOCKING THE BEACHFRONT,Using Operations Research To Repurpose Spectrum

• The Federal Communications Commission (FCC)

recently completed the world’s first two-sided spectrum

auction

• The auction was one of the most successful in FCC

history

• Repurposed 84 MHZ of spectrum

• Generated revenue of nearly $20 billion

• Optimization was essential to the design and

implementation of the auction

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INFORMS Optimization Society Meeting

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2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021

Introduction

U.S. Mobile Data Traffic Growth Forecast

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iPhone

June 2007

iPad

April 2010

*Cisco VNI Forecasts; FCC Analysis

2017

INFORMS Optimization Society Meeting

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Introduction

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Introduction Modeling Interference Optimizations Solution Approaches Lessons Learned

What is the broadcast Incentive Auction?

470MHz

Ch. 14

698MHz

Ch. 51

470MHz

Ch. 14

698MHz

Ch. 51

• Repack the television stations onto fewer channels• Auction newly freed spectrum to wireless bidders

Repacking the stations is the equivalent of a graph coloring problem (with side constraints and objectives!)

Introduction

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Introduction Modeling Interference Optimizations Solution Approaches Lessons Learned

Other Graph Coloring Problems

• Map coloring

• Creating a schedule or time table• Crew Scheduling• Round Robin Sport Scheduling• Security Camera Scheduling• Scheduling Software Updates

• Air Traffic Flow Management

Pre-Auction vs. Post-Auction 600 MHz Band

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

• Simple market solutions insufficient– International coordination of frequency uses.– Different license rights for broadcast versus

broadband. – Need to move non-sellers and provide guard

bands.– Computational challenges in reassigning

stations.

• “Middle Class Tax Relief and Job Creation Act of 2012” gave FCC the right to retune holdouts, turning stations on different frequencies into substitutes. Thus, Broadcasters have the right to continue broadcasting on some channel with no increase in interference, but not to remain on the current channel!

The Reallocation Problem

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Broadband Incentive Auction: Key Components

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Reverse

Auction

Forward

Auction

Broadcasters• Offer to relinquish

spectrum usage rights

Mobile Broadband

Providers• Offer to purchase

spectrum licenses

Integration

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

The Auction Framework

• Stations voluntary participate, accept bids (payments) to vacate their channel

• Cease operation and go off the air

• Move to a lower TV band

• If more stations bid than are needed to clear the spectrum, the bid value decreases until a station no longer accepts a bid and gets placed back on air

• Auction closes once no more stations can be individually packed into the new TV bands

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

The Reallocation Problem

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2197 US Stations

1709 UHF Stations

793 Canadian Stations

348 UHF Stations

U.S. AND CANADIAN STATIONS

Many Optimizations within the Auction Framework…The two highlighted required optimizations solutions really quickly or provable really good; All used HYBRID approaches.

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Reverse Auction Bidder

Auction System

Forward Auction Bidder

Set Clearing Target for

Stage

Final TV ChannelAssignment

Final StageRule not met Final Stage Rule

met

InitialCommitment

Reverse Clock

Auction

Forward Clock

Auction

Forward Assignment

Rounds

Set Transition Schedule

(1)

(2)

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Optimization Descriptions

4/1/2018

• Determines the maximum amount of spectrum to be auctioned.• Allows stations to be assigned in wireless band to prevent the most congested

market from determining the amount of spectrum available

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37

Low-VHF High-VHF UHFUHFG H JI JIHGFEDCBAFEDCBA

Clearing Target Optimization

Participation determines where we put this line!

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Final Channel Assignment

4/1/2018

• After the auction is complete, determines what channels the stations will use.

• Assign stations to their current channel when possible and minimize new interference

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

The Major Modeling Issue:How to Model Interference Protections

• New station assignments cannot interfere more than 0.5% of another station’s current population

• Potential pairwise interference must be studied for stations assigned to the same channel or to adjacent channels

• 2,694,283 pairwise interference restrictions

• Near (but not exact) symmetric restrictions across different channels because of the adjacent channel requirements

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Protecting Stations From Interference

This created over 2.5 MILLION pairwise interference protections to ensure that stations still reach their viewers

5/15/2018

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Clearing Target Optimizations:

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• Determined where the auction started• Potentially assigned stations to wireless bands• Poor performance here could result in BILLIONS $ lost

• Less revenue in forward auction due to less spectrum• Lower prices for stations due to less competition• Less benefit to economy due to inefficient spectrum use

Critically Important Model

Goals of Clearing Target Determination

• Goal was to repurpose as much spectrum as possible on a nationwide basis, this presented the FCC with a choice:

• Either:a. limit the amount of spectrum repurposed in every market to the amount

made available in the most constrained market (Lowest Common Denominator) or

b. allow some TV stations to be assigned channels in the new wireless services band if broadcaster clearing in a market was not sufficient.

The FCC chose to allow TV stations to be assigned in the new wireless services band.

Optimization determined:The Stations to be put in the Wireless Band and the Channel Assignments for those Station.

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# of

Wireless

l icenses

offerred

Highest

Channel10 29

9 31

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7 36

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36 37 38

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A B

27 28 29

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29

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27 28 29

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30 31

30

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27 28 29

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30 31 32 33 34 35

33 34 3530

A

41 42 43 44

37

A

H

41 42 43

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39 40

H I

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A B C D

B C D E F

E F G H I

A B CJG

H I

700 MHz

700 MHz

700 MHz

700 MHzF G

B C D E F G

F G H I JD E

700 MHz

700 MHz

700 MHz

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700 MHz

C D E F G

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F G H

A B C D E F G

A B C D E

D EA B C D E F

A B C D E

D

A B C A B C

D E

A B C D A B C

F

A B C

A B A B

23 24 25 26 39 40 41 42

30 31 32 33 34 35 36

27 28 29 37

3728 29

Clearing Target

(in MHz)126

114

108

84

43 44 45 46 47 48

D E A B C

A B C

Before

Auction

78

72

60

48

42

49 50 51 700 MHz33 34 35 36 37 3827 28 29 30 31 3221 22

Clearing Target Optimization: Band PlansThe FCC proposed a collection of band plans, that would protect TV stations from interference with wireless through Guard Bands (the gray areas on the chart below).

The Channel Assignment chose the highest clearing target as long as the impairments to wireless licenses was not above a specified threshold.

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Clearing Target Optimizations

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What makes it difficult

TV 2 TV Interference• Over 2 million pairwise protections

Inter Service Interference• Protecting interference between

different services (Wireless vs TV)• Over 6 million additional protections!!• Protects on an aggregate, county level

Introduction The Reallocation Problem OPTIMIZATION Solution Approaches Lessons Learned

How the Clearing Target Worked:

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• All optimizations included:• The TV Repacking constraints • Inter-service interference constraints

• Canadian constraints were added to allow a larger guard band between TV and Wireless

• Other constraints assured that US TV stations who participated in the auction would have as much flexibility in bidding as possible

• Primary Objective for Clearing Target Optimization: Determine the Minimum sum of impaired weighted pops across all licenses in both the US and Canada

• Secondary Objective: Determine the maximum number of unimpaired licenses

Formulation Matters

Analyze the constraints to identify more efficient formulations

• The millions of pairwise constraints could be consolidated into better clique constraints

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𝑠1, 𝑐1

𝑠3, 𝑐3

𝑠2, 𝑐2

In addition, we ran different tests to determine the best set of cliques;2.5 million pairwise constraints reduced to 220,000 clique constraintsFor the inter-service interference constraints, >6million to about 680,000

𝑥1 + 𝑥2 ≤ 1𝑥2 + 𝑥3 ≤ 1𝑥1 + 𝑥3 ≤ 1

or 𝑥1 + 𝑥2 + 𝑥3 ≤ 1

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Exploit Problem Structure

Identify an alternative formulation or decomposition

INFORMS Optimization Society Meeting

470MHz

Ch. 14

698MHz

Ch. 51

For the clearing target problem, we were identifying stations to be assigned to the wireless band.

We decomposed the problem into a feasible packing problem in the TV Band and then a best assignment problem in the wireless band.

Assignments to the wireless band would only be in congested areas so we could also separate geographic regions.

We approached the problem as a Logic-based Benders Decomposition where the “cuts” generated told the Master Problem (wireless assignments) about the TV band solution possibilities.

TV BAND

WIRELESS BAND

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Build your own heuristics

Gurobi’s heuristics are great!!!! But they approach the problem from the formulation provided

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• Use your knowledge of what you are modeling to identify other potential heuristics

• We did local optimizations by geography to improve regions of the country

• Similarly, locally optimize over specific stations or channels

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Take advantage of the features of the solver

Our goal was near optimal solutions in a reasonable amount of time.

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• We used Gurobi 6.5 and created our own distributed solver.• Some solvers used Gurobi with different parameter

configurations• Some solvers used Gurobi with our custom heuristics

• Using callbacks, we communicated solutions between solvers.

• Where appropriate, lazy constraints can make the problem much smaller and easier to solve

Another Difficult Optimization Problem: FINAL CHANNEL ASSIGNMENT

• Once the auction ended, the Final Channel Assignment determined the channel assignments for all stations remaining on the air.

• FCC chose to optimize the following goals

• Assign stations to their current channel when possible

• Minimize new aggregate interference

• Avoid moving stations whose move would be exceptionally difficult and/or expensive

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HOW WE OBTAINED GOOD SOLUTIONS IN REASONABLE TIMES:• Clique constraints

• 6 Million constraints 500,000 constraints

• Custom heuristic: used a distance-based metric via Delaunay triangulation. (Used lat./long information)

• Benders decomposition: • Master problem: choose stations to remain on current channel)• Sub-problems: (Find where assignment not possible and add cuts to

Master)

• Again Used a Distributed Computing Environment

Final Channel Optimization Problems

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

For all optimizations: Used distributed solver technology

Our goal was near optimal solutions in a reasonable amount of time

We used Gurobi 6.5 and created our own distributed solver• Some solvers used Gurobi with different parameter

configurations• Some solvers used Gurobi with our custom heuristics

Using callbacks, we communicated solutions between solvers

.

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Results of Hybrid Optimization Approach

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• We solved each Clearing Target to proven optimality!• Allowed the auction to start at the highest clearing target (126 MHz)• Starting at a high clearing target encouraged stations to participate• Ability to get good solutions helped policy analysis

• In Final Channel Assignment, far less stations needed to move than industry expected• Nearly 1700 stations remained on their original channel• Maximum aggregate interference for any station was 1.1% • Only 12 stations received more than 1% aggregate interference

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Results of FCC Auction

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Auction Results: • Final band plan had 84 MHz of completely clear

spectrum• Wireless providers paid almost $20 Billion for the

spectrum• Broadcasters received over $12 Billion• Over $7 Billion to the US Treasury

Optimization contributions:• We solved each Clearing Target to proven

optimality!• In Final Channel Assignment, far less stations

needed to move than industry expected

And…• Optimization team helped design the Transition

Schedule for the remaining stations!

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Lessons Learned From FCC Application

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• Understand your problem– What are you modeling?– What results are your stakeholders looking for?

• Be creative in solving your problem• In policy modeling, confidence building is essential and takes

time and effort…need to give insights rather than answers • When the problem seems unsolvable:

– Often a smaller version of the problem is solvable so exploit this fact for heuristic search and for improving other bound (using multiple optimizations to solve a single optimization)

– Exploit the inherent structure.

A related Problem: How to assign channels to radios on the battlefield

We consider three challenges:

•Minimizing the number of required channels • Minimum-order channel assignment problem (MO-CAP)

•Minimizing interference, given a fixed number of channels• Minimum-interference channel assignment problem (MI-CAP)

•Minimizing the number of channel changes over time• Minimum-cost channel assignment problem over time (MC-CAP-T)

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Goal: To solve realistic, full-sized instances of

various channel assignment problems (CAPs) in a

reasonable amount of time

Model of MANET Operations

• A military unit (e.g., a company or battalion headquarters) uses a MANET to communicate on an assigned channel. (Units indicated by blue and green.)

• Each unit’s MANET may comprise up to 30 radios (indicated by circles)

• Only radios within a particular unit may communicate (indicated by solid arrows); there is no MANET backhaul networkbetween units (in reality, this is provided via other transmission systems)

• All other radios assigned to a different unit but operating on the same channel provide co-channel interference (gray dashed arrows)

• Interference is cumulative at a receiver (e.g., radio r)

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r

INFORMS Optimization Society Meeting

Solution Approach

• Develop an integer programming model of MANET operations that captures the most important aspects of communications• Mobile units on rough terrain

• Cumulative co-channel interference

• Generate realistic input data based on high-fidelity combat scenarios• Derived from Defense Planning Scenarios

• Scenarios instantiated and radios simulated using AGI Systems Toolkit (STK)1

• Radio propagation simulated using the Terrain Integrated Rough Earth Model (TIREM)2

• Develop several formulations of each CAP

• Use heuristics, integer optimization, and constraint programming to solve each of the three problem

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How to overcome computational difficulties?

• Substitute pair-wise interference constraints for aggregate interference constraints.

• For every integer solution found in the process, check if aggregate interference violated… • Whenever a “feasible” integer solution is found, callback checks the

aggregate feasibility (in polynomial time)

• Higher-order packing constraints dynamically added as needed to reduce / eliminate infeasibility:

• Strengthen pairwise formulation through clique constraint

• With these “tricks”, CPLEX gets good solutions BUT… lower bound still bad

• SOLUTION: Use constraint programming to get better lower bound

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1c

u

u S

X S

What if there is not enough spectrum?

36

• Again, we formulate as an IP, considering only pairwise interference to capture a substantial portion of interference

• We attempt to minimize the number of pairwise constraint violations

• We implement using Python and CPLEX

RESULTS:

• Tested optimization versus constraint programming for same formulation

• We run each method with varying levels of channel availability

• We calculate network availability to estimate the operational impact upon the ability to use each MANET• Percentage of radios able to communicate with network control radio

• Overall, Constraint Programming provides superior performance in considerably less time, and provides good bounds

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Results

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ClusteringTotal excess. interf: 22.88 dBm

# radios inoperable: 174

IPTotal excess. interf: 25.52 dBm

# radios inoperable: 224

Max allowable interference

CPTotal excess. interf: 17.53 dBm

# radios inoperable: 96

FINAL APPLICATION: Using Optimization in the App EconomyThe Dynamic Pickup & Delivery Problem for Restaurant Delivery Services

Given:Orders that need to be picked up from restaurants and deliveredTarget pickup times for the ordersTarget service levels for getting food to dinersGeographically distributed couriers who may already have

assignments

Determine:Which driver gets each orderWhich route each driver should takeWhen to dispatch each order, given the current plan

38INFORMS Optimization Society Meeting4/1/2018

Dynamic Pickup & Delivery for Restaurants

Orders are received dynamically and must be responded to quickly.

There are hard constraints on plan time.

Decomposing assignment and routing and iteratively improving the plan ensures planning can stop when it hits the time budget.

Caveat: The route solving has to be really fast, and has its own time budget.

39

AssignmentSolver

Assign-ments

Region State

Route Solver Routes

INFORMS Optimization Society Meeting4/1/2018

Pickup & Delivery: Single Courier Problems

40

People, Meals, Perishable Goods Groceries, Packages, Non-Perishable Goods n

Courier

Pickup

Delivery

Pickup & Delivery: Single Courier Solutions

41

People, Meals, Perishable Goods Groceries, Packages, Non-Perishable Goods n

Courier

Pickup

Delivery

Solution Method Depends on Problem Size & Urgency

Methods Considered

• Enumeration: Depth-First Search + Greedy Node Ordering + Fathoming

• Hybrid CP: Circuit Constraint + Precedence + Sequential GreedyBranching + AP Reduced Cost Domain Filtering

• MIP: 2-Matching Relaxation + Subtour Elimination + Precedence

• MIP+Hybrid CP: MIP Warm Started with Hybrid CP

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How quickly can I act on new information?

43

How quickly can I make a really good decision?

44

How quickly can I be sure I have the best solution?

45

Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Solution Methods

When solving hard problems:

1. Formulation matters2. Exploit problem structure3. Build your own heuristics4. Take advantage of the features of the solver5. Consider multiple optimization approaches

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned

Questions

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Introduction THE REALLOCATION PROBLEM Optimization Solution Approaches Lessons Learned