Frank Edward Curtis Northwestern University Joint work with Richard Byrd and Jorge Nocedal February...

Frank Edward CurtisNorthwestern University

Joint work with Richard Byrd and Jorge Nocedal

February 12, 2007

Inexact Methods for PDE-Constrained Optimization

Emory University

Nonlinear Optimization

“One” problem

0)(

0)(..

)(min

xh

xgts

xfx

Circuit Tuning Building blocks:

Transistors (switches) and Gates (logic units) Improve aspects of the circuit – speed, area, power – by choosing transistor widths

AT1

AT3

AT2

d1

d2

w1 w2

(A. Wächter, C. Visweswariah, and A. R. Conn, 2005)

Circuit Tuning Building blocks:

Transistors (switches) and Gates (logic units) Improve aspects of the circuit – speed, area, power – by choosing transistor widths

Formulate an optimization problem

,...d2AT2,d1AT1maxAT3..

)dw,AT,(min

ts

f

AT1

AT3

AT2

d1

d2

w1 w2

(A. Wächter, C. Visweswariah, and A. R. Conn, 2005)

Strategic Bidding in Electricity Markets Independent operator collects bids and sets production

schedule and “spot price” to minimize cost to consumers

(Pereira, Granville, Dix, and Barroso, 2004)

demand..

bidsmin

ePts

PT



demand..

bidsminminarg..

cost)spot(max

ePts

PPts

P

T


demand..

bidsmin

ePts

PT

Electricity production companies “bid” on how much they will charge for one unit of electricity



demand..

bidsminminarg..

cost)spot(max

ePts

PPts

P

T

0

0

0

xy

y

x Bilevel problem Equivalent to MPCC Hard geometry!

y

x


demand..

bidsmin

ePts

PT

Electricity production companies “bid” on how much they will charge for one unit of electricity

Challenges for NLP algorithms Very large problems Numerical noise Availability of derivatives Degeneracies Difficult geometries Expensive function evaluations Real-time solutions needed Integer variables Negative curvature

Outline Problem Formulation

Equality constrained optimization Sequential Quadratic Programming

Inexact Framework Unconstrained optimization and nonlinear equations Stopping conditions for linear solver

Global Behavior Merit function and sufficient decrease Satisfying first order conditions

Numerical Results Model inverse problem Accuracy tradeoffs

Final Remarks Future work Negative curvature

Equality constrained optimization

e.g., minimize the difference between observed and expected behavior, subject to atmospheric flow equations (Navier-Stokes)

Goal: solve the problem

0)(s.t.

)(min

xcxf

x

*x

Goal: solve the problem

Equality constrained optimization

)()(),( xcxfxL T

0)(s.t.

)(min

xcxf

x ),(:),(

)(:)(

)(:)(

2 xLxW

xcxA

xfxg

xx

i

0)(

)()(

xc

xAxg T

Define: the Lagrangian

Define: the derivatives

Goal: solve KKT conditions

Sequential Quadratic Programming (SQP)

0s.t.

min 21

Adc

Wdddg TT

d

c

Agd

A

AW TT 0

Algorithm: Newton’s method

Algorithm: the SQP subproblem

Two “equivalent” step computation techniques


c

Agd

A

AW TT 0 0s.t.

min 21

Adc

Wdddg TT

d




0A

AW T KKT matrix• Cannot be formed• Cannot be factored


c

Agd

A

AW TT 0 0s.t.

min 21

Adc

Wdddg TT

d




0A

AW T KKT matrix• Cannot be formed• Cannot be factored

Linear system solve• Iterative method• Inexactness

Unconstrained optimization

)(min xfx

)()(2kkk xfdxf

Goal: minimize a nonlinear objective

Algorithm: Newton’s method (CG)

Unconstrained optimization

)(min xfx

kx

)()(2kkk xfdxf

Goal: minimize a nonlinear objective

Algorithm: Newton’s method (CG)

Note: choosing any intermediate step ensures global convergence to a local solution of NLP

(Steihaug, 1983)

Nonlinear equations

0)( xF

)()( kkk xFdxF

Goal: solve a nonlinear system


any step with

and

ensures descent

Nonlinear equations

kkkk rxFdxF )()(

0)( xF

kx

)()( kkk xFdxF

Goal: solve a nonlinear system


10 ,)( kk xFr

(Eisenstat and Walker, 1994)

(Dembo, Eisenstat, and Steihaug, 1982)

0s.t.

min 21

Adc

Wdddg TT

d

)()()( xcxfx

Line Search SQP Framework

Define “exact” penalty function

c

Agd

A

AW TT 0

0s.t.

min 21

Adc

Wdddg TT

d

)()()( xcxfx

Line Search SQP Framework

Define “exact” penalty function

c

Agd

A

AW TT 0

)( dx

1

)( dx

10

for k = 0, 1, 2, … Compute step by…

Set penalty parameter to ensure descent on…

Perform backtracking line search to satisfy…

Update iterate

Algorithm Outline (exact steps)

c

Agd

A

AW TT 0

)()()( xcxfx

)()()( dDxdx

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case

kx

c

Agd

A

AW TT 0

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case

Exact step minimizes the objective on the linearized constraints

kx

c

Agd

A

AW TT 0

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case

kxExact step minimizes the objective on the linearized constraints

… which may lead to an increase in the model objective

c

Agd

A

AW TT 0

)(

)()0()(

21 AdccWdddg

dmmdmredTT

)()( 21 AdcWdddgfdm TT

Quadratic/linear model of merit function

Create model

Quantify reduction obtained from step

0s.t.

min 21

Adc

Wdddg TT

d

c

Agd

A

AW TT 0

)()( 21 AdcWdddgfdm TT

Quadratic/linear model of merit function

Create model

Quantify reduction obtained from step

0s.t.

min 21

Adc

Wdddg TT

d

)(

)()0()(

21 AdccWdddg

dmmdmredTT

rc

Agd

A

AW TT 0

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case



c

Agd

A

AW TT 0

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case

kx

10 ,)( cdmred

c

Agd

A

AW TT 0

Exact step minimizes the objective on the linearized constraints


… but this is ok since we can account for this conflict by increasing the penalty parameter

0s.t.

min 21

Adc

Wdddg TT

d

Exact Case



… but this is ok since we can account for this conflict by increasing the penalty parameter

10 ,)1(

21

c

Wdddg TT

c

Agd

A

AW TT 0

for k = 0, 1, 2, … Compute step by…

Set penalty parameter to ensure descent on…

Perform backtracking line search to satisfy…

Update iterate


c

Agd

A

AW TT 0

)()()( xcxfx

)()()( dDxdx

First attempt

10 ,),(),( cAgr T

Proposition: sufficiently small residual

1e-8 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1

Success 100% 100% 97% 97% 90% 85% 72% 38%

Failure 0% 0% 3% 3% 10% 15% 28% 62%

Test: 61 problems from CUTEr test set

0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

First attempt… not robust

10 ,),(),( cAgr T

Proposition: sufficiently small residual

0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

… not enough for complete robustness We have multiple goals (feasibility and optimality) Lagrange multipliers may be completely off

… may not have descent!

Recall the line search condition

Second attempt

rcdgdD T )(

Step computation: inexact SQP step

)()()( dDxdx

rc

Agd

A

AW TT 0

We can show

Recall the line search condition

Second attempt

rcdgdD T )(

Step computation: inexact SQP step

)()()( dDxdx

rc

Agd

A

AW TT 0

We can show

... but how negative should this be?

for k = 0, 1, 2, … Compute step

Set penalty parameter to ensure descent

Perform backtracking line search

Update iterate


c

Agd

A

AW TT 0

)()()( xcxfx

)()()( dDxdx

for k = 0, 1, 2, … Compute step and set penalty parameter to ensure descent and a stable algorithm

Perform backtracking line search

Update iterate

Algorithm Outline (inexact steps)

rc

Agd

A

AW TT 0

)()()( xcxfx

)()()( dDxdx

0s.t.

min 21

Adc

Wdddg TT

d

Inexact Case

rc

Agd

A

AW TT 0

crcWdddg TT 21

0s.t.

min 21

Adc

Wdddg TT

d

Inexact Case

rc

Agd

A

AW TT 0

kx

0s.t.

min 21

Adc

Wdddg TT

d

Inexact Case

rc

Agd

A

AW TT 0

kx

TAg

r

,

:0 ,10Step is acceptable if for

crcWdddg TT 21

0s.t.

min 21

Adc

Wdddg TT

d

Inexact Case

rc

Agd

A

AW TT 0

kx

c

cr

,

:0 ,10

Step is acceptable if for

crcWdddg TT 21

0s.t.

min 21

Adc

Wdddg TT

d

Inexact Case

rc

Agd

A

AW TT 0

kx

c

cr

,

:0 ,10

Step is acceptable if for

10 ,)1(

21

rc

Wdddg TT

crcWdddg TT 21

for k = 0, 1, 2, … Iteratively solve

Until

Update penalty parameter Perform backtracking line search Update iterate

Algorithm Outline

cdmred

Ag

rT

)(

10 ,

0 ,

0 ,

10 ,

c

cr

rc

Agd

A

AW TT 0

or

Observe KKT conditions

Termination Test

10 , )(,1 max

10 , ,1 max

0

feasfeas

optoptT

xcc

gAg

The sequence of iterates is contained in a convex set and the following conditions hold:

the objective and constraint functions and their first and second derivatives are bounded the multiplier estimates are bounded the constraint Jacobians have full row rank and their smallest singular values are bounded below by a positive constant the Hessian of the Lagrangian is positive definite with smallest eigenvalue bounded below by a positive constant

Assumptions

Sufficient Reduction to Sufficient Decrease

rcdgdD T )(

cWddrcdg TT 21

cd

cWdddD T

2

21)(

10 ,)( cdmred

Taylor expansion of merit function yields

Accepted step satisfies

Intermediate Results

0 ,

10 ,

c

cr

10 ,)1(

21

rc

Wdddg TT

cdmred

Ag

rT

)(

0 ,

0 ,

d

is bounded below by a positive constant

is bounded above

is bounded above

Sufficient Decrease in Merit Function

cdxdD 2

),;(

0lim k

Tk

kgZ

0lim2

kkk

cd

cddxx 2);();(

Step in Dual Space

10 , TTAgAg

|||| c |||| d(for sufficiently small and )

0lim k

kc

0lim k

Tkk

kAg

Therefore,

We converge to an optimal primal solution, and

Problem Formulation Tikhonov-style regularized inverse problem

Want to solve for a reasonably large mesh size Want to solve for small regularization parameter

SymQMR for linear system solves Input parameters:

c

Ag

r

T

0.1 ,1

,1 ,1.0

cdmred

Ag

rT

)(

10 ,

0 ,

0 ,

10 ,

c

crorRecall

:

(Curtis and Haber, 2007)

Numerical Results

c

Ag

r

T

Iters. Time Total LS Iters.

Avg. LS Iters.

Avg. Rel. Res.

0.5 29 29.5s 1452 50.1 3.12e-1

0.1 12 11.37s 654 54.5 6.90e-2

0.01 9 11.60s 681 75.7 6.27e-3

n 1024

m 512

1e-6


Numerical Results


Avg. LS Iters.

Avg. Rel. Res.

1e-6 12 11.40s 654 54.5 6.90e-2

1e-7 11 14.52s 840 76.4 6.99e-2

1e-8 8 10.57s 639 79.9 6.15e-2

1e-9 11 18.52s 1139 104 8.65e-2

1e-10 19 44.41s 2708 143 8.90e-2

n 1024

m 512

1e-1


Numerical Results


Avg. LS Iters.

Avg. Rel. Res.

1e-6 15 264.47s 1992 133 8.13e-2

1e-7 11 236.51s 1776 161 6.89e-2

1e-8 9 204.51s 1567 174 6.77e-2

1e-9 11 347.66s 2681 244 8.29e-2

1e-10 16 805.14s 6249 391 8.93e-2

n 8192

m 4096

1e-1


Numerical Results


Avg. LS Iters.

Avg. Rel. Res.

1e-6 15 5055.9s 4365 291 8.46e-2

1e-7 10 4202.6s 3630 363 8.87e-2

1e-8 12 5686.2s 4825 402 7.96e-2

1e-9 12 6678.7s 5633 469 8.77e-2

1e-10 14 14783s 12525 895 8.63e-2

n 65536

m 32768

1e-1


Review and Future Challenges Review

Defined a globally convergent inexact SQP algorithm Require only inexact solutions of primal-dual system Require only matrix-vector products involving

objective and constraint function derivatives Results also apply when only reduced Hessian of

Lagrangian is assumed to be positive definite Numerical experience on model problem is promising

Future challenges (Nearly) Singular constraint Jacobians Inexact derivative information Negative curvature etc., etc., etc….

Negative CurvatureBig question

What is the best way to handle negative curvature (i.e., when the reduced Hessian may be indefinite)?

Small question What is the best way to handle negative

curvature in the context of our inexact SQP algorithm?

We have no inertia information! Smaller question

When can we handle negative curvature in the context of our inexact SQP algorithm with NO algorithmic modifications?

When do we know that a given step is OK? Our analysis of the inexact case leads to a few

observations…

0s.t.

min 21

Adc

Wdddg TT

d

Why Quadratic Models?

rc

Agd

A

AW TT 0

kx kx

0s.t.

min 21

Adc

Wdddg TT

d


rc

Agd

A

AW TT 0

kx kx

Provides a good…• direction? Yes• step length? Yes

Provides a good…• direction? Maybe• step length? Maybe

0s.t.

min 21

Adc

Wdddg TT

d


rc

Agd

A

AW TT 0

kx kx

One can use our stopping criteria as a mechanism for determining which are good directions

All that needs to be determined is whether the step lengths are acceptable

Hdddg TT

d 21 min

Unconstrained Optimization

gHd

Direct method is the angle test

Indirect method is to check the conditions

or

dgdgT

2 , dHddg T

gdgdgT ,2

Hdddg TT

d 21 min

Unconstrained Optimization

gHd

Direct method is the angle test

Indirect method is to check the conditions

or

dgdgT

2 , dHddg T

gdgdgT ,2

step quality step length

Constrained Optimization

Step quality determined by

Step length determined by

0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

cdmred

Ag

rT

)(

10 ,

0 ,

0 ,

10 ,

c

cror

rcd , max2dWdd T or

Thanks!

Actual Stopping Criteria

Stopping conditions:

Model reduction condition

0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

crcdmred

Agc

crT

,max)(

,max

1 ,max

c

cr

or

crcrcWdddg TT ,max2

,0 ,1,0

Constraint Feasible Case

If feasible, conditions reduce to

0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

kx

Wdddgr

Ag

r

TT

T

21



0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

Wdddgr

Ag

r

TT

T

21

kx

Some region around the exact solution



0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

kx

Wdddgr

Ag

r

TT

T

21

Ellipse distorted toward the linearized constraints



0s.t.

min 21

Adc

Wdddg TT

d

rc

Agd

A

AW TT 0

Wdddgr

Ag

r

TT

T

21

kx

Frank Edward Curtis Northwestern University Joint work with Richard Byrd and Jorge Nocedal February...

Documents

Transcript of Frank Edward Curtis Northwestern University Joint work with Richard Byrd and Jorge Nocedal February...