1. The method of moments in dynamic optimization
Mathematics Seminar. Institute of Mathematics Charles University. Prague, May 2005
R. Meziat
Departamento de Matemáticas
Universidad de los Andes
Colombia, 2005

2. Index Conic programming. Convex envelopes Microstructures
Optimal control
The method of moments in dynamic optimization, R. Meziat, 2005.

3. Conic programming Matrix of m rows and n columns:
P and V are closed convex cones in Rn
Polar cone:
System of homogenous inequalities:
Basic results:
Positive cone generated by the rows of the matrix A
The method of moments in dynamic optimization, R. Meziat, 2005.

4. Conic programming Farkas Lemma:
Theorem of the alternative I: Ax=b has a solution in x 0 of Rn or excluding:
has solution in
The method of moments in dynamic optimization, R. Meziat, 2005.

5. Conic programming
Theorem of the alternative II: Ax b has a solution in x 0 of Rn or excluding:
Theorem of the alternative III: Ax b has a solution in x  Rn or excluding:
has solution in:
has solution in:
The method of moments in dynamic optimization, R. Meziat, 2005.

6. Conic programming Corollary: Consequences of the Farkas Lemma:
Kuhn-Tucker conditions in programs under restrictions in form of inequality.
Duality in convex programming.
The method of moments in dynamic optimization, R. Meziat, 2005.

7. Conic programming
Duality gap: Given a couple of feasible solutions (x,y)
Primal program (P)
The system of lineal inequalities:
Dual program (D)
Has a solution (y,x,t)  Rm x Rn x R1 that satisfies:
The method of moments in dynamic optimization, R. Meziat, 2005.

8. Conic programming One of the following affirmations is always truth:
There is a couple of optimal solutions for the primal (P) and dual (D) that satisfies:
One of the problems is not limited and it is not feasible.
One couple of feasible solutions is a couple of optimal solution if the complementariness relations are fulfilled:
The method of moments in dynamic optimization, R. Meziat, 2005.

9. Conic programming
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:
Reflexivity
Antisymetric
Transitive
Homogenous
Additive
Continuity
The method of moments in dynamic optimization, R. Meziat, 2005.

10. Conic programming
Relationship between cones 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:
Positives octants in Euclidean Spaces:
Lorentz cones:
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
The method of moments in dynamic optimization, R. Meziat, 2005.

11. Conic programming Primal program (P)
(P) and (D) are dual conic programs.
The duality gap is always positive: c x - b y  0 for every couple of feasible solutions (x, y)
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
A couple of feasible solution (x, y) is composed by optimal solutions:
Dual program (D)
The method of moments in dynamic optimization, R. Meziat, 2005.

12. Conic programming Examples of conic programs:
Steiner Min-Max problem:
Weighted Steiner problem:
General form of the duality in the conic programming:
Vi if the family of convex cones.
Primal (P)
Dual (D)
The method of moments in dynamic optimization, R. Meziat, 2005.

13. Conic programming Global optimization in bidimensional polynomials:
be positive semidefinite
Semidefinite relaxation:
A necessary condition is that the values:
be moments is that the matrix:
The method of moments in dynamic optimization, R. Meziat, 2005.

14. Conic programming Primal program (P)
We suppose that the non negative polynomial:
Dual program (D)
can be expressed as:
The method of moments in dynamic optimization, R. Meziat, 2005.

15. Conic programming Solution: We take the coefficient of the expression:
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:
As the feasible solution for the problem (D) which value of the feasible function of (D) is the same with m*
The method of moments in dynamic optimization, R. Meziat, 2005.

16. Convex envelopes
We find the convex envelope of one-dimensional coercive polynomials given in the general form:
Primal problem:
The convex envelope in the point t is:
Dual problem:
The method of moments in dynamic optimization, R. Meziat, 2005.

17. Convex envelopes
We use the truncated Hamburger moment problem and we transform the problem in a semidefinite problem:
The optimal measure has two forms:
We characterize the moments using the Hankel matrix.
The method of moments in dynamic optimization, R. Meziat, 2005.

18. Convex envelopes Example 1: For t=0 For t=0.5 For t=1
The method of moments in dynamic optimization, R. Meziat, 2005.

19. Convex envelopes Example 2: For t= 0 For t= 2
The method of moments in dynamic optimization, R. Meziat, 2005.

20. Convex envelopes Example 3: For t= -0.5
The method of moments in dynamic optimization, R. Meziat, 2005.

21. Microstructures 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.
This method is used to determine the microstructure in unidimensional elastic bars, which deformation potential is non-convex:
Schematic curve of a typical potential of deformation for a steel
The method of moments in dynamic optimization, R. Meziat, 2005.

22. Microstructures
We make an analysis of general models where the non-convex dependence of  in u’ can be written with a polynomial expression:
The method of moments in dynamic optimization, R. Meziat, 2005.

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

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

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

26. Convex envelopes 2D
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
The convex envelope can be defined as:
And we define the probability distribution supported in t1, …,tn with weights 1,…, n,
Bidimensional polynomial:
The method of moments in dynamic optimization, R. Meziat, 2005.

27. Convex envelopes 2D For a fourth order polynomial: Where:
We compute the convex envelope in (a,b) solving the SPD program with m10=a , m01 =b and m00=1
M: Restriction matrix that characterize the moments
The method of moments in dynamic optimization, R. Meziat, 2005.

28. Convex envelopes 2D We change the problem by a semidefinite program.
The solution of the semidefinite program has the moments of the optimal measure.
Convex envelope
calculus in (a,b)
We take
marginal moments
CASE I
CASE II
CASE III NO
YES
The method of moments in dynamic optimization, R. Meziat, 2005.

29. Convex envelopes 2D Weights: CASE I: Support: tx=a, ty=b Weight: =1
CASE II:
Support:
tx=roots(P(tX)), ty=roots(P(ty))
The method of moments in dynamic optimization, R. Meziat, 2005.

30. Convex envelopes 2D CASE III: Support:
tx=roots(P(tX)), ty=roots(P(ty))
Weights:
The method of moments in dynamic optimization, R. Meziat, 2005.

31. Convex envelopes 2D Example 1:
The method of moments in dynamic optimization, R. Meziat, 2005.

32. Convex envelopes 2D Example 1:
The method of moments in dynamic optimization, R. Meziat, 2005.

33. Convex envelopes 2D
We construct the measure for the polynomial f(x,y) in the point (0.5,0)
Marginal measures:
Jointed measure:
The method of moments in dynamic optimization, R. Meziat, 2005.

34. Convex envelopes 2D
We construct the measure for the polynomial f(x,y) in the point (0,0.1)
Marginal measures:
Jointed measure:
The method of moments in dynamic optimization, R. Meziat, 2005.

35. Optimal control BOLZA FORM MAYER FORM
Non linear, optimal control problems with Bolza form or Mayer form:
BOLZA FORM
MAYER FORM
The method of moments in dynamic optimization, R. Meziat, 2005.

36. Optimal control Linearity problems: Convexity problems:
1. NON LINEAR:
Integration.
Stability
Chaos
Convexity problems:
2. NON CONVEX :
The Classical Theory of Optimal Control does not apply for proving the existence of the solution
Search Routines of Numerical Optimization fail to attain the global optimum.
The method of moments in dynamic optimization, R. Meziat, 2005.

37. Optimal control
We introduce the linear and convex relaxation with moments.
The Hamiltonian H has a polynomial form:
The global optimization of a polynomial:
m: New variable of control
We use the probability moments
The method of moments in dynamic optimization, R. Meziat, 2005.

38. Optimal control
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*
Theorem: Let H(u) be an even degree algebraic polynomial whose leader coefficient k is positive, we can express its convex hull as:
The method of moments in dynamic optimization, R. Meziat, 2005.

39. Optimal control
When H(u) is a coercive polynomial with a single global minimum u*, the solution is the vector of moments m.
Hankel Positive semi definite
The method of moments in dynamic optimization, R. Meziat, 2005.

40. Optimal control Discretization of the problem.
The method of moments in dynamic optimization, R. Meziat, 2005.

41. Optimal control Example 1: t vs X Control signal
The method of moments in dynamic optimization, R. Meziat, 2005.

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

43. Optimal control Example 2: Control signal t vs X
The method of moments in dynamic optimization, R. Meziat, 2005.

44. Optimal control Example 3: t vs X t vs Y Control signal
The method of moments in dynamic optimization, R. Meziat, 2005.

45. Optimal control Example 4: THERE IS NO MINIMIZERS
The method of moments in dynamic optimization, R. Meziat, 2005.