The method of moments in dynamic optimization

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
P and V are closed convex cones in Rn
Matrix of m rows and n columns
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 Axb 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.
• VSn 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 t0
For t0.5
For t1
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 r1
  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 m10a , m01 b and m001
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.

The method of moments in dynamic optimization, R.
Meziat, 2005.
29
Convex envelopes 2D

• Weights

• CASE I
• Support txa, tyb
• Weight ?1
• CASE II
• Support
• txroots(P(tX)), tyroots(P(ty))

The method of moments in dynamic optimization, R.
Meziat, 2005.
30
Convex envelopes 2D

• CASE III
• Support
• txroots(P(tX)), tyroots(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

• 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
• 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.