1. Introduction
In this article we establish a duality principle and a related convex dual formulation suitable for the local optimization of the primal formulation for a large class of models in non-convex optimization.
The main duality principle is applied to the Ginzburg-Landau system in superconductivity in the absence of a magnetic field.
Such results are based on the works of J.J. Telega and W.R. Bielski [
2,
3,
13,
14] and on a D.C. optimization approach developed in Toland [
15].
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,
6,
7,
9,
12]. Finally, similar models on the superconductivity physics may be found in [
4,
11].
Remark 1. It is worth highlighting, we may generically denote simply by where denotes a concerning identity operator.
Other similar notations may be used along this text as their indicated meaning are sufficiently clear.
Finally, denotes the Laplace operator and for real constants and , the notation means that is much larger than
At this point we start to describe the primal and dual variational formulations.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
For the primal formulation, consider a functional
where
Here , and
Moreover, and we denote
At this point we define
,
and
by
and
We define also
and
and
by
and
if
where
Define also
where
and
for some
to be specified,
Finally, we also define
Assume now
,
Observe that, by direct computation, we may obtain
for
.
Considering such statements and definitions, we may prove the following theorem.
Theorem 1.
Let be such that
Under such hypotheses, we have
Proof. Observe that
so that, since
,
and
is quadratic in
, we may infer that
Therefore, from a standard saddle point theorem, we have that
Now we are going to show that
Finally, denoting
from
we have
so that
The solution for this last system of equations (
8) and (
9) is obtained through the relations
and
so that
and
Moreover, from the Legendre transform properties
so that
Joining the pieces, we have got
The proof is complete.
□
2. Another Duality Principle Suitable for a Local Optimization of the Primal Formulation
In this section we develop a second duality principle which the dual formulation is concave.
We start by describing the primal formulation.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
For the primal formulation, consider a functional
where
Here , and
Moreover, and we denote
Define the functionals
,
by
and
We define also
and
by
and,
Here we denote
for an appropriate real constant
.
Furthermore, we define
and
by
Assuming
(through a re-scaling, if necessary) and
by directly computing
we may easily obtain that for such specified real constants,
in concave in
on
2.1. The Main Duality Principle and a Concerning Convex Dual Formulation
Considering the statements and definitions presented in the previous section, we may prove the following theorem.
Theorem 2.
Let be such that
Under such hypotheses, we have
Proof. Observe that
so that, since
is concave in
on
, we obtain
Now we are going to show that
Observe now that denoting
there exists
such that
and
so that
Also, denoting
from
we have
so that
From such results, we may infer that
Moreover, from
we have
so that
From such last results we get
and thus
Furthermore, also from such last results and the Legendre transform properties, we have
so that
Joining the pieces, from a concerning convexity in
u, we have got
The proof is complete.
□
3. A Third Duality Principle also Suitable for the Primal Formulation Local Optimization
In this section we establish one more duality principle and related convex dual formulation suitable for a local optimization of the primal variational formulation.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
For the primal formulation, we define
and consider a functional
where
Here we assume
, and define
and
for an appropriate constant
to be specified.
Define also the functionals
and
by
and
for appropriate positive constants
to be specified.
Moreover, define
and
by
and
Furthermore, we define
where
Define also
for an appropriate real constant
to be specified, and
by
Moreover, we assume .
By directly computing
denoting
we may also obtain,
on
At a critical point we have
and
With such results, for a sufficiently small
, we may define the restrictions
and
At this point, we prove that is a convex set.
Firstly, fixing
observe that for
sufficiently large
is convex in
on
.
Observe also that
is equivalent to
which is equivalent to
Moreover, this last inequality is equivalent to
Since for we have , we may obtain that is a convex function on , so that is convex in on
From such results, we may easily infer that is a convex set, .
Similarly, we may prove that is a convex set, .
On the other hand, clearly we have
3.1. A Concerning Duality Principle and a Related Convex Dual Formulation
Considering the statements and definitions presented in the previous section, we may prove the following theorem.
Theorem 3.
Let be such that
and be such that
Under such hypotheses, denoting , we have
Proof. Observe that
so that, since
and
, from the results in the previous lines, for a sufficiently small
, we have that
Consequently, from this and the Saddle Point Theorem, we have
Now we are going to show that
Denoting
there exists
such that
and
so that
Summarizing, we this last equation is satisfied through the relation
Hence from the variation of
in
, we obtain
so that
On the other hand, from the variation of
in
, we have
From such results, since
we get
Consequently, from such last results and from
we obtain
Furthermore, also from such last results and the Legendre transform properties, we have
so that
Summarizing, we have obtained
The proof is complete.
□
4. A Convex Primal Dual for a Local Optimization of the Primal Formulation
In this section we develop a convex primal dual formulation corresponding to a non-convex primal formulation.
We start by describing the primal formulation.
Let be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by
For the primal formulation, consider a functional
where
Here , and
Moreover, and we denote
Define the functional
, by
We define also
for an appropriate real constant
.
Furthermore, we define
for an appropriate real constant
and
Now observe that denoting
, we have
and
Denoting
we have also that
In such a case, we obtain
Observe that at a critical point
so that we may set the non-active restriction
for a small parameter
.
Now we are going to prove that is a convex subset of .
For a
observe that
is equivalent to
which is equivalent to
Observe that since for
we have
, we have also that
is convex on
.
Moreover, for
sufficiently large, the function
is also convex on
.
Summarizing, is convex on so that from such results, we may infer that is a convex set.
On the other hand, at a critical point we have also
Now define the non-active constraint
Similarly as it was made for we may prove that is convex in .
We are going to prove that is convex as well.
Observe that
is equivalent to
which is equivalent to
At this point we highlight that, for
sufficiently large, the function
is convex on
.
Moreover, since for
, we have
the function
is also convex on
. Therefore, we have got the function
is convex on
.
From such results we may infer that is a convex set.
At this point, we define the convex set
Finally, observe that for
, we have that
is positive definite on
From such results we may infer that is convex in and concave in on
With such results in mind, we may prove the following theorem.
Theorem 4.
Let be such that
Under such hypotheses, we have
Proof. The proof that
and
may be done similarly as in the previous sections.
Observe that is convex in and concave in on , where and are convex sets.
From such results and Min-Max Theorem, we may infer that
In particular for
we obtain
Joining the pieces, we have got
The proof is complete. □
5. Conclusion
In this article we have developed convex dual variational formulations suitable for the local optimization of non-convex primal formulations.
It is worth highlighting, the results may be applied to a large class of models in physics and engineering.
We also emphasize the duality principles here presented are applied to a Ginzburg-Landau type equation. In a future research, we intend to extend such results for some models of plates and shells and other models in the elasticity theory.
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