This article develops duality principles and numerical results for a large class of non-convex variational models. The main results are based on fundamental tools of convex analysis, duality theory and calculus of variations. More specifically the approach is established for a class of non-convex functionals similar as those found in some models in phase transition. Finally, in some sections we present concerning numerical examples and the respective softwares.
Keywords:
Subject: Computer Science and Mathematics - Applied Mathematics
1. Introduction
In this section we establish a dual formulation for a large class of models in non-convex optimization. It is worth highlighting 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 [1,2,3,4] and on a D.C. optimization approach developed in Toland [5]. About the other references, details on the Sobolev spaces involved are found in [6]. Related results on convex analysis and duality theory are addressed in [7,8,9,10,11,12,13].
Similar models on the superconductivity physics may be found in [14,15,16].
At this point we recall that the duality principles are important since the related dual variational formulations are either convex (in fact concave) or have a large region of convexity around their critical points. These features are relevant considering that, from a concerning strict convexity, the standard Newton, Newton type and similar methods are in general convergent. Moreover, the dual variational formulations are also relevant since in some situations, it is possible to assure the global optimality of some critical points which satisfy certain specific constraints theoretically established.
Among the main results here developed, we highlight the duality principles for the quasi-convex formulations in the context of the vectorial calculus of variations. An important example in non-linear elasticity is addressed along the text in details.
Also, for the applications in physics in the final sections, we believe to have found a path to connect the quantum approach with a more classical one in a unified framework.
Indeed, we have presented a path to model a great variety of chemical reactions through such a connection between the atomic and classical worlds.
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.
Definition1
(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
is a three times Fréchet differentiable function not necessarily convex. Moreover,
and
We assume there exists such that
Furthermore, suppose G is 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.
Define and
where
Moreover, , , so that at this point we define, , , , and by
and
and
Define now ,
Observe that
.
From the general results in [5], we may infer that
On the other hand
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
Finally, observe that
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
Observe that
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
Observe that
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.
Remark1.
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
Define
and define also
At this point we define
and
where
Moreover, we propose the relaxed functional
Observe that clearly
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.
Remark2.
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, [17].
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.
5.1. A duality principle for the concerning quasi-convex envelope
In this section, denoting
we define the functional , where
where
and,
We define also
and
It is a well known result from the modern Calculus of Variations theory (please, see [18] for details) that
At this point we denote
and
Observe that
, where
and
Also
and
in an appropriate tensor sense.
Here it is worth highlighting we have denoted,
where we recall that
in an appropriate tensorial sense.
Summarizing, defining by
we have got
Remark3.
This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.
We emphasize such results are extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].
Moreover, if is such that
it is a well known result from the Legendre transform proprieties that the corresponding such that
and
is also such that
and
From this and
we obtain
Also, from the modern calculus of variations theory, there exists a sequence such that
and
From this and the Ekeland variational principle, there exists such that
and
so that
and
Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.
In such a case we have
for an appropriate
From continuity we obtain
Summarizing, we have got
Here we highlight such last results are valid just for this finite-dimensional model version.
6. 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
Now observe that
, where
Here we have denoted
where and where
Furthermore, for , we have
Therefore, denoting by
we have got
Finally, we highlight such a dual functional is convex (in fact concave).
6.1. An example in finite elasticity
In this section we develop an application of results obtained in the last section to a model in non-linear elasticity.
Let be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by
Concerning a standard model in non-linear elasticity, consider a functional where
where and
Here is a fourth-order and positive definite symmetric tensor (in an appropriate standard sense). Moreover, is a field of displacements resulting from the f load field action on the volume comprised by .
At this point, we define the functional , where
where
We define also the quasi-convex envelop of J, denoted by , as
It is a well known result from the modern calculus of variations theory (please see [18] for details), that
Observe now that, denoting , and
we have that
, where ,
Hence, denoting
we have obtained
Remark4.
This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.
We emphasize again such results are also extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].
Moreover, if is such that
it is a well known result from the Legendre transform proprieties that the corresponding such that
and
is also such that
and
From this and
we obtain
Also, from the modern calculus of variations theory, there exists a sequence such that
and
From this and the Ekeland variational principle, there exists such that
and
so that
and
Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.
In such a case we have
for an appropriate
From continuity we obtain
Summarizing, we have got
Here we highlight such last results are valid just for this finite-dimensional model version.
7. 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
Furthermore, we define
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.
Therefore
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
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
8. 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
Remark5.
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.
Theorem1.
Assume and suppose is such that
Under such hypotheses, there exists such that is convex in ,
and
9. 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.
Remark6.
Define now
For an appropriate function (or, in a more general fashion, an appropriate bounded operator) define
for some small parameter
Moreover, define
Since for we have , so that for we have
we may infer that is a convex set.
Moreover if , then
so that
and
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.
Theorem2.
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
Moreover, observe that
Summarizing, we have got
From such results, we may infer that
The proof is complete. □
10. 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 [8], Chapter 21.
10.1. 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 [1,2]. Similar theoretical results have been developed in [7], 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 [6]. In addition, the primal variational development of the topology optimization problem has been described in [7].
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.
10.2. Mathematical formulation of the topology optimization problem
Our mathematical topology optimization problem is summarized by the following theorem.
Theorem3.
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.
Observe that
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
Moreover
This completes the proof. □
10.3. About a concerning algorithm and related numerical method
For numerically solve this optimization problem in question, we present the following algorithm
(1)
Set and .
(2)
Calculate such that
(3)
Calculate such that
(4)
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.
**************************************
(1)
clear all
global P m8 d w u v Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3