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On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications
- Ning Ruan^{1} and
- David Yang Gao^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13663-018-0648-x
© The Author(s) 2018
- Received: 25 January 2018
- Accepted: 25 September 2018
- Published: 15 October 2018
Abstract
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.
Keywords
- Fixed point
- Properly-posed problem
- Nonconvex optimization
- Canonical duality theory
- Mathematical modeling
- Multidisciplinary studies
MSC
- 47H10
- 47H14
- 55M05
- 65K10
1 Introduction
Lemma 1
Proof
First we assume that \({F}(\mathbf {x})\) is a potential operator, then x is a stationary point of \(\Pi (\mathbf {x})\) if and only if \(\nabla \Pi (\mathbf {x}) = \nabla {P}(\mathbf {x}) - \mathbf {x}= 0\), thus, x is also a solution to \((\mathcal {{P}}_{0})\) since \({F}(\mathbf {x}) = \nabla {P}(\mathbf {x})\).
Now we assume that \({F}(\mathbf {x})\) is not a potential operator. By the fact that \(\Pi (\mathbf {x}) = \frac{1}{2} \Vert {F}(\mathbf {x}) - \mathbf {x}\Vert ^{2} \ge 0 \ \forall \mathbf {x}\in \mathcal {X}\), the vector \({\bar {\mathbf {x}}}\) is a global minimizer of \(\Pi (\mathbf {x})\) if and only if \({F}({\bar {\mathbf {x}}}) - {\bar {\mathbf {x}}}= 0 \). Thus, \({\bar {\mathbf {x}}}\) must be a solution to \((\mathcal {{P}}_{0})\). □
Definition 1
(Objectivity)
Geometrically speaking, an objective function does not depend on rigid rotation of the system considered, but only on certain measure of its variable. In the Euclidean space \(\mathcal {{W}}\subset {\mathbb {R}}^{m}\), the simplest objective function is the \(\ell _{2}\)-norm \(\Vert \mathbf {w}\Vert \) in \({\mathbb {R}}^{m}\) as we have \(\Vert {\mathbf{R}} \mathbf {w}\Vert ^{2} = \mathbf {w}^{T} {\mathbf{R}}^{T} {\mathbf{R}} \mathbf {w}= \Vert \mathbf {w}\Vert ^{2} \ \forall {\mathbf{R}} \in {\mathcal{R}}\). For general \(F(\mathbf {x})\), we can see from (4) that \(\frac{1}{2} \Vert {F}(\mathbf {x}) \Vert ^{T} \) and \(\frac{1}{2} \Vert \mathbf {x}\Vert ^{2}\) are objective functions. By the fact that \(\mathbf {x}= F(\mathbf {x})\), we know that 〈x, \(F(\mathbf {x}) \rangle \) is also an objective function. Therefore, for a given fixed point problem, the corresponding \(\Pi (\mathbf {x})\) is naturally an objective function.
Physically, an objective function is governed by the intrinsic physical law of the system, which does not depend on observers. Because of Noether’s theorem, the objective function \({W}(\mathbf {w})\) should be a SO(n)-invariant and this invariant is equivalent to a certain conservation law (see Sect. 6.1.2 [11]). Therefore, objectivity is essential for any real-world mathematical models. It was emphasized by P.G. Ciarlet that the objectivity is not an assumption, but an axiom [6].
From the theory of nonconvex analysis, any nonconvex function can be written as a d.c. (deference of convex) function [35]. Therefore, the fixed point problem is actually equivalent to a d.c. programming problem. By the fact that \(\mathcal {X}\) and \(\mathcal {{W}}\) are two different spaces with different scales (dimensions), the problem \((\mathcal {{P}})\) can be used to study general problems in multi-scale complex systems.
For a potential operator, a fixed point is just a stationary point, which can be easily found by traditional linear iteration methods. For a non-potential operator, the fixed point must be a global minimizer. Due to the lack of global optimality condition in the traditional theory of nonlinear optimization, to solve a general nonconvex minimization problem is considered to be NP-hard in global optimization and computer science. However, this paper will show that many of these nonconvex problems can be solved in an elegant way.
2 Methods
According to the Brouwer fixed point theorem, we know that any continuous function from the closed unit ball in an n-dimensional Euclidean space to itself must have a fixed point. Generally speaking, for any given nontrivial input, a well-defined system should have at least one nontrivial response.
Definition 2
(Properly- and well-posed problems [18])
The problem \((\mathcal {{P}})\) is called properly posed if, for any given nontrivial input \(\mathbf {f}\neq 0 \), it has at least one nontrivial solution. It is called well-posed if the solution is unique.
Clearly, this definition is more general than Hadamard’s well-posed problems in dynamical systems since the continuity condition for the solution is not required. Physically speaking, any real-world problems should be well-posed since all natural phenomena exist uniquely. But practically, it is difficult to model a real-world problem precisely. Therefore, properly posed problems are allowed for the canonical duality theory. This definition is important for understanding challenging problems in complex systems.
Example 1
(Manufacturing/production systems)
Example 2
(Lagrange mechanics)
Example 3
(Post-buckling of nonlinear Gao beam)
However, the well-defined objectivity in nonlinear analysis and physics has been seriously misused in optimization and mathematical programming, where the so-called objective function is allowed to be any arbitrarily given function. As a consequence, Gao–Strang’s work has been mistakenly challenged by M.D. Voisei and C. Zalinescu [48]. By oppositely choosing linear functions as the objective function W (see Example 3.1 in [48]) and nonlinear functions as the external energy \(U(\mathbf {u})\) (see Examples 3.2, 3.4, 3.5, and 3.6 in [48]), they produced a series of “counter examples” that led to absurd conclusions including “The hope for reading an optimization theory with diverse applications is ruined by the manner in which [30] is written and the fact that the majority of the results in [30] are false.” These conceptual mistakes verified Arnold’s declaration [1]: “A teacher of mathematics, who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz, will then become a relict like the one nowadays who does not know the difference between an open and a closed set.” A comprehensive review on the canonical duality theory and breakthrough from the recent challenges are given in [24].
The goal of this paper is to apply the canonical duality theory for solving the challenging fixed point problem. The rest of this paper is arranged as follows. Based on the concept of objectivity, the canonical dual for the fixed point problem, its analytical solution, and global optimality condition are presented in the next section. Applications to a general fixed point problem with sum of exponential functions and nonconvex polynomial are discussed in Sect. 4.1. Analytical solutions for a general fixed point problem with a sum of logarithmic and quadratic functions are given in Sect. 4.2. The paper ends with conclusions and future work.
3 Results and discussion
According to the canonical duality, the linear measure \(\epsilon = D \mathbf {x}\) cannot be used directly for studying duality relation due to the objectivity. Also, the linear operator cannot change the nonconvexity of \(W(D\mathbf {x})\). We first introduce the canonical transformation.
Definition 3
(Canonical function and canonical transformation)
A real-valued function \(V:\mathcal {E}_{a} \rightarrow {\mathbb {R}}\) is called canonical if the duality mapping \(\partial V: \mathcal {E}_{a} \rightarrow \mathcal {E}_{a}^{*}\) is one-to-one and onto.
Theorem 1
(Analytic solution and complementary-dual principle)
Proof
Theorem 2
(Triality theorem)
4 Applications
4.1 Exponential and polynomial functions
Example 1
4.2 Logarithmic and quadratic function
Example 2
Example 3
5 Conclusions
Based on the canonical duality theory, a unified model is proposed such that the general fixed point problems can be reformulated as a global optimization problem. This model is directly related to many other challenging problems in variational inequality, d.c. programming, chaotic dynamics, nonconvex analysis/PDEs, post-buckling of large deformed structures, phase transitions in solids, computer science, etc. (see [24] and the references cited therein). By the complementary-dual principle, all the fixed points can be obtained analytically in terms of the canonical dual solutions. Their stability and extremality are identified by the triality theory. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic functions. Our examples show that both globally stable and locally stable/unstable fixed point problems in \({\mathbb {R}}^{n}\) can be obtained easily by solving the associated canonical dual problems in \({\mathbb {R}}^{m}\) with \(m< n\). However, the local stability condition for those fixed points \({\bar {\mathbf {x}}}(\bar {\boldsymbol {\varsigma }})\) with indefinite \(\mathbf {G}(\bar {\boldsymbol {\varsigma }})\) still remains unknown, and it deserves serious study in the future. Also, the results presented in this paper can be generalized to problems with nonsmooth potential functions.
Declarations
Acknowledgements
The authors are thankful to the editors and the anonymous referees for their valuable comments, which reasonably improved the presentation of the manuscript.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The research was supported by US Air Force Office of Scientific Research under the grants (AOARD) FA2386-16-1-4082 and FA9550-17-1-0151.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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