2. Hearn - Optimization

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Optimization and

Discrete Mathematics14 March 2011

Don Hearn

Program Manager

AFOSR/RSL

Air Force Research Laboratory

AFOSR

Distribution A: Approved for public release; distribution is unlimited. 88ABW-2011-0779





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Outline

• Portfolio Overview

• Optimization Research

• Program Goals and Strategies

•

AF Applications Identified; Relation to Subareas• Program Trend

• Transitions

• Scientific Opportunities and Transformation

Opportunities; Relation of Other Agency Funding• Examples from the portfolio

• Summary



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Optimization Research

• Starts with an important hard problem

• Optimization models are proposed

• Theoretical properties – existence anduniqueness of solutions, computationalcomplexity, approximations, duality, …

•

Solution methods – convergence theory, erroranalysis, bounds, …

• If successful, generalize!



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Maximize (or Minimize) f(x)

Subject tog(x) ≤ 0

h(x) = 0

where x is an n-dimensional vector and g, h are vector functions.Data can be deterministic or stochastic and time-dependent.

The simplest (in terms of functional form) is the Linear Program:

Minimize 3x + 5y + z + 6u + ……..

Subject to3 x + y + 2z + …….. <= 152

2x + 7y + 4z +u ….. >= 130

12x + 33y + 2u ….. = 217

etc

Optimization Models



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Textbook Nonlinear Example

• Local optima are a problem:



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Program Goals & Strategies

• Goals –

• Cutting edge optimization research• Of future use to the Air Force

• Strategies –

• Stress rigorous models and algorithms , especially for

environments with rapidly evolving and uncertain data• Engage more recognized optimization researchers,

including more young PIs• Foster collaborations –

• Optimizers, engineers and scientists•

Between PIs and AF labs• With other NX programs• With NSF, other agencies



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Future AF Applications Identified

• Engineering design, including peta-scale methods

• Risk Management, Experimental Design, T&E

• Real-time logistics

•Optimal learning/machine learning

• Meta-material design

• Satellite and target tracking

• Embedded optimization for control

• Biological and social network analysis

• …….. (to be continued)



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Scientific Challenges andPotential Impacts

Some Scientific Challenges• Non-differentiable Optimization • Nonlinear Optimization for Integer and Combinatorial

Models • Simulation Optimization • Optimization for analysis of Biological and Social

Networks • Logistics • Risk and Uncertainty in Engineering

Potential Transformational Impacts• Meta-material design • Engineering optimization • AI, Machine Learning, Reinforcement Learning • Air Force Logistics



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Program Trends

Analysis based optimization – Emphasis on theoretical resultsto exploit problem structure and establish convergenceproperties supporting the development of algorithms

Search based optimization – New methods to address very largecontinuous or discrete problems, but with an emphasis on the

mathematical underpinnings, provable bounds, etc

Dynamic, stochastic and simulation optimization– Newalgorithms that address data dynamics and uncertainty andinclude optimization of simulation parameters

Combinatorial optimization – New algorithms that addressfundamental problems in networks and graphs such asidentifying substructures



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Recent Transition: Model Adaptive Searchfor detoxification process design

Dennis (Rice), Audet (École Polytechnique de Montréal), Abramson (Boeing)

•

The MADS algorithm,based on (Clarke)generalized derivatives,reduced the solutiontimes by 37% on this

problem in whichexpensive simulationswere need to evaluatethe objective function.

• There are a number ofother transitions for airfoil design, surgerydesign, etc.



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New Meta Algorithms for EngineeringDesign Using Surrogate Functions

Dennis (Rice), Audet (École Polytechnique de Montréal), Abramson (Boeing)

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8

0.25 Overlap to Diameter r

atio

21 ’ Cabin

Length

Gross Weight (lbs)

0.10 0.15 0.20 0.25 0.30

Rotor Overlap to Diameter Ratio

36.5

39.8

43.1

46.4 49.6 56.1 52.9 59.4

92000

90750

89500

88250

87710

Cabin Length (ft)

InfeasibleRegion


atio

21 ’ Cabin

Length


atio

21 ’ Cabin

Length

Gross Weight (lbs)

0.10 0.15 0.20 0.25 0.30

Rotor Overlap to Diameter Ratio

Rotor Disk Loading,

psf

36.5

39.8

43.1

46.4 49.6 56.1 52.9 59.4

92000

90750

89500

88250

87710

Cabin Length (ft)

InfeasibleRegion

Helicopter rotor tilt design at BoeingOriginal airfoil

Optimized airfoil, 89% noise reduction

Optimization of angle and graft size forbypass surgery

Current design Proposed y-graft design

Snow water equivalent estimation

Highly fragmenteddomainFeasible domainrepresented by thedark pixels.





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Other OrganizationsThat Fund Related Work

• NSF Several projects jointly funded in the 4

subareas• ONR

•

2 projects (3 PIs) jointly funded –applications are in military logisticsand emergency planning

• DOE

• No joint grants. One PI has additionalfunding in energy management

• ARO • No joint project funding. One PI is on

Army MURI



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International Collaborations

• Ecole Polytechnique de Montreal • Claude Audet (with US researchers at

Rice, Boeing, AFIT)

•

University of Waterloo • S. Vavasis & H. Wolkowicz

• SOARD

• Claudia Sagastizabal, RJ Brazil (with R.Mifflin, Wash. St.)

(plus many unsponsored collaborations)



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Examples from the Portfolio

Di t S h



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Discrete SearchTheory and Methods

– Jacobson, Howe and Whitley

• Study of landscapes in combinatorial problems

• Applications in radar scheduling, routing, general NP Hard problems such as TSP, Air Fleet assignment,homeland security, …

Ongoing research: Search Algorithms



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Approach/Technical Challenges• Disconnect between theory and practice in search.• Current theory needs to be extended to account for

complex structures of real world applications.• Uncertainty and dynamism are not well captured by

current theory.

Accomplishments

Algorithms and complexity proof for a scheduling Proofs about plateaus in Elementary Landscapes Predictive models of plateau size for some classes of

landscapes based on percolation theory Recent results on next slide…

Ongoing research: Search Algorithmsfor Scheduling Under Uncertainty

Colorado State University, Howe & Whitley

orbital path

track

signal

L d A l i d Al ith D l t f



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Landscape Analysis and Algorithm Development forPlateau Plagued Search Spaces

Colorado State University, Howe & Whitley

Traveling Salesman Problem(e.g., supply chain, logistics,circuit designs, vehicle routing)

• Generalized PartitionCrossover (GPX): technique forcombining two locally optimalsolutions to construct better

locally optimal solution in O(n) time

• Results: Analytical andempirical analysis shows GPXcan dominate state of the artmethods

Implications: Can dramatically change how TSP instances are

solved

O ti l D i A t All ti P bl



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Optimal Dynamic Asset Allocation Problems:Addressing the Impact of Information Uncertainty & Quality

University of Illinois, Sheldon Jacobson

Payoff to Stakeholders:• Theory: New fundamental understanding of how data deficienciesand uncertainties impact sequential stochastic assignment problemsolutions.•Long-term payoff: Ability to solve problems that more closely mimic real-world systems driven by real-time data.

Goal: Create a general theoretical framework for addressing optimal dynamic asset

allocation problems in the presence of information uncertainty and limited information quality.

Technical Approaches:

Applications

Assignment Problems Screening and Protection Scheduling Problems

• Sequential Stochastic Assignment• Surrogate Model Formulation and Analysis• Dynamic Programming Algorithm Design

A l i B d O ti i ti



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Analysis-Based OptimizationTheory and Methods

– Mifflin and Sagastizabal

• New theory leading to rapid convergence in non- differentiable optimization

– Freund and Peraire

• SDP for meta-material design

–

Srikant• Wireless Network Operation

I t t B kth h S li



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Important Breakthrough: Superlinearconvergence for nonsmooth problems

Bob Mifflin (Washington State) andCladia Sagastizabal (CEPEL, Rio de Janeiro)

jointly supported with NSF and SOARD

Band gap optimization for wave



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Band-gap optimization for wavepropagation in periodic media

MIT, Freund, Parrilo & Peraire

Design of materials that give

very precise control of light orsound waves

New in this work is the use ofSemi-Definite Optimization forband structure calculation and

optimization in phononiccrystals

Applications includedesign of elasticwaveguides, beamsplitters, acoustic lasers,perfect acoustic mirrors,vibration protectiondevices



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Optimization of photonic band gapsMIT, Freund & Peraire

Goal: Design of photonic crystals withmultiple and combined band gaps

Approach: Solution Strategies

• Pixel-based topology optimization

• Large-scale non-convex eigenvalueoptimization problem

• Relaxation and linearization

Methods: Numerical optimization tools

• Semi-definite programming

• Subspace approximation method

•

Discrete system via Finite Elementdiscretization and Adaptive meshrefinement

Results: Square and triangular lattices

• Very large absolute band gaps

• Multiple complete band gaps

• Much larger than existing results.



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Distributed SchedulingUIUC, R. Srikant

• Wireless Networks:

Links may not be able totransmit simultaneouslydue to interference.

• Collision: Twointerfering users cannot

transmit at the sametime

• Medium Access Control(MAC) Protocol:Determines which links

should be scheduled totransmit at each timeinstant.

• (Relates to RSL Complex Networks portfolio)

Main Result: The first, fully distributedalgorithm which maximizes networkthroughput and explicitly takes

interference into account.Theoretical Contribution: An optimalityproof that blends results from convexoptimization, stochastic networks andmixing times of Markov chains.



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Dynamic and Stochastic Methods

• Stochastic Optimization –

– Marcus and Fu

• Simulation-based optimization

• Dynamic Optimization

– Powell, Sen, Magnanti and Levi

• Fleet scheduling, Refueling, ADP, Learning methods, Logistics

Ongoing (high-risk) research: Simulation-Based



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Ongoing (high-risk) research: Simulation-BasedMethodologies for Estimation and Control

UMD (with NSF), S. Marcus & M. Fu

Convergence of Distribution toOptimum in Simulation-Based

Optimization

ACCOMPLISHMENTS

• Simulation-based optimizationalgorithms with probable convergence &efficient performance

• Evolutionary and sampling-basedmethods for Markov Decision processes

The Knowledge Gradient for Optimal Learning



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Technical challenge

• Optimal information collection policies arecomputationally intractable.

• Efficient algorithm for capturing correlated beliefs,making it possible to solve problems withthousands of choices on small budgets.

Impact

• Used to find new drugs, improve manufacturingprocesses, and optimize policies for stochastic

simulators.

Challenge

•

Finding the best material, technology, operatingpolicy or design when field or laboratoryexperiments are expensive.

Breakthrough

• The knowledge gradient for correlated beliefs(KGCB) is a powerful method for guiding theefficient collection of information. Proving to be

robust across broad problem classes.

The Knowledge Gradient for Optimal LearningIn Expensive Information Collection Problems

Princeton University, Warren B. Powell

A knowledge gradient surface

An Optimization Framework for



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An Optimization Framework forAir Force Logistics of the Future

MIT, Magnanti & Levi

•

Air Force Logistics –– 20 new initiatives to

revolutionize supply chainsand logistics

• DOD logistics + inventory costs

–$130 + $80 billion

• Magnanti and Levi (MIT) - fast real-time algorithms for

– Depot to base supply and jointreplenishment

• Future war space example -sensor asset allocation

• Workshop in 2010 onidentification of specific AFmodels for the future



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Combinatorial Optimization

Butenko (Texas A&M), Vavasis (U. Waterloo),Prokopyev (U. Pittsburgh), Krokhmal (U. Iowa)

Underlying structures are often graphs; analysis is

increasingly continuous non-linear optimization

C h i b i h / t k



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Cohesive subgroups in graphs/networksTexas A&M, S. Butenko

• Data set generated by a certain complex system –

Represent using graphs or networks:

Nodes = agents, Links = interactions

• Interested in detecting cohesive (tightly knit) groupsof nodes representing clusters of the system'scomponents

• (The cohesive subgroup notion was first introduced insocial network analysis; this relates to the RSLCollective Behavior and Socio-cultural Modelingportfolio.)

• Earliest mathematical models of cohesive subgroupswere based on the graph-theoretic concept of a clique

• Other subgroups include k-clubs

A li ti



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Applications

• Acquaintance networks - criminal network analysis

• Wire transfer database networks - detecting moneylaundering

• Call networks - organized crime detection

• Protein interaction networks - predicting proteincomplexes

• Gene co-expression networks - detecting networkmotifs

• Internet graphs - information search and retrieval

A l f l t k



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An example of a complex network

Protein-protein interaction map of H. Pylori

Maximum 2- and 3- Club



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Maximum 2- and 3- ClubSubstructures

H. Pylori

Saccharomyces cerevisiae - Yeast

Finding hidden structures with convex optimization



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Finding hidden structures with convex optimizationU. Waterloo, S. Vavasis & H. Wolkowicz

The following apparently random graph contains a subset of 5

nodes all mutually interconnected:

Recent data mining discoveries fromWaterloo:•Finding hidden cliques and bi-cliques inlarge graphs (see example)•Finding features in images from an imagedatabase•Clustering data in the presence ofsignificant background noise using convexoptimization

5-clique found via convex

optimization

Polyomino Tiling



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Polyomino TilingU. Pittsburgh, O. Prokopyev

• Application is time delay control of phased arraysystems

• Jointly supported with Dr. Nachman of

AFOSR/RSE, motivated by work at the SensorsDirectorate:

• Irregular Polyomino-Shaped Subarrays for Space-Based Active Arrays (R.J. Mailloux, S.G. Santarelli, T.M. Roberts,

D. Luu)

Moti ation and Initial Approach



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Motivation and Initial Approach

•Introducing time delay into phased array systemscauses significant quantization side lobes whichseverely degrade obtained pattern.

• Mailloux et al. [2006] have shown that an array of

`irregularly` packed polyomino-shaped subarrays haspeak side lobes suppressed more than 14db whencompared to an array with rectangular subarrays.

• We aim to model and solve the problem of designing

and arranging polyomino shaped subarrays usingoptimization techniques. First attempt maximizesinformation entropy.

16x16 Board Tiled with Tetrominoes



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Maximize entropy. Minimize entropy.(Observe how centers of gravity

are aligned: order = less entropy)

16x16 Board Tiled with Tetrominoes

Summary



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Summary

• Goals –

• Cutting edge optimization research• Of future use to the Air Force

• The primary strategy is to identify future AF applicationsthat will require important improvements in optimization

theory and new algorithms.

Questions?

Documents

2. Hearn - Optimization