1. Introduction
In this section we establish a dual formulation for a large class of models in non-convex optimization.
The main duality principle is applied to double well models similar as those found in the phase transition theory.
Such results are based on the works of J.J. Telega and W.R. Bielski [
2,
3,
15,
16] and on a D.C. optimization approach developed in Toland [
17].
About the other references, details on the Sobolev spaces involved are found in [
1]. Related results on convex analysis and duality theory are addressed in [
5,
7,
8,
10,
14].
Finally, in this text we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated.
In order to clarify the notation, here we introduce the definition of topological dual space.
Definition 1.1 (Topological dual spaces).
Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined on U. We suppose such a dual space of U, may be represented by another Banach space , through a bilinear form (here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given linear and continuous, we assume the existence of a unique such that
The norm of f , denoted by , is defined as
At this point we start to describe the primal and dual variational formulations.
2. A general duality principle non-convex optimization
In this section we present a duality principle applicable to a model in phase transition.
This case corresponds to the vectorial one in the calculus of variations.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
Consider a functional
where
and where
and
We assume there exists
such that
Moreover, suppose F and G are Fréchet differentiable but not necessarily convex. A global optimum point may not be attained for J so that the problem of finding a global minimum for J may not be a solution.
Anyway, one question remains, how the minimizing sequences behave close the infimum of J.
We intend to use duality theory to approximately solve such a global optimization problem.
Denoting
,
,
,
at this point we define,
,
,
,
and
by
and
and
Define now
,
.
From the general results in [
17], we may infer that
From these last two results we may obtain
Moreover, from standards results on convex analysis, we may have
where
and
Thus, defining
we have got
This last variational formulation corresponds to a concave relaxed formulation in concerning the original primal formulation.
3. Another duality principle for a simpler related model in phase transition with a respective numerical example
In this section we present another duality principle for a related model in phase transition.
Let
and consider a functional
where
and where
and
A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.
Anyway, one question remains, how the minimizing sequences behave close the infimum of J.
We intend to use duality theory to approximately solve such a global optimization problem.
Denoting
, at this point we define,
and
by
and
In order to restrict the action of
on the region where the primal functional is non-convex, we redefine a not relabeled
and define also
and
by
and
Denoting
we also define the polar functional
by
With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by
, where
Having defined such a functional, we may obtain numerical results by solving a sequence of convex auxiliary sub-problems, through the following algorithm (in order to obtain the concerning critical points, at first we have neglected the constraint ).
Set and and
Choose such that and
Set
Calculate
solution of the system of equations:
and
that is
and
so that
and
Calculate
by solving the system of equations:
and
that is
and
If , then stop, else set and go to item 4.
At this point, we present the corresponding software in MAT-LAB, in finite differences and based on the one-dimensional version of the generalized method of lines.
Here the software.
***********************
-
clear all
m8=300;
d=1/m8;
K=0.1;
K1=120;
for i=1:m8
vo(i,1)=i*d/10;
yo(i,1)=sin(i*d*pi)/2;
end;
k=1;
b12=1.0;
while and
k=k+1;
for i=1:m8-1
duo(i,1)=(uo(i+1,1)-uo(i,1))/d;
dvo(i,1)=(vo(i+1,1)-vo(i,1))/d;
end;
m9=zeros(2,2);
m9(1,1)=1;
i=1;
m80(1,1,i)=-f1-K;
m80(1,2,i)=-f1;
m80(2,1,i)=-f1;
m80(2,2,i)=-f1-K1;
m50(:,:,i)=m80(:,:,i)*inv(m12);
z(:,i)=inv(m12)*y11(:,i)*;
for i=2:m8-1
;
m80(1,1,i)=-f1-K;
m80(1,2,i)=-f1;
m80(2,1,i)=-f1;
m80(2,2,i)=-f1-K1;
m50(:,:,i)=inv(m12)*m80(:,:,i);
end;
U(1,m8)=1/2;
U(2,m8)=0.0;
for i=1:m8-1
U(:,m8-i)=m50(:,:,m8-i)*U(:,m8-i+1)+z(:,m8-i);
end;
for i=1:m8
u(i,1)=U(1,i);
v(i,1)=U(2,i);
end;
b12=max(abs(u-uo))
uo=u;
vo=v;
u(m8/2,1)
end;
for i=1:m8
y(i)=i*d;
end;
plot(y,uo)
**************************************
For the case in which
, we have obtained numerical results for
and
. For such a concerning solution
obtained, please see
Figure 1. For the case in which
, we have obtained numerical results also for
and
. For such a concerning solution
obtained, please see
Figure 2.
Remark 3.1. Observe that the solutions obtained are approximate critical points. They are not, in a classical sense, the global solutions for the related optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.
3.1. A general proposal for relaxation
Let be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by
Consider a functional
where
where
,
is convex and Fréchet differentiable, and
where
is also Fréchet differentiable.
Assume there exists
such that
where for each
is an open connected set such that
is regular. We also suppose
At this point we define
and
where
Moreover, we propose the relaxed functional
4. A convex dual variational formulation for a third similar model
In this section we present another duality principle for a third related model in phase transition.
Let
and consider a functional
where
and where
and
A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.
Anyway, one question remains, how the minimizing sequences behave close to the infimum of J.
We intend to use the duality theory to solve such a global optimization problem in an appropriate sense to be specified.
At this point we define,
and
by
and
Denoting
we also define the polar functional
and
by
and
Observe this is the scalar case of the calculus of variations, so that from the standard results on convex analysis, we have
Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some .
Moreover, the corresponding solution
is obtained from the equation
Finally, the Euler-Lagrange equations for the dual problem stands for
where
if
if
and
if
We have computed the solutions and corresponding solutions for the cases in which and
For the solution
for the case in which
, please see
Figure 3.
For the solution
for the case in which
, please see
Figure 4.
Remark 4.1. Observe that such solutions obtained are not the global solutions for the related primal optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.
4.1. The algorithm through which we have obtained the numerical results
In this subsection we present the software in MATLAB through which we have obtained the last numerical results.
This algorithm is for solving the concerning Euler-Lagrange equations for the dual problem, that is, for solving the equation
Here the concerning software in MATLAB. We emphasize to have used the smooth approximation
where a small value for
is specified in the next lines.
*************************************
clear all
(number of nodes)
-
-
-
(we have fixed the number of iterations)
-
-
-
-
-
********************************
5. An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation
In this section we develop an approximate primal dual formulation suitable for a large class of variational models.
Here, the applications are for the Kirchhoff-Love plate model, which may be found in Ciarlet, [
11].
At this point we start to describe the primal variational formulation.
Let be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of , which is assumed to be regular (Lipschitzian), is denoted by . The vectorial basis related to the cartesian system is denoted by , where (in general Greek indices stand for 1 or 2), and where is the vector normal to , whereas and are orthogonal vectors parallel to Also, is the outward normal to the plate surface.
The displacements will be denoted by
The Kirchhoff-Love relations are
Here
so that we have
where
It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.
We also define the operator
, where
, by
The constitutive relations are given by
where:
and
, are symmetric positive definite fourth order tensors. From now on, we denote
and
.
Furthermore
denote the membrane force tensor and
the moment one. The plate stored energy, represented by
is expressed by
and the external work, represented by
, is given by
where
are external loads in the directions
,
and
respectively. The potential energy, denoted by
is expressed by:
Define now
by
where
In such a case for
,
,
in
and
we get
and
This new functional has a relevant improvement in the convexity conditions concerning the previous functional J.
Indeed, we have obtained a gain in positiveness for the second variation which has increased of order
Moreover the difference between the approximate and exact equation
is of order
which corresponds to a small perturbation in the original equation for a load of
for example. Summarizing, the exact equation may be approximately solved in an appropriate sense.
6. An exact convex dual variational formulation for a non-convex primal one
In this section we develop a convex dual variational formulation suitable to compute a critical point for the corresponding primal one.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
Consider a functional
where
and
Here we denote and
Defining
for some appropriate
, suppose also
F is twice Fréchet differentiable and
Define now
and
by
and
where here we denote
Moreover, we define the respective Legendre transform functionals
and
as
where
are such that
and
where
are such that
Here
is any function such that
Observe that through the target conditions
we may obtain the compatibility condition
Define now
for some appropriate
such that
is convex in
Consider the problem of minimizing subject to
Assuming
is large enough so that the restriction in
r is not active, at this point we define the associated Lagrangian
where
is an appropriate Lagrange multiplier.
The optimal point in question will be a solution of the corresponding Euler-Lagrange equations for
From the variation of
in
we obtain
From the variation of
in
we obtain
From the variation of
in
we have
From this last equation, we may obtain
such that
and
From this and the previous extremal equations indicated we have
and
so that
and
From this and equation (
26) and (
27) we have
Replacing the expressions of
and
into this last equation, we have
so that
Observe that if
then there exists
such that
u and
are also such that
and
The boundary conditions for must be such that
From this and equation (
29) we obtain
Summarizing, we may obtain a solution of equation by minimizing on .
Finally, observe that clearly is convex in an appropriate large ball for some appropriate
7. Another primal dual formulation for a related model
Let be an open, bounded and connected set with a regular boundary denoted by
Consider the functional
where
,
and
Denoting
, define now
by
Define also
and
for some appropriate
to be specified.
Moreover define
for some appropriate
to be specified.
Observe that, denoting
we have
and
so that
Observe now that a critical point and in .
Therefore, for an appropriate large
, also at a critical point, we have
Remark 7.1. From this last equation we may observe that has a large region of convexity about any critical point , that is, there exists a large such that is convex on
With such results in mind, we may easily prove the following theorem.
Theorem 7.2.
Assume and suppose is such that
Under such hypotheses, there exists such that is convex in ,
8. A third primal dual formulation for a related model
Let be an open, bounded and connected set with a regular boundary denoted by
Consider the functional
where
,
and
Denoting
, define now
by
where
is a small real constant.
Define also
and
for some appropriate
to be specified.
Moreover define
and
for some appropriate real constants
to be specified.
Remark 8.1.
For an appropriate function (or, in a more general fashion, an appropriate bounded operator) define
for some small parameter
Since for we have , so that for we have
we may infer that is a convex set.
Moreover if , then
so that
Such a result we will be used many times in the next sections.
Observe that, defining
we may obtain
and
so that
However, at a critical point, we have
so that, for a fixed
we define the non-active but convex restriction
for a small parameter
From such results, assuming
and
, we have that
for
and
With such results in mind, we may easily prove the following theorem.
Theorem 8.2.
Suppose is such that
Under such hypotheses, we have that
and
Proof. The proof that
and
may be easily made similarly as in the previous sections.
Moreover, observe that for
sufficiently large, we have
so that this and the other hypotheses, we have also
and
From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that
From such results, we may infer that
The proof is complete. □
9. An algorithm for a related model in shape optimization
The next two subsections have been previously published by Fabio Silva Botelho and Alexandre Molter in [
5], Chapter 21.
9.1. Introduction
Consider an elastic solid which the volume corresponds to an open, bounded, connected set, denoted by
with a regular (Lipschitzian) boundary denoted by
where
Consider also the problem of minimizing the functional
where
subject to
Here
denotes the outward normal to
and
where
and
denotes the Lebesgue measure of
Moreover is the field of displacements relating the cartesian system , resulting from the action of the external loads and
We also define the stress tensor
by
and the strain tensor
by
Finally,
where
corresponds to a strong material and
to a very soft material, intending to simulate voids along the solid structure.
The variable t is the design one, which the optimal distribution values along the structure are intended to minimize its inner work with a volume restriction indicated through the set B.
The duality principle obtained is developed inspired by the works in [
2,
3]. Similar theoretical results have been developed in [
10], however we believe the proof here presented, which is based on the min-max theorem is easier to follow (indeed we thank an anonymous referee for his suggestion about applying the min-max theorem to complete the proof). We highlight throughout this text we have used the standard Einstein sum convention of repeated indices.
Moreover, details on the Sobolev spaces addressed may be found in [
1]. In addition, the primal variational development of the topology optimization problem has been described in [
10].
The main contributions of this work are to present the detailed development, through duality theory, for such a kind of optimization problems. We emphasize that to avoid the check-board standard and obtain appropriate robust optimized structures without the use of filters, it is necessary to discretize more in the load direction, in which the displacements are much larger.
9.2. Mathematical formulation of the topology optimization problem
Our mathematical topology optimization problem is summarized by the following theorem.
Theorem 9.1. Consider the statements and assumptions indicated in the last section, in particular those refereing to Ω and the functional
Define by
where
and where
Define also by
Assume there exists such that
and
Finally, define by
where
where
and
Under such hypotheses, there exists such that
where
and where
and
Proof.
Also, from this and the min-max theorem, there exist
such that
Finally, from the extremal necessary condition
we obtain
and
so that
Hence so that and
This completes the proof. □
9.3. About a concerning algorithm and related numerical method
For numerically solve this optimization problem in question, we present the following algorithm
Set and .
Calculate
such that
Calculate
such that
If or then stop, else set and go to item 2.
We have developed a software in finite differences for solving such a problem.
Here the software.
**************************************
-
clear all
global P m8 d w u v Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3
m8=27;
m9=24;
c3=0.95;
d=1.0/m8;
d1=0.5/m9;
Ea=; (stronger material)
Eb=1000; (softer material simulating voids)
w=0.30;
P=-42000000;
z1=(m8-1)*(m9-1);
A3=zeros(z1,z1);
for i=1:z1
A3(1,i)=1.0;
end;
b=zeros(z1,1);
uo=0.000001*ones(z1,1);
u1=ones(z1,1);
b(1,1)=c3*z1;
for i=1:m9-1
for j=1:m8-1
Lo(i,j)=c3;
end; end;
for i=1:z1
x1(i)=c3*z1;
end;
for i=1:2*m8*m9
xo(i)=0.000;
end;
xw=xo;
xv=Lo;
for k2=1:24
c3=0.98*c3;
b(1,1)=c3*z1;
k2
b14=1.0;
k3=0;
while and
k3=k3+1;
b12=1.0;
k=0;
while and
k=k+1;
k2
k3
k
X=fminunc(’funbeam’,xo);
xo=X;
b12=max(abs(xw-xo));
xw=X;
end;
for i=1:m9-1
for j=1:m8-1
ex=du1(i,j);
ey=dv2(i,j);
exy=1/2*(dv1(i,j)+du2(i,j));
Sxy=E1/(2*(1+w))*exy;
dc3(i,j)=-(Sx*ex+Sy*ey+2*Sxy*exy);
end;
end;
for i=1:m9-1
for j=1:m8-1
f(j+(i-1)*(m8-1))=dc3(i,j);
end;
end;
for k1=1:1
k1
X1=linprog(f,,,A3,b,uo,u1,x1);
x1=X1;
end;
for i=1:m9-1
for j=1:m8-1
Lo(i,j)=X1(j+(m8-1)*(i-1));
end;
end;
b14=max(max(abs(Lo-xv)))
xv=Lo;
colormap(gray); imagesc(-Lo); axis equal; axis tight; axis off;pause(1e-6)
end;
end;
****************************************************
Here the auxiliary Function ’funbeam’
function S=funbeam(x)
global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2
for i=1:m9
for j=1:m8
u(i,j)=x(j+(m8)*(i-1));
v(i,j)=x(m8*m9+(i-1)*m8+j);
end;
end;
for i=1:m9
end;
u(m9-1,1)=0;
v(m9-1,1)=0;
u(m9-1,m8-1)=0;
v(m9-1,m8-1)=0;
for i=1:m9-1
for j=1:m8-1
du1(i,j)=(u(i,j+1)-u(i,j))/d;
du2(i,j)=(u(i+1,j)-u(i,j))/d1;
dv1(i,j)=(v(i,j+1)-v(i,j))/d;
dv2(i,j)=(v(i+1,j)-v(i,j))/d1;
end;
end;
S=0;
for i=1:m9-1
for j=1:m8-1
ex=du1(i,j);
ey=dv2(i,j);
exy=1/2*(dv1(i,j)+du2(i,j));
Sxy=E1/(2*(1+w))*exy;
S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);
end;
end;
S=S*d*d1-P*v(2,(m8)/3)*d*d1;
***********************************************
For a two dimensional beam of dimensions and we have obtained the following results:
10. A duality principle for a general vectorial case in the calculus of variations
In this section we develop a duality principle for a general vectorial case in variational optimization.
Let
be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by
. Let
be a functional where
where
and
Here we have denoted
and
so that we may also denote
Assume
where
is a differentiable function such that
as
. Moreover, suppose there exists
such that
Under some mild hypotheses, from convexity, we have that
where
Now observe that the restriction
for some
is equivalent to the restriction
where
,
with appropriate boundary conditions, so that with an appropriate Lagrange multiplier
, we obtain
where we have denoted
and
Joining the pieces, we have got
where we recall that
We emphasize such a dual formulation in is convex (in fact concave).
11. A note on the Galerkin Functional
Let be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by .
Consider the functional
where
Here ,
At this point we define
for some appropriate real constant
and
Observe that
so that we define the Galerkin functional
by
At this point, for an appropriate small real constant
and bounded constant operator
, we set the intended non-active restriction
and define
Observe that since for
we have
in
so that if
then
we may infer that
is a convex set.
Furthermore, if
, then
so that
and hence
For a small parameter
we define the intended non-active restriction
and define
Observe that for and sufficiently large is convex in (positive definite Hessian) so that is a convex set. Assuming , define , which is a convex set.
Summarizing, if
, then
With such results in mind, we define the following convex optimization problem for finding a critical point of J.
Observe that a critical point
of
, from such a concerning convexity of
on the convex set
, is also such that
Finally, we may also define the convex optimization problem of minimizing
subject to
Here is a large real constant.
Such a functional
is also convex on
so that a critical point
of
J is also a critical point of
, and thus
12. A note on the Legendre-Galerkin functional
Let be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by .
Consider the functional
where
Here ,
We denote also
and
,
and
by
Moreover, we define
by
Observe now that these three last suprema are attained through the equations,
From such results, at a critical point, we obtain the following compatibility conditions
From such relations we have
and
so that
and
Moreover, we define the functional
by
Hence, a critical point of
J corresponds to the solution of the following system of equations
and
From this last equation we may obtain
so that the final equations to be solved are
and
with the boundary conditions
With such results in mind, we define the Legendre-Galerkin functional
, where
At this point, defining
we obtain
From such results we may infer that
Observe that a critical point so that at a neighborhood of any critical point.
At this point we define
for an appropriate real constant
.
Define now
,
for a small real constant
and
Similarly as done in the previous section, we may prove that is a convex set.
Furthermore, for we have that is convex on .
Summarizing, we may define the following convex optimization problem to obtain a critical point of the primal functional
J,
We call the Legendre-Galerkin functional associated to J.
12.1. Numerical examples
We have obtained numerical solutions for two one-dimensional examples.
-
For
For the respective solution please see
Figure 7.
-
For
For the respective solution please see
Figure 8.
13. A general concave dual variational formulation for global optimization
Let be an open, bounded and connected set a regular (Lipschitzian) boundary denoted by
Consider a functional
where
Here , and we also denote
Assume there exists
such that
Furthermore, suppose
G is three times Fréchet differentiable and there exists
such that
Define now
where,
where
and
Moreover, we define the polar functionals
and
, where
and
At this point we define the functional
by
With such results in mind we define
and
for appropriated real constants
and
Moreover, we define also the penalized functional
where
Finally, we remark that for
sufficiently small and
sufficiently large,
is concave in
around a concerning critical point. We recall that a critical point
14. A related restricted problem in phase transition
In this section we develop a convex (in fact concave) dual variational for a model similar to those found in phase transition problems.
Let
Consider the functional
where
We also denote and
Furthermore, we define the functionals
G and
by
and
Moreover we define
by
and consider the problem of minimizing
on the set
Already including the Lagrange multiplier
concerning such restrictions, we define
From this we may infer that for some
Summarizing, denoting
, and
we have got
We have developed numerical results by maximizing the dual functional for two examples, namely.
-
Example A: In this case, we consider , .
-
Example B: In this case, we consider , .
15. One more dual variational formulation
In this section we develop one more dual variational formulation for a related model.
Let
and consider the functional
defined by
where
We define also the relaxed functional
, already including a concerning restriction and corresponding non-negative Lagrange multiplier
, where
where
Here, we highlight , for some real constant c.
Hence, denoting
and
we have obtained
Finally, for
we emphasize
is concave on
.
Here is a small regularizing real constant.
Remark 15.1. The constraint is included to restrict the action of v on the region where the primal functional is non-convex, through an appropriate constant
16. A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations
In this section we develop a duality principle for a related vectorial model in the calculus of variations.
Let be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by
For
, consider a functional
where
where
and
We assume and are Fréchet differentiable and F is also convex.
Also
where
it is supposed to be Fréchet differentiable. Here we have denoted
.
We define also
by
where
and
Moreover, we define the relaxed functional
by
where