A Fast Interior Point Method for FreeFem++ · Interior-Point Methods ... •Some non-linear optimization tools mardi 3 juillet 12. Introduction FreeFem++ : a short presentation Optimization

A Fast Interior Point Method for FreeFem++

Application to a laplacian problem with Signorini type boundary conditions (Z. Belhachmi, F. Ben Belgacem & F. Hecht)

Sylvain Auliac - Laboratoire Jacques-Louis Lions

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Outline

mardi 3 juillet 12

Outline

Introduction• FreeFem++ : a short presentation• Optimization tools in FreeFem++

mardi 3 juillet 12

Outline


Interior-Point Methods • A short overview• The IPOPT software• Using IPOPT through FreeFem++

mardi 3 juillet 12

Outline


Interior-Point Methods • A short overview• The IPOPT software• Using IPOPT through FreeFem++

«Signorini-like» Laplacian Problem• Definition of the problem• Numerical results

Closing...

mardi 3 juillet 12

Introduction FreeFem++ : a short presentation Optimization tools with FreeFem++

The FreeFem++ Software (F. Hecht) An open-source finite element PDEsolver with :

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The FreeFem++ Software (F. Hecht) An open-source finite element PDEsolver with :

•An automatic mesh generator

⌦ ⇢ R2s.t.

@⌦ = � = {(cos(t), sin(t)) , t 2 [0, 2⇡]}

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The FreeFem++ Software An open-source finite element PDEsolver with :

•An automatic mesh generator

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•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language

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find u 2 H10 (⌦) s.t. 8v 2 H1

0 (⌦)Z

⌦rurv �

Z

⌦fv = 0

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•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one

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•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)

mardi 3 juillet 12



•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)•An MPI interface

mardi 3 juillet 12



•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)•An MPI interface•An IDE interface (by A. Le Hyaric)

mardi 3 juillet 12



•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)•An MPI interface•An IDE interface (by A. Le Hyaric)•Some non-linear optimization tools

mardi 3 juillet 12



•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)•An MPI interface•An IDE interface (by A. Le Hyaric)•Some non-linear optimization tools•An active community of users

mardi 3 juillet 12



•An automatic mesh generator•A FEM/PDE/Variational Forms specific user language •Automatic interpolation from a mesh to another one•Some sparse sequential linear solvers (LU, Cholesky, UMFPack, etc...), and some parallel ones (MUMPS, HYPRE, SUPERLU, PASTIX, etc...), eigenvalues and vectors (ARPACK)•An MPI interface•An IDE interface (by A. Le Hyaric)•Some non-linear optimization tools•An active community of users

➡ http://www.freefem.org/ff++/mardi 3 juillet 12

http://www.freefem.org/ff++/

http://www.freefem.org/ff++/


Optimization Tools of FreeFem++ :

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Quadratic problems solvers: LinearCG, LinearGMRESGeneral non-linear optimizers: •NLCG, BFGS•CMAES (derivative free evolution strategy, MPI version available)•NEWUOA (derivative free)

Unconstrained Optimization :

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•Derivative free / stochastic algorithms : DIRECT, CRS, ISRES, NEWUOA, BOBYQA, COBYLA, SBPLX, etc...

•Gradient-based algorithms : LBFGS, MMA, SLSQP, T-Newton, etc...

•Augmented Lagrangian MethodWhole list available on the NLopt web page

NLopt - Constrained non-linear optimizers :

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http://ab-initio.mit.edu/wiki/index.php/NLopt_Algorithms






•Derivative free / stochastic algorithms : DIRECT, CRS, ISRES, NEWUOA, BOBYQA, COBYLA, SBPLX, etc...

•Gradient-based algorithms : LBFGS, MMA, SLSQP, T-Newton, etc...

•Augmented Lagrangian MethodWhole list available on the NLopt web page

NLopt - Constrained non-linear optimizers :

IPOPT :Constrained non-linear optimization using sparse matrices

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Interior Point Method A short overviewThe IPOPT softwareUsing IPOPT through FreeFem++

Interior Point Methods : a short overview

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Objective function :

f : Rn �! R

Find : x0 = argmin8i, ci(x)�0

f(x)

c = (c1, . . . , cm) : Rn �! Rm

Constraints :Classical minimization under constraints problem :

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f : Rn �! R


f(x)

c = (c1, . . . , cm) : Rn �! Rm


Barrier Function :For a given µ > 0, solve the unconstrained problem of finding

x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)

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IP Methods : given a decreasing sequence with find a a sequence of associated minimizers.

µk �!k!+1

0(µk)(xµk)



f : Rn �! R


f(x)

c = (c1, . . . , cm) : Rn �! Rm



x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)

mardi 3 juillet 12



µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)

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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)

Illustration : minimize on a trianglef(x, y) = (x + 0.1)2 + (y � 0.5)2

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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)


Isovalues of f

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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)


Isovalues of flog(µ) = 2

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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)



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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)



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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)


Isovalues of flog(µ) = �1

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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)



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µk �!k!+1

0(µk)(xµk)


x

µ

= argminx2Rn

f(x)� µ

mX

i=1

ln c

i

(x)



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IP Methods : given a decreasing sequence withfind a sequence such that :

µk �!k!+1

0(µk)(xk)

8k, x

k

= argminx2Rn

f(x)� µ

k

mX

i=1

ln c

i

(x)

Links with the original problem :

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µk �!k!+1

0(µk)(xk)

8k, x

k

= argminx2Rn

f(x)� µ

k

mX

i=1

ln c

i

(x)


Extremum point : rf(xk)�mX

i=1

µk

ci(xk)rci(xk) = 0

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µk �!k!+1

0(µk)(xk)

8k, x

k

= argminx2Rn

f(x)� µ

k

mX

i=1

ln c

i

(x)

Links with the original problem :Extremum point :

rf(xk)� Jc(xk)T�k = 0

�k = µk/c(xk), where

«Dual variables»

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µk �!k!+1

0(µk)(xk)

Links with the original problem :Extremum point : rf(xk)� Jc(xk)T

�k = 0

8k, rf(xk)� Jc(xk)T�k = 0, �k = µk/c(xk) 2 Rm

ci(xk)�k,i = µk

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µk �!k!+1

0(µk)(xk)


�k = 0

KKT Conditions :


ci(xk)�k,i = µk

rf(x⇤)� Jc(x⇤)T�

⇤ = 0c(x⇤) � 08i, �

⇤i � 0

8i, ci(x⇤)�⇤i = 0

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µk �!k!+1

0(µk)(xk)


�k = 0

KKT Conditions :


ci(xk)�k,i = µk

rf(x⇤)� Jc(x⇤)T�

⇤ = 0c(x⇤) � 08i, �

⇤i � 0

8i, ci(x⇤)�⇤i = 0

µk �! 0

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µk �!k!+1

0(µk)(xk)


�k = 0

Under some favorable assumptions :xk �!

k!+1x

⇤


ci(xk)�k,i = µk

With even more assumptions : �k �!k!+1

�⇤

KKT point and multipliers such that, 9(x⇤, �⇤)

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IP main step: given a decreasing sequence withfind a sequence such that :

µk �!k!+1

0(µk)(xk)



k!+1x

⇤

With even more assumptions :


�k �!k!+1

�⇤

Extremum point : rf(xk)� Jc(xk)T�k = 0 �k = µk/c(xk)


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µk �!k!+1

0(µk)(xk)



k!+1x

⇤

With even more assumptions :


�k �!k!+1

�⇤

Extremum point : rf(xk)� Jc(xk)T�k = 0 �k = µk/c(xk)


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µk �!k!+1

0(µk)(xk)


xk �!k!+1

x

⇤


KKT point and corresponding multipiers of theoriginal problem, such that, under some assumptions :

�k �!k!+1

�⇤

9(x⇤, �⇤)

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µk �!k!+1

0(µk)(xk)



xk �!k!+1

x

⇤

KKT point and corresponding multipiers of theoriginal problem, such that, under some assumptions :9(x⇤, �⇤)

�k �!k!+1

�⇤

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µk �!k!+1

0(µk)(xk)


Links with the original problem : KKT point and corresponding multipliers of the original problem, such that, under some assumptions :

xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

mardi 3 juillet 12



µk �!k!+1

0(µk)(xk)



xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

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µk �!k!+1

0(µk)(xk)


Primal-dual IP method - Newton’s method is applied to the non-linear system :⇢

rf(x)� Jc(x)T� = 0

ci(x)�i = µ , 8i m


xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

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µk �!k!+1

0(µk)(xk)


Primal-dual IP method - Newton’s method is applied to the non-linear system : ⇢

rf(x)� Jc(x)T� = 0

ci(x)�i = µ , 8i m

H = r2f �mX

i=1

�ir2ci

Newton update :(�x

,��

)✓

H �JT

c

⇤Jc

C

◆ ✓�

x

��

◆=

✓rf � JT

c

�C�� µ1

◆

Where and (as for ).C = (�ijci)1i,jm ⇤


xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

mardi 3 juillet 12



µk �!k!+1

0(µk)(xk)



rf(x)� Jc(x)T� = 0

ci(x)�i = µ , 8i m

H = r2f �mX

i=1

�ir2ci


,��

)✓

H �JT

c

⇤Jc

C

◆ ✓�

x

��

◆=

✓rf � JT

c

�C�� µ1

◆

Where and (as for ).C = (�ijci)1i,jm ⇤H = r2f �mX

i=1

�ir2ci


xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

mardi 3 juillet 12



µk �!k!+1

0(µk)(xk)



rf(x)� Jc(x)T� = 0

ci(x)�i = µ , 8i m

H = r2f �mX

i=1

�ir2ci


,��

)✓

H �JT

c

⇤Jc

C

◆ ✓�

x

��

◆=

✓rf � JT

c

�C�� µ1

◆

Where and (as for ).C = (�ijci)1i,jm ⇤


xk �!k!+1

x

⇤

9(x⇤, �⇤)

�k �!k!+1

�⇤

mardi 3 juillet 12


The IPOPT software :• C++ open source implementation of a primal-dual interior point method for non-linear smooth optimization, from the COIN-OR project ☞ https://projects.coin-or.org/Ipopt

☞ the mailing list

• Uses sparse matrices and sparse symetric linear solvers (mumps, PARDISO, etc... ) for the Newton’s steps.

• Handles both inequality and/or equality constraints• A fortran variant for variationnal inequalities...?

Bibliography : •A. Wächter and L. T. Biegler, On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, Mathematical Programming 106(1), pp. 25-57, 2006

•Some IPOPT related papers : https://projects.coin-or.org/Ipopt/wiki/IpoptPapers

mardi 3 juillet 12

https://projects.coin-or.org/Ipopt

https://projects.coin-or.org/Ipopt

http://list.coin-or.org/mailman/listinfo/ipopt

http://list.coin-or.org/mailman/listinfo/ipopt

https://projects.coin-or.org/Ipopt/wiki/IpoptPapers

https://projects.coin-or.org/Ipopt/wiki/IpoptPapers


The FreeFem++ Interface :8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

⬅ Simple bounds constraints

mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi ⬅ «True» constraints

mardi 3 juillet 12


The FreeFem++ Interface : 8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs f, rf, c, Jc, x

lb, x

ub,

clb, cub and...

H : Rn ⇥ Rm �! Mn (R)

(x , �) 7�! r2f(x)�mX

i=1

�ir2ci(x)

mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs f, rf, c, Jc, x

lb, x

ub,

clb, cub and...

H : Rn ⇥ R⇥ Rm �! Mn (R)

(x , � , �) 7�! �r2f(x) +mX

i=1

�ir2ci(x)

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c

mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c

: func real f(real[int] &X) {...}

mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c


: func real[int] df(real[int] &X) {...} func real[int] c(real[int] &X) {...} (optional)

mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c



: func matrix Jc(real[int] &X) {...} Needed only if there are constraint functions

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c




: func matrix H(real[int] &X, real s, real[int] &L) In case of affine constraints, the prototype may be : func matrix H(real[int] &X) {...} If H is omitted Newton is replaced by a BFGS algorithm.

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c





: real[int] arrays •size n for x bounds•size m for c bounds•x bounds are optionnal•Set components to ±1e19

for unboundedness in particular directions.

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c







Call IPOPT : full featuredint status = IPOPT(f, dF, c, Jc, H, xstart, lb=xlb,ub=xub,clb=clb,cub=cub, ... ... );

Remark: requires load ″ff-Ipopt″; earlier in the script

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>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c







Call IPOPT : unconstrainedint status = IPOPT(f, dF, H, xstart, lb=xlb,ub=xub, ... ... );


mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c







Call IPOPT : with BFGSint status = IPOPT(f, dF, c, Jc, xstart, lb=xlb,ub=xub,clb=clb,cub=cub ... ... );


mardi 3 juillet 12



>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

IPOPT needs

x

lb, x

ub, clb, cub

H

f

Jc

rf, c







Call IPOPT : unconstrained with BFGSint status = IPOPT(f, dF, xstart, lb=xlb,ub=xub, ... ... );


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The FreeFem++ Interface : case of quadratic objective and affine constraints

8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

mardi 3 juillet 12



8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1

2 hAx, xi + hb, xic(x) = Cx + d

(A, b) 2Mn,n(R)⇥ Rn, (C, d) 2Mn,m(R)⇥ Rm

mardi 3 juillet 12



8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1



➡ Possibility to directly pass the matrices and vectors

mardi 3 juillet 12

Quadratic objective and affine constraints :... //begining of the scriptmatrix A = ... ;matrix C = ... ;real[int] b = ... , d = ... ;IPOPT([A,b] , [C,d] , xstart , /*named parameters*/ );



8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1




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8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1




Homogeneous quadratic objective and affine constraints :... //begining of the scriptmatrix A = ... ;matrix C = ... ;real[int] d = ... ;IPOPT(A , [C,d] , xstart , /*named parameters*/ );

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8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1




Standard objective and affine constraints :... //begining of the scriptfunc real f(real[int] &X) {...}func real[int] df(real[int] &X) {...}func matrix H(real[int] &X) {...} //no need for the full prototypematrix C = ... ;real[int] d = ... ;IPOPT(f, df, H, [C,d] , xstart , /*named parameters*/ );

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8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1




Quadratic objective and standard constraints : hazardous case... //begining of the scriptfunc real[int] c(real[int] &X) {...}func matrix Jc(real[int] &X) {...}matrix A = ... ;real[int] b = ... ;

IPOPT([A,b] , c, Jc, xstart , /*named parameters*/ );

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8>>><

>>>:

x

⇤ = argminx2Rn

f(x)

8i n, x

lbi

x

⇤i

x

ubi

8i m, c

lbi

c

i

(x⇤) c

ubi

8x 2 Rn, f(x) = 1




Quadratic objective and standard constraints : hazardous case... //begining of the scriptfunc real[int] c(real[int] &X) {...}func matrix Jc(real[int] &X) {...}matrix A = ... ;real[int] b = ... ;

IPOPT([A,b] , c, Jc, xstart , /*named parameters*/ );

Here, is always equal to A➡ erroneous hessian in some cases

H(x,�, �)

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The FreeFem++ Interface : some hints and tips

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Matrix returning functions : must return a global object

func matrix H(real[int] &X) { matrix M; ... //compute M return M;}

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M is cleaned after closing the function block, so the returned object does not exist anymore...

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Sparse matrix returning functions : must return a global objectmatrix M; //just declare a global matrixfunc matrix H(real[int] &X) { ... //do something to fill M return M; //M will not be deleted before H goes out of scope}

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Sparse matrix returning functions : must return a global objectmatrix M; //just declare a global matrixfunc matrix H(real[int] &X) { ... //do something to fill M return M; //M will not be deleted before H goes out of scope}

The structure of the matrices : has to be constant through the optimization process...•Use varf as much as possible to build matrices, matrices built with the same varf always have the same structure even if some coefficients becomes null.

•Remember that zeros are discarded when building sparse matrices from arrays : real[int] c=...; real[int,int] tmp=...; int[int] I=..., J=...; matrix M = tmp; matrix M = [I, J ,c]; //two dangerous methods

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The FreeFem++ Interface : some hints and tips (2)

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Named parameters : subset of the IPOPT parameters, plus a few FreeFem++ specific ones, which can be used for :• Stopping criteria (tol, maxiter, maxcputime, etc...)• Getting final values (objvalue, lm, uz, lz, etc...)• Perform a warm start• Call a derivatives checker• Force the BFGS mode (add bfgs=1)• Specify an extern option file (see the IPOPT documentation)• etc...

Check the documentation for more informations.

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http://www.coin-or.org/Ipopt/documentation/




Named parameters : subset of the IPOPT parameters, plus a few FreeFem++ specific ones, which can be used for :• Stopping criteria (tol, maxiter, maxcputime, etc...)• Getting final values (objvalue, lm, uz, lz, etc...)• Perform a warm start• Call a derivatives checker• Force the BFGS mode (add bfgs=1)• Specify an extern option file (see the IPOPT documentation)• etc...

Check the documentation for more informations.

Returned value : the IPOPT function returns an int revealing what happens during the optimization. Positive values often means IPOPT performed well and negative values are relevant of troubles.

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Examples Quadratic problemsA non quadratic example

A Quadratic Example :

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Stokes Cavity Flow :8<

:

�u�rp = fr.u = 0

in ⌦

u = gon @⌦

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Stokes Cavity Flow :

Find : u = argminv2K

14

Z

⌦

��rv +t rv��2 �

Z

⌦f.v

Where : K =�v 2 H1(⌦)2, v � g 2 H1

0 (⌦)2, r.v = 0

8<

:

�u�rp = fr.u = 0

in ⌦

u = gon @⌦

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14

Z

⌦

��rv +t rv��2 �

Z

⌦f.v

Where : K =�v 2 H1(⌦)2, v � g 2 H1

0 (⌦)2, r.v = 0



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14

Z

⌦

��rv +t rv��2 �

Z

⌦f.v

Where : K =�v 2 H1(⌦)2, v � g 2 H1

0 (⌦)2, r.v = 0



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14

Z

⌦

��rv +t rv��2 �

Z

⌦f.v

Where : K =�v 2 H1(⌦)2, v � g 2 H1

0 (⌦)2, r.v = 0



Jh(u) = (Au, u)� (b, u) , u 2 Kh

Discret Problem : find minimizer of

+ Dirichlet boundary conditionsKh =�v 2 P2(Th)2 | Chv = 0

(Where ) Chv = 0 , 8q 2 P1(Th),Z

⌦qr.vh = 0

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14

Z

⌦

��rv +t rv��2 �

Z

⌦f.v

Where : K =�v 2 H1(⌦)2, v � g 2 H1

0 (⌦)2, r.v = 0



Jh(u) = (Au, u)� (b, u) , u 2 Kh




⌦qr.vh = 0

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Jh(u) = (Au, u)� (b, u) , u 2 Kh




⌦qr.vh = 0



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Jh(u) = (Au, u)� (b, u) , u 2 Kh




⌦qr.vh = 0



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Jh(u) = (Au, u)� (b, u) , u 2 Kh




⌦qr.vh = 0



To get back the dual variables and store them in a P1 finite elements function.

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Jh(u) = (Au, u)� (b, u) , u 2 Kh




⌦qr.vh = 0



➡Dual variables converge toward the pressure field

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Examples Quadratic problemA non quadratic example

A Constrained Minimal-Surface Example :

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Given g, let , and :

8v 2 E, S(v) =Z

⌦

p1 + |rv|2 , V (v) =

Z

⌦v

And, given V0 >0, try to find : u = argminv 2 E,

V (v) � V0

S(v)

E =�v 2 L2(⌦) | v � g 2 H1

0 (⌦)

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Given g, let , and :

8v 2 E, S(v) =Z

⌦

p1 + |rv|2 , V (v) =

Z

⌦v

And, given V0 >0, try to find : u = argminv 2 E,

V (v) � V0

S(v)

E =�v 2 L2(⌦) | v � g 2 H1

0 (⌦)

Computations withg = x

2y

2, ⌦ = {(x, y) 2 R2 | x

2 + y

2 1}V0 = 0 V0 =

12

V0 = 1 V0 = 2

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«Signorini-like» Laplacian Problem Definition of the problemNumerical results

«Signorini-like» Laplacian Problem : Conjoint work with Z. Belhachmi, F. Ben Belgacem & F. Hecht ➡ Quadratic Finite Elements with Non-Matching Grids for the Unilateral Boundary Contact

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⌦s ⌦i

⌦s, ⌦i : contiguous domains of R2

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⌦s ⌦i

�


� = @⇣⌦s [ ⌦i

⌘

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⌦s ⌦i

� �C


� = @⇣⌦s [ ⌦i

⌘�C = ⌦s \ ⌦i

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Domain :



⌦s ⌦i

� �C


� = @⇣⌦s [ ⌦i

⌘�C = ⌦s \ ⌦i

Problem :

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Domain :



⌦s ⌦i

� �C


� = @⇣⌦s [ ⌦i

⌘�C = ⌦s \ ⌦i

Problem : find such thatpi, ps

• • Dirichlet or Neumann conditions on •

�p↵ = f↵ in ⌦↵ (↵ = s, i)�

[p] � 0, @np↵ � ⇠ and (@np� ⇠)[p] = 0 on �C

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Domain :


«Signorini-like» Laplacian Problem :

⌦s ⌦i

� �C


� = @⇣⌦s [ ⌦i

⌘�C = ⌦s \ ⌦i

Problem : find such thatpi, ps

(ps, pi) = argmin(us,ui)2K

X

↵2{s,i}

12

Z

⌦↵

|ru↵|2 �Z

⌦↵

f↵u↵ �Z

�↵n

g↵nu↵

Where :

K =⇢

(us, ui) 2 H1(⌦s)⇥H1(⌦i)��

u↵ � g↵d 2 H1

0 (⌦↵) ↵ = i, s[u] � 0

�

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Numerical Results

T ihT s

h

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Numerical Results

T ihT s

hJh(u) = (Au, u)� (b, u) , u 2 Kh

Kh =�uh 2 P2(T s

h )⇥ P2(T ih ) | Chuh � 0


+ Dirichlet boundary conditions

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Numerical Results

T ihT s

hJh(u) = (Au, u)� (b, u) , u 2 Kh

Kh =�uh 2 P2(T s

h )⇥ P2(T ih ) | Chuh � 0



Building the needed objects :

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Numerical Results

T ihT s

hJh(u) = (Au, u)� (b, u) , u 2 Kh

Kh =�uh 2 P2(T s

h )⇥ P2(T ih ) | Chuh � 0



Building the needed objects :• A, b are directly obtained from the separated Poisson matrices and second membersmatrix A = [ [As , 0] , [0 , Ai] ];

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Numerical Results

T ihT s

hJh(u) = (Au, u)� (b, u) , u 2 Kh

Kh =�uh 2 P2(T s

h )⇥ P2(T ih ) | Chuh � 0




•Dirichlet boundary conditions are treated as simple bounds, using a trick allowed by the varf syntax

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Numerical Results

T ihT s

hJh(u) = (Au, u)� (b, u) , u 2 Kh

Kh =�uh 2 P2(T s

h )⇥ P2(T ih ) | Chuh � 0




•Dirichlet boundary conditions are treated as simple bounds, using a trick allowed by the varf syntax

•C depends on the unilateral conditions treatment

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Numerical Results :

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Numerical Results :Two exact solutions :

⇢p

i(x, y) = xy + x

pr + x� y

pr � x

p

s(x, y) = xy � x

pr + x� y

pr � x , r =

px

2 + y2

⇢p

i(x, y) = xy + [2 max(x, 0)�min(x, 0) ⇤ sin(2⇡y) ]sin(3x)

p

s(x, y) = xy + [�2 max(x, 0)�min(x, 0) ⇤ sin(2⇡y) ]sin(3x)

Computations : we numerically evaluate with the and norms, with decaying h in the three following cases :• Conforming meshes, corresponding degrees of freedom are directly compared.• Non-conforming meshes with simple interpolation process.• Non-conforming meshes with mortar projection (C is not sparse)

kph � pk H1, L2

L1

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Numerical Results : First solution

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Numerical Results : First solution⇢

p


p


H1norm

L2norm

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Numerical Results : First solution⇢

p


p


H1norm

L2norm

L1 norm

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Numerical Results : Second solution

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⇢p

i(x, y) = xy + x

pr + x� y

pr � x

p

s(x, y) = xy � x

pr + x� y

pr � x , r =

px

2 + y2

L2normH1

norm

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⇢p

i(x, y) = xy + x

pr + x� y

pr � x

p

s(x, y) = xy � x

pr + x� y

pr � x , r =

px

2 + y2

L2norm

H1norm

L1 norm

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Outlook

• A state of the art software for large-scale smooth optimization directly available into FreeFem++• Succesfully tested on several quadratic convex problems, outperforming any of the previously available optimizers• Limits : ➡ local optimization ➡ needs order two derivatives for full efficiency

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Outlook

• A state of the art software for large-scale smooth optimization directly available into FreeFem++• Succesfully tested on several quadratic convex problems, outperforming any of the previously available optimizers• Limits : ➡ local optimization ➡ needs order two derivatives for full efficiency

Sequel :• 3D computations for the Signorini problem• Optimal shape design• Development of (smart) tools to ease the computation of derivatives within FreeFem++

mardi 3 juillet 12

Documents

A Fast Interior Point Method for FreeFem++ · Interior-Point Methods ... •Some non-linear optimization tools mardi 3 juillet 12. Introduction FreeFem++ : a short presentation Optimization