The method of moments in dynamic optimization R. Meziat Departamento de Matemáticas Universidad de...

The method of The method of moments in dynamic moments in dynamic optimizationoptimization

R. MeziatDepartamento de Matemáticas

Universidad de los AndesColombia, 2005

Mathematics Seminar. Institute of Mathematics. Charles University. Prague, May 2005

Conic programming.

Convex envelopes

Microstructures

Optimal control

IndexIndex

The method of moments in dynamic optimization, R. Meziat, 2005.

Conic programmingConic programming

: 0nP x R Ax

m nA

: 0t mV A en R

P and V are closed convex cones in Rn

Polar cone:

Matrix of m rows and n columns:

System of homogenous inequalities:

Positive cone generated by the rows of the matrix A

: 0,nA x R a x a A

Basic results:

P V

V P



1

11

, ,

0 1, , 0

IFF

0 1, ,

m n

i

mm i

a a R

a x i m b x

b a a i m

0

0

tA y

b y

Farkas Lemma:Farkas Lemma: Theorem of the alternative I: Theorem of the alternative I: Ax=b has a solution in x 0 of Rn or excluding:

has solution in

my


Theorem of the alternative II: Theorem of the alternative II: Ax b has a solution in x 0 of Rn or excluding:

Theorem of the alternative III: Theorem of the alternative III: Ax b has a solution in x Rn or excluding:


0

0

tA y

b y

has solution in:

0 my in

0

0

tA y

b y

has solution in:

0 my in


Corollary:

0, 0

0

n n

n

K R anti symmetric

exists x R

Kx x

Kx x


Consequences of the Farkas Lemma:Consequences of the Farkas Lemma:

Kuhn-Tucker conditions in programs under restrictions in form of inequality.

Duality in convex programming.



Primal program (P)

min

. .

0 n

c x

s t

Ax b

x en R

Dual program (D)

max

. .

0

t

m

b y

s t

A y c

y en R

Duality gap: Given a couple of feasible solutions (x,y)

0 . .c x b y The system of lineal inequalities:

0

0

0

t

Ax tb

A y tc

b y c x

0

0

0

t

Ax tb y

A y tc x

b y c x t

Has a solution (y,x,t) Rm x Rn x R1 that satisfies:



One of the following affirmations is always truth:

1. There is a couple of optimal solutions for the primal (P) and dual (D) that satisfies:

2. One of the problems is not limited and it is not feasible.

.b y c x

One couple of feasible solutions is a couple of optimal solution if the complementariness relations are fulfilled:

0

0

tA y c x

Ax b y



Equivalence between convex cones and relations of order in linear spaces with inner product.

Relation of compatible linear order with the topology and the operations of the underlying linear space

Order properties:Order properties:ReflexivityAntisymetricTransitiveHomogenousAdditiveContinuity



Relationship between cones Relationship between cones and orders:and orders: Given a closed cone pointed V and a lineal space E, the relation a b defined as a-b V fulfills all the properties.

Example of cones:Example of cones:

Positives octants in Euclidean Spaces:

,n nE R V R

Lorentz cones:

2 21 1:

n

nn n n

E R

V L x R x x x

Positive semidefinite matrix cones.

E: Space of n x n symmetric matrix with the interior product of Frobenius.

V=S+n: Cone of positive semidefinite symmetrical matrix



Primal program (P)

min

. .

c x

s t

Ax b V

Dual program (D)

*

max

. .t

b y

s t

A y c y V

1. (P) and (D) are dual conic programs.

2. The duality gap is always positive: c x - b y 0 for every couple of feasible solutions (x, y)

3. When one of the problems (P) or (D) is limited and feasible, then the other one has a solutions and the optimal solution is the same

4. A couple of feasible solution (x, y) is composed by optimal solutions:

)

) 0

a b y c x

b Ax b y



Examples of conic programs:Examples of conic programs: Steiner Min-Max problem:

Weighted Steiner problem:

21

1

min

, ,

n

N

i ix Ri

nN

x b

b b R

1, , 2min maxn i N ix R

x b

General form of the duality in the conic General form of the duality in the conic programming:programming:

0 0

min

. .

i i i i ii

c x

s t

A x b

A x b V A x b

0 01

0 01

*

max

. .

0

l

i ii

l

i ii

i i i i

b y b y

s t

A y A y c

y V y

Vi if the family of convex cones.

Primal (P)

Dual (D)



Global optimization in Global optimization in bidimensional polynomials:bidimensional polynomials:

2

*

1 2

2 ,0 0,2

min ,

, 0

i jijR

i j n

n n

m q x y c x y

c c


0 ;́0 ' ' ´

with ´ .

i i', j j' i j n i j nm

n n

A necessary condition is that the values:

: 0 2ijm i j n

be moments is that the matrix:

be positive semidefinite

Semidefinite relaxation: Semidefinite relaxation:

1 2 '

'; ' 0 ';0 ' ' '

min

. .

0

ij iji j n

i i j j i j n i j n

c m

s a

m



We suppose that the non negative polynomial:

0, * myxq

can be expressed as:

*

2

1

' '' '

1 0 ' 0 ' ' '

,

,r

kk

rk k i i j j

ij i jk i j n i j n

q x y m

q x y

c c x y

Primal program (P)

1 2 '

00

', ' 0 ;0 ' ' '

min

. .

1

0

ij iji j n

i i j j i j n i j n

c m

s t

m

m

Dual program (D)

00

', ' 0 ';0 ' ' '

max

. .

, :1 2 '

0

ij ij

i i j j i j n i j n

s t

c i j i j n


Solution:Solution:

We take the coefficient of the expression:

'201

2* ,,nji

jiij

r

kk yxyxqmyxq

As the feasible solution for the problem (D) which value of the feasible function of (D) is the same with m*


The feasible solution for (D) give us a inferior cote for (P) and we have that the inferior cote is m* for the relaxation of the global problem:

*

'''0;'0';'

00

'21

*

S.D.P.

1

..

min

m

m

m

as

mc

m

njinjijjii

njiijij

Convex envelopesConvex envelopes

2 2 2 '

00 10 01

2', ' 0 ';0 ' ' '

min min

, , 1, ,

0

ij iji j nR

c

i i j j i j n i j n

q d c m

q a b a b m m a m b

mP R

00 10 01

', ' 0 ';0 ' ' '

max

. .

, : 2 2 '

0

ij ij

i i j j i j n i j n

a b

s a

c i j i j n

2

0

ni

ii

f t c t

Primal problem:

Dual problem:

We find the convex envelope of one-dimensional coercive polynomials given in the general form:


mincf t f s d s

The convex envelope in the point t is:

sd s t


1 2

*1 2t t

1 1 1 2 2 21, , 1, , 1, ,ct f t t f t t f t

*t


The optimal measure has two forms:

1.

2.

cf t f t

We use the truncated Hamburger moment problem and we transform the problem in a semidefinite problem:

2

0

0 1

, 0

min

. . 0, 1,

n

i im

i

n

i j i j

c m

s t H m m t

H m

We characterize the moments using the Hankel matrix.


2 41 2f t t t 2 4

2

2 3

2 3 4

min 1 2

1

. . 0

m m

t m

s t t m m

m m m

Example 1:Example 1:

*1 10.5 0.5


For t=0

For t=0.5*

1 10.7503 0.2497

For t=1 *1


2 3 47 5 3f t t t t t 1 2 3 4

2

2 3

2 3 4

min 7 5 3

1

. . 0

m m m m

t m

s t t m m

m m m



For t= 0*

2.7961 1.29670.3168 0.6832

For t= 2

*1.2967 2.7960.1945 0.8055


8 7 6 5 4 3 23 2.2 3 0.1 1.4f t t t t t t t t t



8 7 6 5 4 3 2 1

1 2 3 4

1 2 3 4 5

1 2 3 4 5 6

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

min 3 2.2 3 0.1 1.4

1

. . 0

m m m m m m m m

t m m m m

t m m m m m

m m m m m ms t

m m m m m m

m m m m m m

m m m m m m

For t= -0.5

*1.0840 0.55890.6445 0.3555

*

1.084 0 0.6445

0.5589 0.6445 0.6986 0.6445 1

x xx

x x

MicrostructuresMicrostructures

This method is used to determine the microstructure in unidimensional elastic bars, which deformation potential is non-convex:

1 ,00

0 s.a.

,,min1

0

uu

u

dxuxuxu

General problem: u is the displacement of

each point respect to the starting point.

u’ is the unitary deformation

f internal energy of deformation.

y potential of external forces.

Schematic curve of a typical potential of deformation for a steel




We make an analysis of general models where the non-convex dependence of in u’ can be written with a polynomial expression:

1 ,0

,,0

Kc

xcx

K

K

k

kk

1 ,00

0 s.a.

,,min1

0

uu

u

dxuxuxu

1 ,00

s.a.

, ,min

10:1

0

uu

dxu

dxuxdx

xv

x

xv

x



The original problem has a minimizer only if it is a Direc Delta

If the original problem does not have a minimizer, there is a region I where the parametrized measure is supported by two points. This solution determines the oscillation of the solutions of the problem.

* 0,1x s x x

1 2

*1 2

1 2

1

0

x s x s x

j

p x p x

p x p x x I

p x



1

00

1

0

2, 0

12

1 , 0

1

2, 0

1

21 , 0

min ,

s.t. x ,

0 0, 1 ,

1,

0,

y 0, for even

ó 0

y 0, for odd

K

k kk

K

i j i j

K

i j i j

K

i j i j

K

i j i j

c x m x x u dx

u x m

u u

m x

m x

m x K

m x

m x K

m

1 2

*1 2

1 2

1

0

x s x s x

j

p x p x

p x p x x I

p x

1

00

min , ,

s.a. 0,

0 0, 1

0,1

xv

x

x

x d x u dx

u x d

u u

x

MicrostructuresMicrostructures The new problem is an convex

optimization problem in the variable m, thus the existence of the minimizer is guaranteed

The non-lineal problem is in the restriction imposed by the moment characterization.

The way that the problem has taken an ideal form in order to solve it with software for non-linear programming.

The solution of the relaxed problem tell us wheater the original problem has solution or not.

K

kkkc mct

0

minm

21 21

*ssx pp


Convex envelopes 2DConvex envelopes 2D

Caratheodory theorem: Caratheodory theorem: Every point in a convex envelope of a coercive function f can be expressed as a convex combination that has r+1 points when the function is defined in r


,0 2

, i ji j

i j n

f x y c x y

Bidimensional polynomial:

The convex envelope can be defined as:

0

n

k kk

Min f t

And we define the probability distribution supported in t1, …,tn with weights 1,…, n,

1

0i

r

i ti


, mincf a b f d

, ,0 2

, min

. . 0

c i j i jm

i j n

f a b c m

s a M

( ) ( )im co


, ,i ji jm x y d x yWhere:

M: Restriction matrix that characterize the moments

For a fourth order polynomial:

, ,0 4

00 10 01 20 11 02

10 20 11 30 21 12

01 11 02 21 12 03

20 30 21 40 31 22

11 21 12 31 22 13

02 12 03 22 13 04

min

0

i j i ji j

c m

m m m m m m

m m m m m m

m m m m m m

m m m m m m

m m m m m m

m m m m m m

We compute the convex envelope in (a,b) solving the SPD program with m10=a , m01 =b and m00=1



We change the problem by a semidefinite program.

The solution of the semidefinite program has the moments of the optimal measure.

1 0M

Convex envelope calculus in (a,b)

We take marginal moments

2 0M CASE I

CASE II CASE III

NOYES

NOYES00 101

10 20

00 01

01 02

m mM

m m

m m

m m

00 10 20

2 10 20 30

20 30 40

00 01 02

01 02 03

02 03 04

m m m

M m m m

m m m

m m m

m m m

m m m


CASE I: Support: tx=a, ty=b Weight: =1

CASE II: Support:

tx=roots(P(tX)), ty=roots(P(ty))

00 10 20 00 01 02

10 20 30 01 02 032 2

( ) ( )

1 1X Y

X X Y Y

m m m m m m

P t m m m P t m m m

t t t t


*

,X Yt t

Weights:

2 11 2

2 1 2 1

2 11 2

2 1 2 1

X XX X

X X X X

Y YY Y

Y Y Y Y

t a a t

t t t t

t b b t

t t t t

1 2

1 2

*1 2

*1 2

X X

Y Y

X X t X t

Y Y t Y t


CASE III: Support:

tx=roots(P(tX)), ty=roots(P(ty)) Weights:

1 1

1 1 2 3 10 1 1 2 3 012 2 2 2 2 2

2 1 2 3 20 2 1 2 3 023 3 3 3 3 3

3 1 2 3 30 3 1 2 3 03

X X X X Y Y Y Y

X X X X Y Y Y Y

X X X X Y Y Y Y

t t t m t t t m

t t t m t t t m

t t t m t t t m


1 2 3

1 2 3

*1 2 3

*1 2 3

X X X

Y Y Y

X X t X t X t

Y Y t Y t Y t

00 10 20 30

10 20 30 40

20 30 40 502 3

00 01 02 03

01 02 03 04

02 03 04 052 3

1

1

X

X X X

Y

Y Y Y

m m m m

m m m mP t

m m m m

t t t

m m m m

m m m mP t

m m m m

t t t


We construct the measure for the polynomial f(x,y) in the point (0.5,0)

Marginal measures:

*2 1 0

*1 1 0

0.1667 0.1667 0.667

0.1667 0.1667 0.667

X

Y

Jointed measure:

*(2, 1) (1,1) (0,0)0.1667 0.1667 0.667



We construct the measure for the polynomial f(x,y) in the point (0,0.1)


Marginal measures:

*0.0469 0.9479

*0.9479 1.0465

0.9529 0.0471

0.9529 0.0471

X

Y

Jointed measure:

*( 0.0469,0.0531) (0.9479,1.0465)0.9529 0.0471

Optimal controlOptimal control


Non linear, optimal control problems with Bolza form or Mayer form:

1

0

0

min , , 1

. . , ,

0

u

n

f x t u F x

s t x g x t u

x x

x u

0

min 1

. . , ,

0

u

n

F x

s t x g x t u

x x

x u

BOLZA FORMBOLZA FORM MAYER FORMMAYER FORM

1

2

0

0

, , ,

, , ,

Nk

kk

Nk

kk

f x t u a x t u

g x t u c x t u


Linearity problems:

1. NON LINEAR: Integration. Stability Chaos

Convexity problems:Convexity problems:

2.2. NON CONVEX :NON CONVEX :• The Classical The Classical

Theory of Optimal Theory of Optimal Control does not Control does not apply for proving apply for proving the existence of the existence of the solutionthe solution

• Search Routines of Search Routines of Numerical Numerical Optimization fail Optimization fail to attain the global to attain the global optimum. optimum.



We introduce the linear and convex relaxation with moments.

1

2

1

00

0

0

min ,

. . ,

0

N

k km t

k

N

k kk

a x t m t

s t x c x t m t

x x

m: New variable of control

We use the probability moments


The Hamiltonian H has a polynomial form:

0

, ,N

kk

k

H x t p u

The global optimization

of a polynomial:

0

min , ,N

kk

uk

H u x t p u

, :R

co graph H u H u d P R


Theorem: Let H(u) be an even degree algebraic polynomial whose leader coefficient k is positive, we can express its convex hull as:

, :R

co graph H u H u d P R


Theorem: Assume that the Hamiltonian is a coercive polynomial with a single global minimum u*, then the optimization problem has an unique solution given by the Dirac measure u*

1

min min

min min

P R R

N

moments k kk

global H H u d

global H c m


When H(u) is a coercive polynomial with a single global minimum u*, the solution is the vector of moments m.

1

2

1

00

0

0

,

. . ,

0

0

0,1

minN

k km t k

N

k kk

a x t m t dt

s t x c x t m t

x x

H m

for every t


* * k

km u

1

2

1

00

0

0

2( ) 0, 0

,

. . ,

0

0 1

0,1

minN

k km t k

N

k kk

K

i j i j

a x t m t dt

s t x c x t m t

x x

m t with m t

for every t

0 1 2

1 2 3 1

2 3

1 2

N

N

N N N

m m m m

m m m m

H m m m

m m m

Hankel Positive semi definite



Discretization of the Discretization of the problem.problem.

1

2

11

,0 0

1

0

0

/ 2

, 0

min

. . 1,...,

1

0

a r h Nv

k km x

r ka rh

Nr r

k kk

K

i j i j

a m r dt

x xs t c m r r v

h

m r

m r

1

2

1

00

0

0

2( ) 0, 0

,

. . ,

0

0 1

0,1

minN

k km t k

N

k kk

K

i j i j

a x t m t dt

s t x c x t m t

x x

m t with m t

for every t


1

2

0

2

10 |

. .

(0) 0

Min x t dt

s t x u ux x

x

1

2

0

2 1

1

1 2

10

. .

10

(0) 0

Min x t dt

s t x m m x x

m

m m

x

22

11

12 1

10

1 2

10 10 12

. .

10 0

N

r rr

r rr r

hMin x rh x r h

x xs t m r m r x x

hm

xm m

t vs X

Control signal





* * ?k

km u



10

22

0

2

|

. .

(0) 0

Min x t dt

s t x u ux x

x

10

22

0

2 1

0 1

1 2

. .

0

(0) 0

Min x t dt

s t x m m x x

m m

m m

x

22 22

11

12 1

0 10

1 2

12

. .

0 0

(0) 0

N

r rr

r rr r

hMin x rh x r h

x xs t m r m r x x

hm m

xm m

x

t vs X

Control signal




12 2

0

22 2 2

min 1 0.7

. .

11

2

0 0 0 0

u tx y dt

s t x u

y u u x u

x y


12 2

,0

1

2 4 6 2

1 2 3

1 2 3 4

2 3 4 5

3 4 5 6

min 1 0.7

. .

22

1

0

0 0 0 0

m xx y dt

s t x m

xy m m m xm

m t m t m t

m t m t m t m t

m t m t m t m t

m t m t m t m t

x y

t vs X

t vs Y

Control signal



12

0

22

min 1

. .

0 0

1

u tu x t dx x

s t x u

x

u u

12

2 4

0

1

min 1 2 1

. .

0 0

u tm m x t dx x

s t x m

x


THERE IS NO MINIMIZERS

The method of moments in dynamic optimization R. Meziat Departamento de Matemáticas Universidad de...

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