Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Exploiting Spatial Structure in Mixed-IntegerPDE-Constrained Optimization

Joey Hart1

with Pierre Gremaud1, Tim Kelley1, Sven Leyffer2,Bart van Bloemen Waanders3

1North Carolina State University

2Argonne National Lab

3Sandia National Lab

May 2, 2017

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

1 Introduction

2 Coupling PEBBL and ROL

3 Heuristic to Spatial Problem Structure

4 Conclusion

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

1 Introduction

2 Coupling PEBBL and ROL

3 Heuristic to Spatial Problem Structure

4 Conclusion

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Mixed-Integer PDE-Constrained Optimization (MIPDECO)

minw ,uJ (w , u)

w ∈ DN , u ∈ H

L(w)u = f (w)

c(w) ≤ 0

• D ⊆ Z• H is a function space

• L is a differential operator

• f is a source term

• c is constrains on w

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

General Approaches for Solving MIPDECO Problems

In principle, tools from

- integer programming

- PDE-constrained optimization

may be combined to solve these large and challenging problems.

Integer programs require solutions of many relaxed problems

=⇒ many PDE-constrained optimization solves

=⇒ a large number of PDE solves

• this talk will focus on two approaches- coupling PEBBL (integer programming) with ROL

(PDE-constrained optimization)- developing a quick heuristic to warm start these more

computationally intensive algorithms

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

General Approaches for Solving MIPDECO Problems

In principle, tools from

- integer programming

- PDE-constrained optimization

may be combined to solve these large and challenging problems.

Integer programs require solutions of many relaxed problems

=⇒ many PDE-constrained optimization solves

=⇒ a large number of PDE solves

• this talk will focus on two approaches- coupling PEBBL (integer programming) with ROL

(PDE-constrained optimization)- developing a quick heuristic to warm start these more

computationally intensive algorithms

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

General Approaches for Solving MIPDECO Problems

In principle, tools from

- integer programming

- PDE-constrained optimization

may be combined to solve these large and challenging problems.

Integer programs require solutions of many relaxed problems

=⇒ many PDE-constrained optimization solves

=⇒ a large number of PDE solves

• this talk will focus on two approaches- coupling PEBBL (integer programming) with ROL

(PDE-constrained optimization)- developing a quick heuristic to warm start these more

computationally intensive algorithms

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

1 Introduction

2 Coupling PEBBL and ROL

3 Heuristic to Spatial Problem Structure

4 Conclusion

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Parallel Enumeration and Branch-and-Bound Library 1

• Branch-and-Bound seeks to efficiently search a binary tree tofind the best integer solution

• in the MIPDECO context, it requires the solution of acontinuous PDE-constrained optimization problem at eachtree node

• for high dimensional problems, the number of nodes in thetree may be very large

• PEBBL provides scalable performance to parallelize the searchover large trees

1PEBBL: an object-oriented framework for scalable parallel branch andbound. Jonathan Eckstein, William E. Hart, Cynthia A. Phillips. 2015.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Parallel Enumeration and Branch-and-Bound Library 1

• Branch-and-Bound seeks to efficiently search a binary tree tofind the best integer solution

• in the MIPDECO context, it requires the solution of acontinuous PDE-constrained optimization problem at eachtree node

• for high dimensional problems, the number of nodes in thetree may be very large

• PEBBL provides scalable performance to parallelize the searchover large trees

1PEBBL: an object-oriented framework for scalable parallel branch andbound. Jonathan Eckstein, William E. Hart, Cynthia A. Phillips. 2015.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Parallel Enumeration and Branch-and-Bound Library 1

• Branch-and-Bound seeks to efficiently search a binary tree tofind the best integer solution

• in the MIPDECO context, it requires the solution of acontinuous PDE-constrained optimization problem at eachtree node

• for high dimensional problems, the number of nodes in thetree may be very large

• PEBBL provides scalable performance to parallelize the searchover large trees

1PEBBL: an object-oriented framework for scalable parallel branch andbound. Jonathan Eckstein, William E. Hart, Cynthia A. Phillips. 2015.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Parallel Enumeration and Branch-and-Bound Library 1

• Branch-and-Bound seeks to efficiently search a binary tree tofind the best integer solution

• in the MIPDECO context, it requires the solution of acontinuous PDE-constrained optimization problem at eachtree node

• for high dimensional problems, the number of nodes in thetree may be very large

• PEBBL provides scalable performance to parallelize the searchover large trees

1PEBBL: an object-oriented framework for scalable parallel branch andbound. Jonathan Eckstein, William E. Hart, Cynthia A. Phillips. 2015.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Rapid Optimization Library 2

• ROL provides state of the art algorithms for PDE-constrainedoptimization

• it must be called repeatedly inside PEBBL to solve aPDE-constrained optimization problem at each node

2ROL: A C++ package for large scale optimization. Drew Kouri, DenisRidzal, Greg von Winckel, Bart van Bloemen Waanders, Wilkins Aquino, TimWalsh .

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Rapid Optimization Library 2

• ROL provides state of the art algorithms for PDE-constrainedoptimization

• it must be called repeatedly inside PEBBL to solve aPDE-constrained optimization problem at each node

2ROL: A C++ package for large scale optimization. Drew Kouri, DenisRidzal, Greg von Winckel, Bart van Bloemen Waanders, Wilkins Aquino, TimWalsh .

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Coupling PEBBL and ROL

• coupling PEBBL and ROL provides a highly scalable parallelframework for MIPDECO

• applications of interest typically involve O(1, 000) orO(10, 000) integer variables

• this may require O(100, 000) or O(1, 000, 000) PDE solves

• developing efficient heuristics to warm start PEBBL (or otherinteger programs) may reduce this

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Coupling PEBBL and ROL

• coupling PEBBL and ROL provides a highly scalable parallelframework for MIPDECO

• applications of interest typically involve O(1, 000) orO(10, 000) integer variables

• this may require O(100, 000) or O(1, 000, 000) PDE solves

• developing efficient heuristics to warm start PEBBL (or otherinteger programs) may reduce this

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Coupling PEBBL and ROL

• coupling PEBBL and ROL provides a highly scalable parallelframework for MIPDECO

• applications of interest typically involve O(1, 000) orO(10, 000) integer variables

• this may require O(100, 000) or O(1, 000, 000) PDE solves

• developing efficient heuristics to warm start PEBBL (or otherinteger programs) may reduce this

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Coupling PEBBL and ROL

• coupling PEBBL and ROL provides a highly scalable parallelframework for MIPDECO

• applications of interest typically involve O(1, 000) orO(10, 000) integer variables

• this may require O(100, 000) or O(1, 000, 000) PDE solves

• developing efficient heuristics to warm start PEBBL (or otherinteger programs) may reduce this

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

1 Introduction

2 Coupling PEBBL and ROL

3 Heuristic to Spatial Problem Structure

4 Conclusion

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Particular Instance with Spatial Structure

minw ,uJ (w , u)

w ∈ DN , u ∈ H

L(w)u = f (w)

c(w) ≤ 0

• D = 1, 2• w ∈ DN encodes a spatial discretization of two materials

• c(w) ≤ 0 encodes a linear volume constraint

• find the optimal arrangement of materials subject to c(w) ≤ 0

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Particular Instance with Spatial Structure

minw ,uJ (w , u)

w ∈ 1, 2N , u ∈ H

L(w)u = f (w)

1

N

N∑i=1

(wi − 1) ≤ β

• D = 1, 2• w ∈ DN encodes a spatial discretization of two materials

• c(w) ≤ 0 encodes a linear volume constraint

• find the optimal arrangement of materials subject to c(w) ≤ 0

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

The Basic Idea: Recursive Partitioning Heuristic

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

1 consider every split along the directions of the coordinate axes

2 for each possible split, consider the four possible materialassignments

3 choose the split and material assignment which yields smallestobjective function value

4 repeat this process on the subdomains

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Example of Splits

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Four Possible Material Assignments for a Given Split

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

• one of these corresponds to the previous state, we do not haveto compute anything for it

• test the other three possible assignments by solving threeforward problems

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continue Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

• once we found an optimal split and material assignment,repeat this process on each subdomain

• termination criteria to come in later

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Continuing Recursively

0.25 0.5 0.75

0.25

0.5

0.75

0 1

01

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

A Few Preliminary Comments

Pros:

• PDE-constrained optimization is not needed

• incorporates spatial structure

• encourages “topologically simple” solutions

• can adaptively act on large spatial blocks

Cons:

• need a way to enforce volume constraint

• bad choice early can be detrimental later

• computational complexity increases with iterations

• convergence and stopping criteria is unclear

Fixes:

• adaptive volume penalty

• subsampling splits

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

A Few Preliminary Comments

Pros:

• PDE-constrained optimization is not needed

• incorporates spatial structure

• encourages “topologically simple” solutions

• can adaptively act on large spatial blocks

Cons:

• need a way to enforce volume constraint

• bad choice early can be detrimental later

• computational complexity increases with iterations

• convergence and stopping criteria is unclear

Fixes:

• adaptive volume penalty

• subsampling splits

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

A Few Preliminary Comments

Pros:

• PDE-constrained optimization is not needed

• incorporates spatial structure

• encourages “topologically simple” solutions

• can adaptively act on large spatial blocks

Cons:

• need a way to enforce volume constraint

• bad choice early can be detrimental later

• computational complexity increases with iterations

• convergence and stopping criteria is unclear

Fixes:

• adaptive volume penalty

• subsampling splits

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Adaptive Volume Penalty

• do not enforce volume constraint at each iteration• rather, minimize a penalized objective

J (w , u) + α1

N

N∑i=1

(wi − 1)

• adapt the penalty parameter α at each iteration• adapt α using the volume constrain ratio

1N

N∑i=1

(wi − 1)

β

• at a given iteration, only consider splitting subdomains whichencourage the ratio to move toward 1

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Adaptive Volume Penalty

• do not enforce volume constraint at each iteration• rather, minimize a penalized objective

J (w , u) + α1

N

N∑i=1

(wi − 1)

• adapt the penalty parameter α at each iteration• adapt α using the volume constrain ratio

1N

N∑i=1

(wi − 1)

β

• at a given iteration, only consider splitting subdomains whichencourage the ratio to move toward 1

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Adaptive Volume Penalty

• do not enforce volume constraint at each iteration• rather, minimize a penalized objective

J (w , u) + α1

N

N∑i=1

(wi − 1)

• adapt the penalty parameter α at each iteration• adapt α using the volume constrain ratio

1N

N∑i=1

(wi − 1)

β

• at a given iteration, only consider splitting subdomains whichencourage the ratio to move toward 1

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Subsampling Splits

• the number of possible splits grows with the spatial mesh

• we do not need to consider every split when many are“nearby” one another

• randomly subsample η% of the possible splits with equalprobability

• η is a “accuracy versus cost knob”

• other sampling schemes may be considered

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Subsampling Splits

• the number of possible splits grows with the spatial mesh

• we do not need to consider every split when many are“nearby” one another

• randomly subsample η% of the possible splits with equalprobability

• η is a “accuracy versus cost knob”

• other sampling schemes may be considered

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Test Problem: Governing PDE

−∇ · (ε∇u) = f in Ω

u = 0 on ∂Ω

[u] = 0 on Γk , k = 1, 2, . . . ,NI

[n · (ε∇u)] = 0 on Γk , k = 1, 2, . . . ,NI

• Ω = (0, 1)2

• ε : Ω→ m1,m2 is piecewise constant defined on N cells

• wi encodes the value of ε on cell i , i = 1, 2, . . . ,N

• Γk , k = 1, 2, . . . ,NI are the interfaces of the cells

• f (x1, x2) = sin(πx1) sin(πx2)

• n is the normal vector to an interfaceJoey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Test Problem: Optimal Material Arrangement

minw ,u||u||22

w ∈ 1, 2N

1

N

N∑i=1

(wi − 1) ≤ .5

(w , u) satisfies PDE

Using the more expensive material in at most half of Ω, find is theoptimal material arrangement.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Test Problem: Optimal Material Arrangement

minw ,u||u||22

w ∈ 1, 2N

1

N

N∑i=1

(wi − 1) ≤ .5

(w , u) satisfies PDE

Using the more expensive material in at most half of Ω, find is theoptimal material arrangement.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Small Example to Get Intuition: Optimize By Enumeration

0 .25 .5 .75 1

1

.75

.5

.25

0

Figure: Optimal spacial distribution of ε on a 4× 4 grid (N = 16). Yellowdenotes material 2, blue denotes material 1.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Test Problem: A Real Example to Test the Method

minw ,u||u||22

w ∈ 1, 210,000

1

10, 000

10,000∑i=1

(wi − 1) ≤ .5

(w , u) satisfies PDE

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Optimal Material Assignment

0 .25 .5 .75 1

1

.75

.5

.25

0

Numerical Result

N = 10, 000

5% of splits

1642 PDE Solves

0 .25 .5 .75 1

1

.75

.5

.25

0

Numerical Result

N = 10, 000

20% of splits

6250 PDE Solves

0 .25 .5 .75 1

1

.75

.5

.25

0

ConjecturedOptimal Solution

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Iteration Histories

Iteration0 10 20 30

J

#10-4

2

3

4

Iteration0 10 20 30

,

#10-4

1.6

1.8

2

2.2

2.4

Iteration0 10 20 30

Vol

ume

Con

stra

int R

atio

0.5

1

1.5

2

Iteration0 10 20 30

J

#10-4

1.5

2

2.5

3

3.5

4

Iteration0 10 20 30

,#10-4

1.2

1.4

1.6

1.8

2

Iteration0 10 20 30

Vol

ume

Con

stra

int R

atio

0.5

1

1.5

2

Figure: Iteration histories for the objective function J (left), penaltyparameter α (center), and volume constraint ratio (right). η = .05 ontop, η = .2 on bottom.

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =1

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =2

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =3

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =4

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =5

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =6

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =7

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =8

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =9

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =10

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =11

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =12

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =13

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =14

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =15

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =16

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =17

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =18

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =19

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =20

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =21

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =22

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =23

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =24

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =25

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =26

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =27

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =28

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =29

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Solution Iteration History

Iteration =30

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

1 Introduction

2 Coupling PEBBL and ROL

3 Heuristic to Spatial Problem Structure

4 Conclusion

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Summary

• introduced MIPDECO in an abstract formulation

• introduced the coupling of PEBBL and ROL for solving largeMIPDECO problems

• presented a heuristic approach to exploit spatial structure fora class of MIPDECO problems

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Summary

• introduced MIPDECO in an abstract formulation

• introduced the coupling of PEBBL and ROL for solving largeMIPDECO problems

• presented a heuristic approach to exploit spatial structure fora class of MIPDECO problems

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Summary

• introduced MIPDECO in an abstract formulation

• introduced the coupling of PEBBL and ROL for solving largeMIPDECO problems

• presented a heuristic approach to exploit spatial structure fora class of MIPDECO problems

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Ongoing Work

• working through challenges with coupling PEBBL and ROL

- software implementation, multilevel parallelism, load balancing- non-convexity of problems with PDE constraints

• generalizing the heuristic and analyzing it on other testproblems

- analysis of the input parameters- analysis of how the volume constraint effects performance

• compare the computational complexity of PEBBL and ROL,the heuristic, and topology optimization

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Ongoing Work

• working through challenges with coupling PEBBL and ROL

- software implementation, multilevel parallelism, load balancing- non-convexity of problems with PDE constraints

• generalizing the heuristic and analyzing it on other testproblems

- analysis of the input parameters- analysis of how the volume constraint effects performance

• compare the computational complexity of PEBBL and ROL,the heuristic, and topology optimization

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Ongoing Work

• working through challenges with coupling PEBBL and ROL

- software implementation, multilevel parallelism, load balancing- non-convexity of problems with PDE constraints

• generalizing the heuristic and analyzing it on other testproblems

- analysis of the input parameters- analysis of how the volume constraint effects performance

• compare the computational complexity of PEBBL and ROL,the heuristic, and topology optimization

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization

Introduction Coupling PEBBL and ROL Heuristic to Spatial Problem Structure Conclusion

Questions?

Joey Hart

jlhart3@ncsu.edu

Collaborators: Pierre Gremaud, Tim Kelley, Sven Leyffer,Bart van Bloemen Waanders

Joey Hart NCSU

Exploiting Spatial Structure in Mixed-Integer PDE-Constrained Optimization