Non convex Non convex variational problems variational problems under restrictions under restrictions on the derivativeon the derivativeR. Meziat, C. Rivera, D. Patiño

Departamento de MatemáticasUniversidad de los AndesColombia, 2005

SIAM Conference on OptimizationSIAM Conference on Optimization

Stockholm, May 2005Stockholm, May 2005

Introduction

Analysis of the problem

Problem of the moments

Examples

IndexIndex

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

IntroductionIntroduction

b b

b a

a a

x dx y y x dx

0

, , ,n

kk

k

f x y x c x y

We will analyze the variational problem:

min , , '

. .

, ' , [ , ]

,

b

y

a

a b

I y x f x y x y x dx

s t

x y x y x x y x x a b

y a y y b y

The integrand f is given by the general expression:

f is a polynomial in the derivative y’ and and are integrable:

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

IntroductionIntroduction

' x

R

y x d

We use Young measure:

min , ,

. .

[ , ] [ , ]

,

b

y xRa

x

a b

I y x f x y x d dx

s t

x x x a b

y a y y b y

And the link between y and v is given by:

Then we can use the moment theory for the parametrized measure and the boundary conditions

The task is to determine a probability measures:

:x a x b

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Analysis of the problemAnalysis of the problem

We take the original problem and we transform it into a problem including its derivative constraints. The new integrand is:

0

, , ,, ,

pk

kk

c x y if x y Af x y

otherwise

We may use the one dimensional Solobev space H1,p(a,b) as the family of admissible functions where p is the degree of the polynomial. A is the set of values (x,y,)

satisfying the inequalities

min , , '

. .

b

y

a

a b

I y x f x y x y x dx

s t y a y y b y

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Analysis of the problemAnalysis of the problem

min , ,

. .

'

,

b

Y xRa

x

R

a b

I f x y x d dx

s t

y x d

y a y y b y

We use the Direct Method if the Integrand is coercive and semi-continuous functional:

1,0 , b ap

a

y yH a b y x a

b a

We assume for the problem:

I u f u x dx

Every minimizing sequence un has a weakly convergent subsequence and we characterize the weak limit of the sequence:

x

R

x f d

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Analysis of the problemAnalysis of the problem

minY

xR

I f dx

*x g x

1 2

*1 2

1 2 1, 0

x g x g x

i

p x p x

p x p x p x

We can determine minimizing sequences for the functional I by solving the problem:

The measure could be composed by one delta:

Or two deltas:

Then:

*

1 1 2 2

x

R

x f d

p x f g x p x f g x

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Analysis of the problemAnalysis of the problem

I

*, , , ,C x

R

f x y x f x y x dx

x

a x

a R

y x y dx

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

min infXY

I I

Then the problem can be relaxed:

The generalized functional

satisfies:

1.

2. The support of every optimal parametrized measure must be contained in the set of points satisfies:

, , , ,cf x y x f x y x

3. The problem has a minimizer if the Young measures is composed of Dirac measures:

*x

a

a

y x y g s ds

Problem of the momentsProblem of the moments

1

0

0

1

1 2

min '

. .

0 ,

1 ,

'

u C

I u f u x dx

s t

u u

u u

c x u x c x

[ , ] : , 1

bn i

a b i

a

M m m t d i n

Problem of the moments of Problem of the moments of Hausdorff:Hausdorff: We characterize the moments of a probability measure supported in the interval [l,)

We define de cone of the moments of measures in the interval [a,b]:

And the cone of the positives function in [a,b] as:

[ , ]1

: 0 [ , ]n

n ia b i

i

P c c t t a b

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Problem of the momentsProblem of the moments

,

1

1 2 , 0

0

0

n

i j i j

n

i j i j i j i j

m

a b m abm m

222 1

0 0 0

0

[ , ]

n n ni i j

i i ji i j

f t c t t t a b t t

t a b

1 2

0 0 0 0

[ ] 0n n n n

i j i j i j i ji j i j

i j i j

t a b t abt t

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

The vector [m0, …,m2n] is in the closure of the cone of algebraic moments of the positive measure in [a,b].

And the following matrix are positive semidefinites.

0c m

t m

Problem of the momentsProblem of the moments

1

0

1 1 2

00 0

inf ' min

min min

u C

n

x k km

k

I u f u x dx I

I f d dx c m x

Theorem of relaxation: We use this theorem for changing the problem and characterize the measure with moments. We transform the problem in a conical program:

1 2,

1

1 2 1 2, 0

. . 0 [ , ]

0 ,

n

i j i j

n

i j i j i j i j

s t m x x c x c x

a b m x abm x m x a c x b c x

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Problem of the momentsProblem of the moments

1 2

1 2

1, 0x x

a x a x x

2

2 32

1

1

x m x

P t x m x m x

t t

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Measure:

If m2(x)=x2, then:

If m2(x)>x2, the values of a1(x), a2(x):

The weights of the probability measure are:

21

2 1

12

2 1

1 2

a x xx

a x a x

x a xx

a x a x

a x a x

Example 1Example 1

1 22 2

0

min 1 '

. . 0 1

'

u t u t dt

s t u a u b

u t

We take the problem:

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

1

2 4

0 0

1 2

1 2 3

2 3 4

1 1 2

1 2 2 3

min 1 2

. . 0 1

1

0

0

t

m m u s ds dt

s t u a u b

m m

m m m

m m m

m m m

m m m m

Example 1Example 1

a=0, b=1, =0.5

0.9469 0.5*

0.9469 0.5

0.3333 0.6666 [0,0.4)

0.2462 0.7538 [0.4,1]

if tt

if t

*0.9485 0.50.3456 0.65544

a=b=0 =0

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 1Example 1

a=0.5, b=0, =0.5

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 2Example 2

1

2 4

0 0

1 2

1 2 3

2 3 4

1 2

2 1 3 2

min 1 2

. . 0 1

1

0

0

t

m m u s ds dt

s t u a u b

m m

m m m

m m m

m m m

m m m m

*0.1 0.84840.9 0.1

1 22 2

0

min 1 '

. . 0 1

'

u t u t dt

s t u a u b

u t

We change the problem to: a=b=0 =-0.1

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 2Example 2 a=b=0 =-x/2

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 3Example 3

14 3 2 2

0

min 0.5 ' 0.8 ' 0.5 ' 0.5 '

. . 0 1

'

u t u t u t u t u t dt

s t u a u b

u t

We take the problem:

21

4 3 2 1

0 0

1 2

1 2 3

2 3 4

1 2 1

2 1 3 2

min 0.5 0.8 0.5 0.5

. . 0 1

1

0

0

t

m m m m u s ds dt

s t u a u b

m m

m m m

m m m

m m m

m m m m

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 3Example 3 a=0 b=0.1 =0

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

Example 4Example 4

1 2 22

0

min 1 '

. . 0 1

'

u t u t g t dt

s t u a u b

u t

21

2 4

0 0

1 2

1 2 3

2 3 4

1 2 1 2 3

1 2 3 2 3 4

min 1 2

. . 0 1

1

0

0

t

m m u s ds g t dt

s t u a u b

m m

m m m

m m m

m m m m m

m m m m m m

We analyse the problem:

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

a=0 b=0.1 =-0.5 =0.5

g(t)=t2

Example 4Example 4

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

a=0.5 b=0.2 =-0.5 =0.5

g(t)=t/4

Example 5Example 5

21

2 4 6

0 0

1 2 3

1 2 3 4

2 3 4 5

3 4 5 6

1 1 2 2 3

1 2 2 3 3 4

2 3 3 4 4 5

min 3.5 4

. . 0 1

1

0

0

t

m m m u s ds dt

s t u a u b

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

*0.0314 0.23210.8821 0.1179

Non convex variational problems under restrictions on the derivativeNon convex variational problems under restrictions on the derivative R. Meziat, C. Rivera. 2005.

12 4 6 2

0

min 3.5 ' 4 ' '

. . 0 1

'

u t u t u t u t dt

s t u a u b

u t

We take the problem: a=0 b=0 =0.2