L₁-L_p Minimization via a Distributed Smoothing Neurodynamic Approach for Robust Multi-View Three-Dimensional Space Localization

Abstract

This paper presents a distributed smoothing neurodynamic approach for solving the $L_1$-$L_p$ minimization problem, with application to robust and collaborative multi-view three-dimensional (3D) space localization. To handle the non-Lipschitz continuity gradients, a smooth approximation technique is introduced, yielding a distributed neurodynamic model that integrates classical smoothing neural networks with multi-agents consensus theory. Theoretical analysis guarantees the global convergence of each agent’s state to the optimal solution. The stability and convergence of the proposed approaches are rigorously proved using Lyapunov theory. Numerical experiments on multi-view 3D space localization in the presence of measurement noise demonstrate the method’s effectiveness and practical value for distributed visual computing.

1. Introduction

The rapid advancement of robotic navigation and autonomous driving has created a pressing demand for accurate environmental perception and modeling. On these intelligent platforms, vision systems based on video and image data have garnered significant academic attention as core perception modules, exemplified by applications such as visual odometry [1], the vision language model [2], and generalized visual localization [3]. The construction of semantic maps [4,5], which integrate geometric and semantic information, represents a significant research direction by facilitating a fundamental transition in environmental perception from a purely geometric to a cognitive level. This process often relies on the accurate estimation of 3D positions from two-dimensional observations [6], and even one-dimensional distances [7], which is 3D space localization, which often relies on triangulation measurements. Modern computer vision techniques can reliably detect objects and extract their corresponding 2D pixel coordinates from video and image data, such as convolutional neural network [8,9,10], YOLO [11,12], and Transformer Networks [13]. Therefore, triangulation emerges as a fundamental technique for solving systems of overdetermined equations derived from multi-view geometric constraints [14], used to reconstruct precise 3D coordinates of objects from their 2D coordinates. Consequently, significant research efforts have been dedicated to enhancing its robustness and accuracy. Reference [15] offers a robust optimization solution to the triangulation problem for uncertain data. Reference [16] proposes an approach based on convex relaxations for certifiably optimal robust multi-view triangulation that is capable of achieving provably optimal reconstructions in the presence of substantial noise and a high proportion of outliers. The applicability of robust triangulation extends to diverse domains. For instance, it has been utilized to reconstruct accurate height profiles of steep terrain [17]. In the context of mobile robots in Reference [18], monocular odometry systems benefit from constraints imposed on feature matching by semantic information and on triangulation by depth, thereby enhancing localization accuracy. Furthermore, triangulation serves as the foundational mechanism for depth estimation in Simultaneous Localization and Mapping (SLAM) [19] and constitutes the core of traditional Structure from Motion (SfM) [20]. In these systems, the 3D space localization of the detection target is well completed based on visual image data and triangulation methods.

Computationally, triangulation can be formulated as an optimization problem, aiming to find the optimal 3D coordinates that best explain the multi-view observations. This paradigm of "recovering original data from observed data" shares a profound mathematical similarity with the framework of Compressed Sensing (CS), which seeks to reconstruct a sparse signal from under-sampled linear measurements. According to CS theory, a sparse signal $\mathit{\upsilon }\in {\mathbb{R}}^n$ can be exactly recovered from its underdetermined linear measurements $\mathit{b}=A\mathit{\upsilon }\in {\mathbb{R}}^m$$(m\ll n)$, where $A\in {\mathbb{R}}^{m\times n}$ is the sensing matrix. Within the vast majority of pragmatic implementations, the measurements are inescapably adulterated by certain noise perturbations. Under such circumstances, the compressed measurements can be typically formulated as

$$\begin{array}{c}\hfill b=A\upsilon +e\end{array}$$

(1)

where $e\in {\mathbb{R}}^m$ denotes measurement noise.

An orthodox tactic to retrieve the sparse vector $\upsilon$ is converted to the ensuing $l_0$-minimization problem [21]:

$$\begin{array}{c}\hfill min{∥\upsilon ∥}_0\hspace{1em}s.t.\hspace{1em}{∥A\upsilon -b∥}_2\le ϵ\end{array}$$

(2)

where ${∥\upsilon ∥}_0$ signifies $l_0$-norm, to wit the quantity of non-vanishing entries within vector $\upsilon$. $ϵ>0$ constrains the residual error of $l_0$-norm and is pre-determined by the noise level. However, addressing Problem (2) is NP-hard [22]. Thus, to address this problem, it is replaced by the succeeding model, i.e., BP denoising (BPDN) [23]:

$$\begin{array}{c}\hfill min{∥\upsilon ∥}_1\hspace{1em}s.t.\hspace{1em}{∥A\upsilon -b∥}_2\le ϵ\end{array}$$

(3)

where ${∥\upsilon ∥}_1 = |{\upsilon }_1| +\dots + |{\upsilon }_n|$, namely $l_1$-norm. Later, Wen et al. focused on the ${\ell }_p-{\ell }_1$ optimization problem to realize the robust sparse recovery under impulsive measurement perturbations [24]:

$$\begin{array}{c}\hfill min{∥\upsilon ∥}_1\hspace{1em}s.t.\hspace{1em}{∥A\upsilon -b∥}_p\le ϵ\end{array}$$

(4)

where ${∥\upsilon ∥}_p={({\displaystyle \sum _{i=1}^n}{\left|{\upsilon }_i\right|}^p)}^{\frac{1}{p}}$. When $p=2$, the optimization problem (4) constitutes a particular instance of (3).

In particular, sparse signal reconstruction with a neurodynamic approach has attracted a large number of scholars [25,26,27,28,29,30,31,32,33,34,35,36]. The $l_0$ minimization problem can be addressed by a neurodynamic approach based on a recurrent neural network (RNN) model, which is capable of attaining the global optimum and has been applied to sparse signal recovery [25]. Two neurodynamic stratagems via $L_1$-minimization were proposed for sparse signal restitution through the integration of augmented Lagrangian multipliers and differential feedback [28]. Using scaled gradient projection, a neural network was presented for ${\ell }_1-{\ell }_2$ minimization by Liu et al., and the stability was proved by constructing a Lyapunov function [29]. Wang et al. developed a $l_{1-p}$ sparse recovery minimization model and proposed a gradient projection smoothing neural network, which applied to the compressed sensing problem [31]. To address the $l_1-l_p$ minimization, a smoothing neural network (SNN) was developed by dint of a smoothing approximate artifice for sparse signal restitution with measurement perturbations [32].

Distributed neurodynamic approaches have gained popularity due to their high reliability, scalability, and privacy protection. In multi-agent systems, while data transmission heavily relies on channel resources and efficient transmission methods have been widely studied [37,38], a distributed strategy keeps observation data localized. This not only reduces communication overhead but also enhances the privacy of each agent. Consequently, numerous distributed neurodynamic methods have been developed for solving distributed optimization problems [39,40,41,42,43,44,45,46,47]. To address the distributed optimization problem featuring globally coupled constraints, Li et al. propounded a distributed neurodynamic stratagem using recurrent neural networks (RNNs) [42]. By Che and Wang, a collaborative neurodynamic stratagem was propounded with an augmented Lagrangian function for the resolution of a global and combinatorial optimization problem [43]. Concerned with constrained global optimization problems, Yan et al. proposed the distributed neuraldynamic approach, which utilized multiple RNNs to ascertain the global optimal solution collectively [44]. In the context of multiple-objective distributed optimization, a collaborative neurodynamic stratagem with objective weighting and decision space decomposition was developed, which focuses on Pareto optimal solutions [45]. Zhao et.al presented a decentralized smoothing projection neurodynamic stratagem with an expedient convergence rate in addressing constrained nonsmooth generalized convex optimization problems. Meanwhile, by dint of the continuous variant of Polyak’s heavy ball methodology, a distributed smoothing projection neurodynamic stratagem also was proposed for a constrained nonsmooth strongly convex optimal problem [46]. Later, by virtue of a projection operator, two decentralized continuous-time and discrete-time neurodynamic stratagems were developed via $L_1$ minimization to sparse recovery [47]. Recently, to address the $L_1$-minimization problem, Zhao et al. tries to propose a distributed neurodynamic approach by smooth approximation and projection operator. Considering that the distributed $L_1$-$L_p$ minimization problem uses the distributed smooth neurodynamic approach, further in-depth research and discussion is necessary for sparse recovery. Furthermore, measurement noise is customarily factored in within most practical applications. In compressed sensing (CS), the $L_1$-$L_p$-minimization is designed with robust sparse signal recovery with measurement noise, which brings both practical and academic value to our research.

Inspired by preliminary work, such as the distributed neurodynamic approach for $L_1$-minimization via smooth approximation [46], we identify a compelling opportunity to extend this paradigm. Therefore, this paper aims to develop a novel distributed neurodynamic algorithm that leverages the $L_1$-$L_p$ minimization framework from compressed sensing to achieve accurate 3D localization from multi-view, noisy observations. It avoids raw data centralization, exchanges only state estimates with neighboring nodes, and keeps sensitive observations local. These features make the approach particularly suitable for scalable multi-agent systems under bandwidth constraints. The main contributions of this article are highlighted as below:

• We propose a robust, distributed computing framework for multi-view 3D localization by innovatively formulating it as a compressed sensing problem. This design inherently leverages the anti-noise ability of sparse recovery theory. Compared to other triangulation-based solutions, this is a distributed approach that can be deployed on multi-agent systems without having to worry about computing power limitations.
• We develop a novel distributed smoothing neurodynamic approach (Equation (14)), featuring a derivative feedback item. A key feature of which is its effective handling of the non-Lipschitz continuity issue through smooth approximation. The stability and convergence of this algorithm are theoretically guaranteed by a constructed Lyapunov function.

An outline of the remainder of this paper is as follows. Section 2 mainly introduces smoothing approximation, the nonnegative quadrant, and other introductory knowledge. In Section 3, a distributed $L_1$-$L_p$ optimization problem is proposed, and the proposal of a distributed smoothing neurodynamic approach is accompanied by formal proofs establishing its global convergence and stability. In Section 4, the multi-view 3D localization problem is formulated as a compressed sensing task and addressed with the proposed distributed smoothing neurodynamic approach for $L_1$-$L_p$ minimization, thereby achieving robust and collaborative recovery of 3D coordinates under measurement noise.

2. Preliminary Knowledge

This section expounds on the smoothing approximation and nonnegative quadrant which can be used in the later parts.

A.

Smoothing approximation [48]:

Definition 1.

Function $\mathrm{\Gamma }:{\mathbb{R}}^n\to \mathbb{R}$ is locally Lipschitz, and $\breve{\mathrm{\Gamma }}:{\mathbb{R}}^n\times (0,+\infty )\to \mathbb{R}$ is a smoothing function corresponding to $\mathrm{\Gamma }$, provided that $\breve{\mathrm{\Gamma }}$ satisfies the ensuing stipulations:

1. 

For any fixed $\varpi >0$ and any fixed $\upsilon \in {\mathbb{R}}^n$, the function $\breve{\mathrm{\Gamma }}(·,\varpi )$ is continuously differentiable on ${\mathbb{R}}^n$, and $\breve{\mathrm{\Gamma }}(\upsilon ,·)$ is differentiable on $(0,+\infty )$, respectively.

2. 

For any prescribed $\upsilon \in {\mathbb{R}}^n,\underset{\varpi \to 0^{+}}{lim}\breve{\mathrm{\Gamma }}(\upsilon ,\varpi )=\mathrm{\Gamma }(\upsilon )$.

3. 

There is a positive constant $k_{\breve{\mathrm{\Gamma }}}>0$ such that

$$\left|{\nabla }_{\varpi }\breve{\mathrm{\Gamma }}(\upsilon ,\varpi )\right|\le k_{\breve{\mathrm{\Gamma }}},\forall \varpi \in (0,+\infty ),\upsilon \in {\mathbb{R}}^n$$

4. 

$$\left\{\underset{z\to \upsilon ,\varpi \to 0}{lim}{\nabla }_z\breve{\mathrm{\Gamma }}(z,\varpi )\right\}\subseteq \partial \mathrm{\Gamma }(\upsilon )$$

From (2) and (3) of Definition 1, the following properties are deduced:

• $\underset{z\to \upsilon ,\varpi \to 0}{lim}\breve{\mathrm{\Gamma }}(z,\varpi )=\mathrm{\Gamma }(\upsilon )$
• $|\breve{\mathrm{\Gamma }}(\upsilon ,\varpi )-\mathrm{\Gamma }(\upsilon )|\le k_{\breve{\mathrm{\Gamma }}}\varpi ,\forall \varpi \in (0,+\infty ),\upsilon \in {\mathbb{R}}^n$
• $|\breve{\mathrm{\Gamma }}(\upsilon ,\varpi )-\breve{\mathrm{\Gamma }}(\upsilon ,\tilde{\varpi })|\le k_{\breve{\mathrm{\Gamma }}}(u-\tilde{\varpi }),\forall \upsilon \in {\mathbb{R}}^n,\varpi \ge \tilde{\varpi }>0$

There are two propositions concerning the compositions of smoothing functions introduced in [33]:

Lemma 1.

1. 

Suppose that ${\breve{h}}_1,\dots ,{\breve{h}}_m$ are smoothing functions of $h_1,\dots ,h_m$, then ${\sum }_{i=1}^m{a}_i{\breve{h}}_i$ is a smoothing function of ${\sum }_{i=1}^m{a}_i{h}_i$ with $k_{{\sum }_{i=1}^m{a}_i{\breve{h}}_i}={\sum }_{i=1}^m{a}_i{k}_{{\breve{h}}_i}$ under the conditions that $a_i\ge 0$ and $h_i$ is regular for each $i=1,2,\dots ,m$.

2. 

For a regular function $\varphi :{\mathbb{R}}^m\to \mathbb{R}$ and a continuously differentiable $\phi :{\mathbb{R}}^n\to \mathbb{R}$ with smoothing function $\breve{\phi }$, it follows that $\breve{\phi }\circ \varphi$ smooths the composition $\phi \circ \varphi$ and satisfies ${\kappa }_{\breve{\phi }\circ \varphi }={\kappa }_{\breve{\phi }}$.

The following smooth approximation function is frequently performed, which will also be used in this paper.

$$\begin{array}{c}\hfill \breve{\mathrm{\Gamma }}(s,\varpi )=\left\{\begin{array}{c}\left|s\right|\hspace{1em}\hspace{1em},\left|s\right|>\varpi \hfill \\ {\displaystyle \frac{s^2}{2\varpi }}+{\displaystyle \frac{\varpi }{2}},\left|s\right|\le \varpi \hfill \end{array}\right.\end{array}$$

(5)

For any $\varpi >0$, the function $\breve{\mathrm{\Gamma }}(·,\varpi )$ is convex. Additionally, for any $s\in {\mathbb{R}}^n$, the function $\breve{\mathrm{\Gamma }}(s,·)$ is continuous and nondecreasing.

Assumption 1.

In ${\mathbb{R}}^n$, both $f(\upsilon )$ and $g(\upsilon )$ functions are convex.

Assumption 2

(Slater Condition). There exists $\breve{\upsilon }\in R^n$, such that $g(\breve{\upsilon })<0$.

B.

Nonnegative Quadrant [49]:

The positive projection of a is denoted as $a^{+}$, namely $a^{+}=max\left\{0,a\right\}$. Enumerated below are two properties:

$$\begin{array}{c}(1) a=a^{+}-{(-a)}^{+}\hfill \\ \\ (2) ⟨a^{+},{\left(-a\right)}^{+}⟩=0\hfill \end{array}$$

3. The Distributed Smoothing Approximation for ${\mathit{L}}_{\mathbf{1}}-{\mathit{L}}_{\mathit{p}}(\mathbf{2}\ge \mathit{p}\ge \mathbf{1})$ Problem

In this part, the smoothing approximation for the distributed $L_1-L_p(2\ge p\ge 1)$ optimization problem with local prior information of the blocked observation matrix (i.e., for any $j(j=1,\dots ,w)$, the evaluation of the noise level ${\epsilon }^j$ is known) is described in detail.

$$\begin{array}{c}\hfill \begin{array}{c}min F(\mathit{\upsilon })={\displaystyle \frac{1}{w}}{\displaystyle \sum _{j=1}^w}{∥{\mathit{\upsilon }}^j∥}_1\hfill \\ \mathrm{s}.\mathrm{t}. G^j({\mathit{\upsilon }}^j)={∥{\mathit{A}}^j{\mathit{\upsilon }}^j-b^j∥}_p^p-{ϵ}^j\le 0,j=1,\dots ,w, L\mathit{x}=0\hfill \end{array}\end{array}$$

(6)

where ${\mathit{A}}^j\in {\mathbb{R}}^{m\times n}$, $L=L_w\times I_w\in {\mathbb{R}}^{wn\times wn}$, Laplacian matrix $L_w\in {\mathbb{R}}^{w\times w}$ of G (an undirected weighted connected graph). ${\mathit{\upsilon }}^j\in R^n$, and $\mathit{\upsilon }\in R^{wn}$ represents the set of ${\mathit{\upsilon }}^j(j=1,2\dots ,w)$), ${\mathit{b}}^j\in R^m$, ${ϵ}^j\in R$. We reformulate the original equality $\left|\right|{\mathit{A}}^j{\mathit{\upsilon }}^j-b^j{\left|\right|}_p^p-{ϵ}^j=0$ as the inequality $\left|\right|{\mathit{A}}^j{\mathit{\upsilon }}^j-b^j{\left|\right|}_p^p-{ϵ}^j\le 0$ in distributed multi-linear-equation solving primarily to handle varying sensor observation errors. Using equality constraints or least squares forces equal influence from each sensor, allowing high-error sensors to negatively affect others. Instead, inequality constraints define norm balls for each sensor; the radii of norm balls are ${ϵ}^j$. The consensus constraint then seeks a common point within the intersection of these balls.

Obviously, ${∥x∥}_1$ is a convex, nonsmooth function, so $F(\mathit{\upsilon })$ is a convex, nonsmooth function. When $1\le p\le 2$, the gradient of $G(\mathit{\upsilon })$ is not Lipschitz continuous (not bounded). When $p=1$, $F(\mathit{\upsilon })$ and $G(\mathit{\upsilon })$ is also a convex, nonsmooth function. However, the traditional proximal gradient methods are no longer appropriate. In order to solve Problem (6) by the neuraldynamic approach, we perform a smooth approximation.

Remark 1.

Note the local prior information of the blocked observation matrix (i.e., for any $j(j=1,\dots ,w)$, the evaluation of the noise level ${ϵ}^j$ is known); thus, one has ${∥\left(\begin{array}{ccc}{\mathit{A}}^1& & \\ & \ddots & \\ & & {\mathit{A}}^w\end{array}\right)\mathit{\upsilon }-\left[\begin{array}{c}{\mathit{b}}^1\\ ⋮\\ {\mathit{b}}^w\end{array}\right]∥}_p^p-ϵ={\sum }_{j=1}^w\left({∥{\mathit{A}}^j{\mathit{\upsilon }}^j-{\mathit{b}}^j∥}_p^p-{ϵ}^j\right)={∥\mathit{A}\mathit{\upsilon }-\mathit{b}∥}_p^p-ϵ$ with ${\sum }_{j=1}^w{ϵ}^j=ϵ$ when ${\mathit{\upsilon }}^j={\mathit{\upsilon }}^i=\mathit{\upsilon },i,j=1,\dots ,w$, and the distributed $L_1$-$L_p$ minimization problem (6) is equivalent to the optimization problem (7) in [32].

$$\begin{array}{c}\hfill minf(\mathit{\upsilon })={∥\mathit{\upsilon }∥}_1,\hspace{1em}s.t.\hspace{1em}g(\mathit{\upsilon })={∥A\mathit{\upsilon }-b∥}_p^p-ϵ\le 0\end{array}$$

(7)

where ${∥A\mathit{\upsilon }-b∥}_p^p={\displaystyle \sum _{i-1}^m}{\left|A_j\mathit{\upsilon }-b_j\right|}^p,(1\le p\le 2)$. Namely, the optimal solution of the distributed $L_1$-$L_p$ optimization problem (6) is the optimal solution of (7).

Assumption 3.

In this paper, ${\varpi }^1=\dots ={\varpi }^j(j=1,\dots ,w)$ are considered.

Since $\underset{z\to \upsilon ,\varpi \to 0}{lim}\breve{\mathrm{\Gamma }}(z,\varpi )=\mathrm{\Gamma }(\upsilon )$, $F(\mathit{\upsilon })$ can be approximated by

$$\begin{array}{cc}\hfill \breve{F}(\mathit{\upsilon },\mathit{\varpi })& ={\displaystyle \frac{1}{w}}\sum _{j=1}^w\sum _{i=1}^n\breve{F}({\upsilon }_i^j,{\varpi }^j),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \breve{F}({\upsilon }_i^j,{\varpi }^j)& =\left\{\begin{array}{c}\left|{\upsilon }_i^j\right|,\hspace{1em}\hspace{1em}\left|{\upsilon }_i^j\right|>{\varpi }^j\hfill \\ {\displaystyle \frac{{({\upsilon }_i^j)}^2}{2{\varpi }^j}}+{\displaystyle \frac{{\varpi }^j}{2}},\left|{\upsilon }_i^j\right|\le {\varpi }^j\hfill \end{array}\right.\hfill \end{array}$$

(9)

$G^j({\upsilon }^j)$ can be approximated by

$$\begin{array}{cc}\hfill {\breve{G}}^j({\mathit{\upsilon }}^j,{\varpi }^j)& =\sum _{k=1}^{m^j}g({\mathit{A}}_k^j{\mathit{\upsilon }}^j-b_k^j)-{\epsilon }^j\le 0,\hfill \\ \hfill g({\mathit{A}}_k^j{\mathit{\upsilon }}^j-b_k^j)& =\left\{\begin{array}{c}{\left|{\mathit{A}}_k^j{\mathit{\upsilon }}^j-b^j\right|}^p,\hspace{1em}\left|{\mathit{A}}_k^j{\mathit{\upsilon }}^j-b^j\right|>{\varpi }^j\hfill \\ {\left({\displaystyle \frac{{({\mathit{A}}_k^j{\mathit{\upsilon }}^j-b^j)}^2}{2{\varpi }^j}}+{\displaystyle \frac{{\varpi }^j}{2}}\right)}^p,\left|{\mathit{A}}_k^j{\mathit{\upsilon }}^j-b^j\right|\le {\varpi }^j\hfill \end{array}\right.\hfill \end{array}$$

(10)

Therefore, the optimization problem (6) is converted to the following problem:

$$\begin{array}{c}\hfill min \breve{F}(\mathit{\upsilon },\mathit{\varpi }) \mathrm{s}.\mathrm{t}. \breve{G}(\mathit{\upsilon },\mathit{\varpi }), L\mathit{\upsilon }=0\end{array}$$

(11)

4. Distributed Smoothing Neurodynamic Approach

In this section, a distributed smoothing neurodynamic approach is proposed for solving Problem (11), which is based on the multi-agent consensus condition and derivative feedback.

For any agent $j,(j=1,\dots ,w)$,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\begin{array}{c}{\dot{\mathit{\upsilon }}}^j=-2({\nabla }_{{\mathit{\upsilon }}^j}\breve{F}({\mathit{\upsilon }}^j,{\varpi }^j)+{\nabla }_{{\mathit{\upsilon }}^j}{\breve{G}}^j({\mathit{\upsilon }}^j,{\varpi }^j)[{\lambda }^j+{\breve{G}}^j{({\mathit{\upsilon }}^j,{\varpi }^j]}^{+}+{\displaystyle \sum _{i\in N_j}}({\mathit{\eta }}^j-{\mathit{\eta }}^i)+{\displaystyle \sum _{i\in N_j}}({\mathit{\upsilon }}^j-{\mathit{\upsilon }}^i))\hfill \end{array}\hfill \\ {\dot{\zeta }}^j=-{\zeta }^j+{[{\zeta }^j+{\breve{G}}^j({\mathit{\upsilon }}^j,{\varpi }^j)]}^{+}\hfill \\ {\dot{\mathit{\eta }}}^j={\mathit{\upsilon }}^j\hfill \end{array}\right.\end{array}$$

(12)

and its vector form can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\mathit{\upsilon }}=-2({\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){[\mathit{\zeta }+\breve{G}(\mathit{x},\mathit{\varpi })]}^{+}+L(\mathit{\eta }+\mathit{\upsilon }))\hfill \\ \dot{\mathit{\zeta }}=-\mathit{\zeta }+{[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}\hfill \\ \dot{\mathit{\eta }}=\mathit{\upsilon }\hfill \end{array}\right.\end{array}$$

(13)

where $\mathit{\zeta } \in {\mathbb{R}}^w$ is a penalty parameter, ${\left[·\right]}^{+}=max\{0,·\}$, ${\varpi }_j={\varpi }_0{e}^{-2t},(j=1,\dots ,w)$, $\mathit{\varpi }={\varpi }_1=\dots ={\varpi }_w$. ${\mathit{\varpi }}^w=[{\varpi }_1,\dots ,{\varpi }_w]$.

${\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })$ is a gradient of the smoothing function $\breve{F}$ in terms of ${\upsilon }_i$

$$\begin{array}{cc}\hfill {\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })& ={\displaystyle \frac{1}{w}}\sum _{j=1}^w{\nabla }_{{\mathit{\upsilon }}^j}\breve{F}({\mathit{\upsilon }}^j,{\varpi }^j)={\displaystyle \frac{1}{w}}\sum _{j=1}^w\sum _{i=1}^n{\nabla }_{{\upsilon }_i^j}\breve{F}({\upsilon }_i^j,{\varpi }^j)\hfill \end{array}$$

$${\nabla }_{{\upsilon }_i^j}\breve{F}({\upsilon }_i^j,{\varpi }^j)=\left\{\begin{array}{c}sign({\upsilon }_i^j),\hspace{1em}\left|{\upsilon }_i^j\right|>{\varpi }^j\hfill \\ {\displaystyle \frac{{\upsilon }_i^j}{{\varpi }^j}},\hspace{1em}\hspace{1em} \left|{\upsilon }_i^j\right|\le {\varpi }^j\hfill \end{array}\right.$$

${\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })$ is a gradient of the smoothing function $\breve{G}$ with respect to ${\upsilon }_i$

$${\nabla }_{\mathit{\upsilon }}{\breve{G}}^j(\mathit{\upsilon },{\varpi }^j)=\sum _{k=1}^d{\nabla }_{\mathit{\upsilon }}g({\mathit{A}}_k^j\mathit{\upsilon }-b_k^j)$$

$${\nabla }_{\mathit{\upsilon }}g({\mathit{A}}_k^j\mathit{\upsilon }-b_k^j)=\left\{\begin{array}{c}p{\left|{\mathit{A}}_k^j\mathit{\upsilon }-b^j\right|}^{p-1}sign({\mathit{A}}_k^j\mathit{\upsilon }-b^j){{\mathit{A}}_k^j}^T,\left|{\mathit{A}}_k^j\mathit{\upsilon }-b^j\right|>{\varpi }^j\hfill \\ p{({\displaystyle \frac{{\mathit{A}}_k^j\mathit{\upsilon }-b^j}{2{\varpi }^j}}+{\displaystyle \frac{{\varpi }^j}{2}})}^{p-1}{\displaystyle \frac{({\mathit{A}}_k^j\mathit{\upsilon }-b_j){{\mathit{A}}_k^j}^T}{{\varpi }^j}},\left|{\mathit{A}}_k^j\mathit{\upsilon }-b^j\right|\le {\varpi }^j\hfill \end{array}\right.$$

Lemma 2.

The proposed distributed smooth neurodynamic approach (13) is tantamount to the ensuing distributed dynamic system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\mathit{\upsilon }}=-2({\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}+L\mathit{\upsilon }+\mathit{\omega })\hfill \\ \dot{\mathit{\zeta }}=-\mathit{\zeta }+{[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}\hfill \\ \dot{\mathit{\omega }}=L\mathit{\upsilon },\mathit{\omega }=L\mathit{\eta }\hfill \end{array}\right.\end{array}$$

(14)

Obviously, the distributed smooth neurodynamic approach (14) can be directly deduced from (13). Thus, if $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\eta }}^{*})$ is the equilibrium point of (13), then $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\eta }}^{*})$ is also the equilibrium point (14), vice versa.

Proof. 

In (14), $\dot{\mathit{\omega }}=L\mathit{\upsilon }$, we have

$$\mathit{\omega }(t)={\int }_0^{t}L\mathit{\upsilon }(z)dz+{\mathit{\omega }}_0$$

Based on $\mathit{\omega }=L\mathit{\eta }$, i.e., ${\mathit{\omega }}_0=L{\mathit{\eta }}_0$, that is

$$\mathit{\omega }(t)=L{\int }_0^t\mathit{\upsilon }(z)dz+L{\mathit{\eta }}_0=L({\int }_0^t\mathit{\upsilon }(z)dz+{\mathit{\eta }}_0)$$

Let $\mathit{\eta }(t)={\int }_0^t\mathit{\upsilon }(z)dz+{\mathit{\eta }}_0$, which fulfills $\mathit{\omega }(t)=L\mathit{\eta }(t),\dot{\mathit{\eta }}(t)=\mathit{\upsilon }(t)$. Therefore, (13) is equivalent to (14), vice versa. The proof is brought to a conclusion. □

Theorem 1.

In Assumptions 1 and 2, if ${\mathit{\upsilon }}^{*}$ is an optimal solution to the sparse optimization problem (11), if and only if there exist ${\mathit{\zeta }}^{*}\in {\mathbb{R}}^w,\mathit{\varpi }\to 0$, then $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\omega }}^{*},{\mathit{\varpi }}^{*})$ constitutes an equilibrium point for (14).

Proof. 

In accordance with the Karush–Kuhn–Tucker (KKT) conditions

$$\{\begin{array}{}0\in \partial {\displaystyle \frac{1}{w}}{\displaystyle \sum _{j=1}^w}{∥*{\mathit{\upsilon }}^j∥}_1+\partial {∥{\mathit{A}}^j{\mathit{\upsilon }}^{*}-b^j∥}_p^p{\mathit{\zeta }}^{*}+L{\mathbf{\omega }}^{*}\hfill & \mathrm{(15a)}{\mathit{\zeta }}^{*}\ge 0,{\mathit{\zeta }}^{*}({∥{\mathit{A}}^j{\mathit{\upsilon }}^{*}-b∥}_p^p-\epsilon )=0\hfill & \mathrm{(15b)}{∥{\mathit{A}}^j{\mathit{\upsilon }}^{*}-b∥}_p^p-\epsilon \le 0\hfill & \mathrm{(15c)}L{\mathit{\upsilon }}^{*}=0\hfill & \mathrm{(15d)}\end{array}$$

where ${\mathbf{\omega }}^{*}\in R^{wn},{\mathit{\zeta }}^{*}\in R^m$ are Lagrange multipliers, and ${\mathit{\upsilon }}^{*}$ denotes an optimal solution to the optimization problem (11). According to (4) in Definition 1, $\left\{\underset{z\to \mathit{\upsilon },\mathit{\varpi }\to 0}{lim}{\nabla }_z\breve{\mathrm{\Gamma }}(z,\varpi )\right\}\subseteq \partial \mathrm{\Gamma }(\upsilon )$, if $\mathit{\varpi }\to 0^{+}$ and ${\mathit{\zeta }}^{*}={[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}$. We have

$$\begin{array}{c}0=-{\nabla }_{{\mathit{\upsilon }}^{*}}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{{\mathit{\upsilon }}^{*}}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}-L{\mathbf{\omega }}^{*}\\ \subseteq \partial {\displaystyle \frac{1}{w}}{\displaystyle \sum _{i=1}^w}∥{\mathit{\upsilon }}_k^{*}∥+{(\partial {∥{\mathit{A}\mathit{\upsilon }}_k^{*}-b∥}_p^p)}^T{\mathit{\zeta }}^{*}+L{\mathbf{\omega }}^{*}\end{array}$$

In conclusion, all the ensuing equations hold true.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}0=-2{\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-2{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\upsilon }}^{*})]}^{+}-2L({\mathit{\omega }}^{*}+L{\mathit{\upsilon }}^{*})\hfill \\ 0=-{\mathit{\zeta }}^{*}+{[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\upsilon }}^{*})]}^{+}\hfill \\ 0=L{\mathit{\upsilon }}^{*}\hfill \end{array}\right.\end{array}$$

Therefore, if ${\mathit{\upsilon }}^{*}$ is the option solution of Problem (6), Approach (14) converges to point $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\omega }}^{*},{\mathit{\varpi }}^{*})$. Conversely, if $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\omega }}^{*},{\mathit{\varpi }}^{*})$ is an equilibrium point of (14), it must acquit the following:

$$\{\begin{array}{}-2{\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-2{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\upsilon }}^{*})]}^{+}-2L({\mathit{\omega }}^{*}+L{\mathit{\upsilon }}^{*})=0\hfill & \mathrm{(16a)}-{\mathit{\zeta }}^{*}+{[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}=0\hfill & \mathrm{(16b)}L{\mathit{\upsilon }}^{*}=0\hfill & \mathrm{(16c)}\end{array}$$

Further, by (16a) and (16c), it may be deduced that ${\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})+{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\mathit{\zeta }}^{*}+L{\mathit{\omega }}^{*}=0$. Define an augment Lagrangian function $L(\mathit{\upsilon },\mathit{\zeta },\mathit{\omega },\mathit{\varpi })=\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\mathit{\zeta }}^T\breve{G}(\mathit{\upsilon },\mathit{\varpi })+{\mathit{\zeta }}^{T}L\mathit{\upsilon }$; one has $(\upsilon -{\mathit{\upsilon }}^{*})({\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})+{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\mathit{\zeta }}^{*}+L{\mathit{\omega }}^{*}=0)$ for any $(\mathit{\upsilon },\mathit{\zeta },\mathit{\omega },\mathit{\varpi })$. Due to (16b), ${\mathit{\zeta }}^{*}={[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}$, we can get ${\mathit{\zeta }}^{*}\ge 0$, and it can definitely get $\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\le 0$, when $\mathit{\varpi }\to 0,\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})=G({\mathit{\upsilon }}^{*},{\mathit{\upsilon }}^{*})\le 0$. Thus, this equation ${\mathit{\zeta }}^{*}={[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\upsilon }}^{*})]}^{+}$ likewise holds. When ${\mathit{\zeta }}^{*}\le 0$, this equation fails to hold. Consequently, one can deduce that ${\mathit{\zeta }}^{*}\ge 0,{∥\mathit{A}{\mathit{\upsilon }}^{*}-b∥}_p^p-\epsilon \le 0$ is held; ${\mathit{\zeta }}^{*}({∥\mathit{A}{\mathit{\upsilon }}^{*}-b∥}_p^p-\epsilon )=0$. In light of the variational analysis theory, the equilibrium point $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\omega }}^{*},{\mathit{\varpi }}^{*})$ is a saddle point of the convex function $L(\mathit{\upsilon },\mathit{\zeta },\mathit{\omega },\mathit{\varpi })$, and moreover, the equilibrium point $({\mathit{\upsilon }}^{*},{\mathit{\zeta }}^{*},{\mathit{\omega }}^{*},{\mathit{\varpi }}^{*})$ of (14) constitutes the optimal solution to Problem (11). Therefore, the proof of Theorem 1 is brought to a conclusion. □

Theorem 2.

Under Assumptions 1 and 2, the trajectory ${({\mathit{\upsilon }}^T,{\mathit{\zeta }}^T,{\mathit{\omega }}^T)}^T\in {\mathbb{R}}^{2wn+w}$ associated with the distributed smooth neurodynamic approach (14) exhibits stability, and x converges to an optimal solution of Problem (11) irrespective of the initial point.

Proof. 

Let ${\mathit{\upsilon }}^{*}\in {\mathbb{R}}^n$ stand as an optimal solution to Problem (6). From Definition 1, there exist ${\mathit{\zeta }}^{*}\in {\mathbb{R}}^m$ and ${\mathit{\varpi }}^{*}$ fulfilling the equations in (16a)–(16c). □

Considering the following Lyapunov function:

$$\begin{array}{cc}\hfill V=& \breve{F}(\mathit{\upsilon },\mathit{\varpi })-\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}){\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-(\mathit{\varpi }-{\mathit{\varpi }}^{*}){\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{∥\mathit{\omega }-{\mathit{\omega }}^{*}∥}_2^2+k_{\breve{F}}\mathit{\varpi }+{\displaystyle \frac{1}{2}}({∥{[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}∥}_2^2-{∥{[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}∥}_2^2)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{∥\mathit{\upsilon }-{\mathit{\upsilon }}^{*}∥}_2^2+{\displaystyle \frac{1}{2}}{∥\mathit{\zeta }-{\mathit{\zeta }}^{*}∥}_2^2+{\displaystyle \frac{1}{2}}{∥\mathit{\omega }-{\mathit{\omega }}^{*}∥}_2^2+{\displaystyle \frac{1}{2}}{∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2\hfill \\ \hfill & +{(L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*})}^T(\mathit{\omega }-{\mathit{\omega }}^{*})-{(\mathit{\zeta }-{\mathit{\zeta }}^{*})}^T{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}+k_{\breve{G}}{\mathit{\varpi }}^w{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\hfill \\ \hfill & -{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\hfill \\ \hfill & -{(\mathit{\varpi }-{\mathit{\varpi }}^{*})}^T{\nabla }_{\mathit{\varpi }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\hfill \end{array}$$

(17)

where $k_{\breve{F}}=k_{{\displaystyle \sum _{i=1}^n}\breve{F}({\upsilon }_i,\varpi )}>0,k_{\breve{G}}=k_{{\displaystyle \sum _{i=1}^m}\omega (A_j\upsilon -b_j)}>0,\mathit{\varpi }={\mathit{\varpi }}_0{e}^{-2t}$. Since convexity of $\breve{F}(\mathit{\upsilon },\mathit{\varpi })$ and $\breve{G}(\mathit{\upsilon },\mathit{\varpi })$ combines $k_{\breve{F}}\mathit{\varpi }+k_{\breve{G}}{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\mathit{\varpi }\ge 0$ and ${\displaystyle \frac{1}{2}}{∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2+(L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*})(\mathit{\omega }-{\mathit{\omega }}^{*})+{\displaystyle \frac{1}{2}}{∥\mathit{\omega }-{\mathit{\omega }}^{*}∥}_2^2\ge 0$, we can get $V\ge 0$.

Based on (14), we can derive the derivative of $V(\upsilon ,\zeta ,\omega )$ in regard to t.

$$\begin{array}{cc}\hfill \dot{V}=& {\displaystyle \frac{dV}{d\mathit{\upsilon }}}·{\displaystyle \frac{d\mathit{\upsilon }}{dt}}+{\displaystyle \frac{dV}{d\mathit{\zeta }}}·{\displaystyle \frac{d\mathit{\zeta }}{dt}}+{\displaystyle \frac{dV}{d\mathit{\varpi }}}·{\displaystyle \frac{d\mathit{\varpi }}{dt}}+{\displaystyle \frac{dV}{d\mathit{\omega }}}·{\displaystyle \frac{d\mathit{\omega }}{dt}}\hfill \\ \hfill \le & -2{∥{\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}+L(\mathit{\omega }+L\mathit{\upsilon })∥}_2^2\hfill \\ \hfill & +2{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T({\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi }))\hfill \\ \hfill & -2\mathit{\varpi }({\nabla }_{\mathit{\varpi }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))-2k_{\breve{F}}\mathit{\varpi }\hfill \\ \hfill & +{(-\mathit{\zeta }+{\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})}^T({\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}\hfill \\ \hfill & -2{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}+\mathit{\zeta })-2{\mathit{\varpi }}^w({\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\hfill \\ \hfill & {\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}-{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})\hfill \\ \hfill & +{∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2+2{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T(L({\mathit{\omega }}^{*}+L{\mathit{\upsilon }}^{*})\hfill \\ \hfill & -L(\mathit{\omega }+L\mathit{\upsilon }))+2{(\mathit{\omega }-{\mathit{\omega }}^{*})}^T(L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*})\hfill \end{array}$$

(18)

For the convenience of subsequent proofs, we decompose $\dot{V}$ into four parts, each of which can be more easily shown to be negative definite.

$$\begin{array}{cc}\hfill F_1=& -2{∥{\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}+L(\mathit{\omega }+L\mathit{\upsilon })∥}_2^2\hfill \\ \hfill F_2=& 2{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T({\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi }))\hfill \\ \hfill & -2\mathit{\varpi }({\nabla }_{\mathit{\varpi }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))-2k_{\breve{F}}\mathit{\varpi }\hfill \\ \hfill F_3=& {(-\mathit{\zeta }+{\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})}^T({\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}\hfill \\ \hfill & -2{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}+\mathit{\zeta })-2{\mathit{\varpi }}^w({\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\hfill \\ \hfill & {\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}-{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})\hfill \\ \hfill F_4=& {∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2+2{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T(L({\mathit{\omega }}^{*}+L{\mathit{\upsilon }}^{*})\hfill \\ \hfill & -L(\mathit{\omega }+L\mathit{\upsilon }))+2{(\mathit{\omega }-{\mathit{\omega }}^{*})}^T(L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*})\hfill \end{array}$$

(19)

And we have

$$\dot{V}=F_1+F_2+F_3+F_4$$

For $F_1$, since it is the negative square of a 2-norm, we have $F_1\le 0$.

For $F_2$, exploiting the convexity of $F(\upsilon ,\varpi )$ about $\upsilon$ and $\varpi$, we have

$$\begin{array}{c}{\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})\le \breve{F}(\mathit{\upsilon },\mathit{\varpi })-\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\le k_{\breve{F}}(\mathit{\varpi }-{\mathit{\varpi }}^{*})+{\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})\hfill \end{array}$$

Taking the upper and lower bounds from this inequality and rearranging them to the same side and multiplying by 2 as a whole, we yield $2{({\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))}^T(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})-2k_{\breve{F}}(\mathit{\varpi }-{\mathit{\varpi }}^{*})\le 0$. Since $\mathit{\varpi }\ge {\mathit{\varpi }}^{*}>0,{\nabla }_{\mathit{\varpi }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })\ge 0,{\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})=0$, we have

$$\begin{array}{cc}\hfill & 2{(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})}^T({\nabla }_{\mathit{\upsilon }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\upsilon }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))-2k_{\breve{F}}\mathit{\varpi }-2\mathit{\varpi }({\nabla }_{\mathit{\varpi }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))\hfill \\ \hfill \le & -2k_{\breve{F}}{\mathit{\varpi }}^{*}-2\mathit{\varpi }({\nabla }_{\mathit{\varpi }}\breve{F}(\mathit{\upsilon },\mathit{\varpi })-{\nabla }_{\mathit{\varpi }}\breve{F}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}))\le 0\hfill \end{array}$$

Thus, we can easily derive that $F_2\le 0$.

For $F_3$, we can use logical manipulations of the type exemplified by $a={[a]}^{+}-{[-a]}^{+}$. Following this logic, we can decompose some of the terms to construct certain ones. Equation $F_3$ is transformed into

$$\begin{array}{cc}\hfill F_3=& -{∥{\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-\mathit{\zeta }∥}_2^2\hfill \\ \hfill & +2{({\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})}^T(\breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\hfill \\ \hfill & -{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}))+{({\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})}^T\hfill \\ \hfill & \breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\left[-\mathit{\zeta }-\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-2\mathit{\varpi }{k}_{\breve{G}}{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\hfill \\ \hfill & -2({\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+})(\breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\hfill \\ \hfill & -{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}))-2({\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+})\hfill \\ \hfill & \breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})+{\left[-\mathit{\zeta }-\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-2{\mathit{\varpi }}^w({\nabla }_{\mathit{\varpi }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })\hfill \\ \hfill & {\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-{\nabla }_{\mathit{\varpi }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*}){\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+})\hfill \end{array}$$

(20)

According to the convexity of $\breve{G}(\mathit{\upsilon },\mathit{\varpi })$ of $\upsilon$ and $\varpi$, we have

$$\begin{array}{c}\hfill {\nabla }_{\mathit{\varpi }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})(\mathit{\varpi }-{\mathit{\varpi }}^{*})\le \breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\le {\nabla }_{\mathit{\varpi }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })(\mathit{\varpi }-{\mathit{\varpi }}^{*})\end{array}$$

(21)

Based on ${\nabla }_{\mathit{\varpi }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })\ge 0$, we have

$$\begin{array}{cc}\hfill & 2{({\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+})}^T(\breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}))\hfill \\ \hfill & -2{\mathit{\varpi }}^w{\nabla }_{\mathit{\varpi }}\breve{G}(\mathit{\upsilon },\mathit{\varpi }){[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}\le 0\hfill \end{array}$$

(22)

It should be noted that due to the convexity of $\breve{G}(\mathit{\upsilon },\mathit{\varpi })$ of $\upsilon$, $(\breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{\mathit{\upsilon }}\breve{G}(\mathit{\upsilon },\mathit{\varpi })(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}))$ is a vector with all negative elements. Due to (21)

$$\begin{array}{cc}\hfill & -2k_{\breve{G}}{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\mathit{\varpi }-2{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\hfill \\ \hfill & (\breve{G}(\mathit{\upsilon },\mathit{\varpi })-\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})-{\nabla }_{\mathit{\upsilon }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})(\mathit{\upsilon }-{\mathit{\upsilon }}^{*}))\hfill \\ \hfill & +2{\nabla }_{\mathit{\varpi }}\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})u{\left[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\right]}^{+}\le 0\hfill \end{array}$$

(23)

Since $⟨a^{+},{\left(-a\right)}^{+}⟩=0$, we have

$$2{[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}{[-\mathit{\zeta }-\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}=0$$

Inasmuch as ${\mathit{\upsilon }}^{*}$ is an optimal solution of Problem (6), we can get $\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})\le 0$. Therefore

$$\begin{array}{cc}\hfill & 2{[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+}(\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})+{[-\mathit{\zeta }-\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+})\hfill \\ \hfill & -2{[{\mathit{\zeta }}^{*}+\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})]}^{+}(\breve{G}({\mathit{\upsilon }}^{*},{\mathit{\varpi }}^{*})+{[-\mathit{\zeta }-\breve{G}(\mathit{\upsilon },\mathit{\varpi })]}^{+})\hfill \\ \hfill \le & 0\hfill \end{array}$$

(24)

This process essentially further decomposes $F_3$ into several smaller terms, and the negative definiteness of each is individually demonstrated. Thus

$$F_3\le -{∥{\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-\mathit{\zeta }∥}_2^2$$

For $F_4$,

$$\begin{array}{cc}\hfill F_4=& {∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2+2(\mathit{\upsilon }-{\mathit{\upsilon }}^{*})(L({\mathit{\omega }}^{*}+L{\mathit{\upsilon }}^{*})\hfill \\ \hfill & -L(\mathit{\omega }+L\mathit{\upsilon }))+2(\mathit{\omega }-{\mathit{\omega }}^{*})(L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*})\hfill \\ \hfill =& -{∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2\hfill \end{array}$$

(25)

Thus

$$\begin{array}{cc}\hfill \dot{V}=& F_1+F_2+F_3+F_3\hfill \\ \hfill \le & -2{∥{\nabla }_{\upsilon }\breve{F}(\mathit{\upsilon },\mathit{\varpi })+{\nabla }_{\upsilon }\breve{G}(\mathit{\upsilon },\mathit{\varpi }){\left[-\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}+L(\mathit{\omega }+L\mathit{\upsilon })∥}_2^2\hfill \\ \hfill & -{∥{\left[\mathit{\zeta }+\breve{G}(\mathit{\upsilon },\mathit{\varpi })\right]}^{+}-\mathit{\zeta }∥}_2^2-{∥L\mathit{\upsilon }-L{\mathit{\upsilon }}^{*}∥}_2^2\hfill \\ \hfill \le & 0\hfill \end{array}$$

(26)

It is noted that $\dot{V}\le 0$ and the Lyapunov function (17) is positive definite. Therefore, the trajectory ${({\mathit{\upsilon }}^T,{\mathit{\zeta }}^T,{\mathit{\omega }}^T)}^T\in {\mathbb{R}}^{2wn+m}$ associated with the distributed neural network approach (14) is stable. And the approach (14) converges to the optimal solution pertaining to Problem (11). The demonstration is thus consummated.

5. Multi-View 3D Localization Using the Distributed Smoothing Neurodynamic Approach via ${\mathit{L}}_{\mathbf{1}}$-${\mathit{L}}_{\mathit{p}}$ Minimization

We apply the proposed algorithm to a multi-view 3D localization problem. While triangulation in multi-view 3D localization and distributed compressed sensing are not conventionally directly related, our recent research focuses on constructing semantic maps from images captured by multiple cameras viewing the same environment from different perspectives. With mature solutions like YOLO available for object recognition and matching, we focus on the subsequent problem: given the projection matrix $P=K[R|T]$ for each camera and the matching relationships of objects across multiple views, we aim to reconstruct 3D object coordinates using a distributed computing approach based on multi-view pixel coordinates.

5.1. Formulating Multi-View 3D Localization as a $L_1$-$L_p$ Minimization Problem

Consider a multi-agent system comprising n cameras indexed by $j=1,\dots ,n\in \mathrm{\Xi }$. Each camera i has a known projection matrix $P^j$. The camera network collectively observes m objects, where ${\mathrm{b}}^{ij}$ denotes the observation of object i (with $i=1,\dots ,m$) by camera j. Let $v^i$ represent the true homogeneous 3D coordinate of object i. Based on camera geometry principles, we establish the fundamental relationship:

$$\begin{array}{c}\hfill b^{ij}=P^j{\mathit{\upsilon }}^i\end{array}$$

(27)

To formulate this comprehensively, let ${\overline{\mathbf{b}}}^j=\{b^{j1},b^{j2},\dots ,b^{jm}\}$ represent the collective observation homogeneous coordinates from camera j, and $\overline{\mathbf{v}}=\{v^1,v^2,\dots ,v^m\}$ denote the complete set of original 3D homogeneous coordinates (homogeneous coordinates use $N+1$ dimensions to represent N dimensional coordinates, and usually, the extra bit is the constant 1). The observation matrix for camera j is constructed as a block diagonal matrix ${\mathbf{A}}^j=\mathrm{diag}\{P^j,P^j,\dots ,P^j\}$, where each diagonal block contains the projection matrix $P^j$. Notably, this observation process inherently constitutes a compressed measurement since it projects 3D homogeneous coordinates onto 2D image planes. Thus, we obtain:

$$\begin{array}{c}\hfill {\overline{\mathbf{b}}}^j={\mathbf{A}}^j\overline{\mathit{\upsilon }}\end{array}$$

(28)

If there exists an appropriate invertible linear transformation that converts $\overline{\mathit{\upsilon }}$ to $\mathit{\upsilon }=R\overline{\mathit{\upsilon }}+T$ such that $\mathit{\upsilon }$ becomes sparse, we can reconstruct it using compressed sensing principles. For instance, in intelligent connected vehicle systems where cameras monitor vehicle positions and all vehicles remain on the road surface, we can employ a coordinate transformation to align the road plane with $z=0$ in the $xyz$ coordinate system. Here the linear transformations $[R|T]$ are priori knowledge.

Additionally, considering that the fourth component of homogeneous coordinates is always 1, we address a more generalized problem by defining $\mathit{\upsilon }=\overline{\mathit{\upsilon }}-{\mathbf{1}}_{4m}$, where ${\mathbf{1}}_{4m}$ is a column vector of all ones. This yields:

$$\begin{array}{c}\hfill {\mathbf{A}}^j\mathit{\upsilon }={\mathbf{A}}^j(\overline{\mathit{\upsilon }}-{\mathbf{1}}_{4m})={\mathbf{A}}^j\overline{\mathit{\upsilon }}-{\mathbf{A}}^j{\mathbf{1}}_{4m}={\overline{\mathbf{b}}}^j-{\mathbf{A}}^j{\mathbf{1}}_{4m}={\mathbf{b}}^j\end{array}$$

(29)

In this formulation, $\mathit{\upsilon }$ contains at least m zero elements. Since all cameras observe the same scene, we require ${\mathit{\upsilon }}^j={\mathit{\upsilon }}^k,\forall j,k\in \mathrm{\Xi }$ for consistency, enforced through the constraint $\mathbf{v}=\left\{{\mathit{\upsilon }}^j\right\}, L\mathit{\upsilon }=0$ or ${\sum }_{k\in N_j}({\mathit{\upsilon }}^j-{\mathit{\upsilon }}^k)=0, j=1,\dots ,n$, where L represents the graph Laplacian matrix.

Thus, the multi-view 3D localization problem is transformed into a $L_1$-$L_p$ minimization problem. We formulate this as the following distributed $L_1$-$L_p$ minimization problem:

$$\begin{array}{cc}\hfill \underset{{\mathit{\upsilon }}^j}{min}& {\displaystyle \frac{1}{n}}\sum _{j=1}^n{\parallel {\mathit{\upsilon }}^j\parallel }_1\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \parallel {\mathbf{A}}^j{\mathit{\upsilon }}^j-{\mathbf{b}}^j{\parallel }_p^p\le {ϵ}^j,\hspace{1em}j=1,\dots ,n\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \sum _{k\in N_j}(v^j-v^k)=0,\hspace{1em}j=1,\dots ,n\hfill \end{array}$$

(32)

This formulation follows the robust sparse recovery framework, where the $L_1$ norm promotes sparsity in the reconstructed 3D coordinates, while the $L_p$ constraint handles measurement noise and outliers. In special cases, it can help restore coordinates on a certain plane or line. The consensus constraint ensures that all cameras collaboratively recover a consistent 3D representation of the scene.

5.2. Numerical Experiments

Based on the content presented earlier, we have established that the multi-view 3D localization problem can be solved via $L_1$-$L_p$ minimization, and the algorithm along with its principles has been detailed in Section 3 and Section 4. Therefore, this subsection will demonstrate the effectiveness of the algorithm through numerical experiments.

5.2.1. Simulation Experiment Design

We consider a scenario where five cameras capture the same scene from different viewpoints, as illustrated in Figure 1.

Figure 1. Camera configuration: five cameras capturing the same scene from different viewpoints.

The cameras communicate in a pairwise ring topology, as follows: $cam1\to cam2\to cam3\to cam4\to cam5\to cam1$.

We model each camera as an ideal camera with unit focal length, where conventional cameras can be equivalently represented through appropriate linear transformations. Therefore, the intrinsic matrix of the camera is:

$$\begin{array}{c}\hfill K=\left[\begin{array}{ccc}1& 0& 0.5\\ 0& 1& 0.5\\ 0& 0& 0\end{array}\right]\end{array}$$

(33)

Let the position of each camera be denoted by $T^j={[3\ast sin({\displaystyle \frac{2(j-0.5)\pi }{j}}),3\ast cos({\displaystyle \frac{2(j-0.5)\pi }{j}}),4]}^T$, with its optical axis pointing toward the origin, so that the rotation matrix $R^j$ describing its orientation can be derived from $T^j$. Subsequently, the projection matrix $P^j$ for each camera is computed based on $P^j=K[R^j|T^j]$. Subsequently, we randomly generate a point cloud consisting of 30 points and convert them into homogeneous coordinates $\overline{\mathit{\upsilon }}$. The observed coordinates ${\overline{\mathit{b}}}^j$ for each camera are then obtained via the camera projection matrices based on Formula (27). Additionally, Gaussian observation noise with amplitude $a_{mp}^j$ is added to the generated coordinates to simulate the noisy recognition results from the cameras j. The actual input ${\mathbf{b}}^j$ for the algorithm is then produced based on the formulation of Equation (29).

In short, the experiment is designed as follows: a set of coordinates $\overline{\mathit{\upsilon }}$ is randomly generated and observed by a group of ideal cameras, yielding precise 2D observation coordinates ${\overline{\mathbf{b}}}^j$ for each camera. Noise is then added to these precise observations to simulate real noisy measurements. Our algorithm is subsequently applied to recover the 3D coordinates of each camera’s local value ${\mathit{\upsilon }}^j$ from the noisy observations. The effectiveness of the algorithm is validated by comparing the reconstructed coordinates ${\mathit{\upsilon }}^j$ or average reconstructed coordinates ${\displaystyle \frac{1}{5}}{\sum }_{j=1,\dots 5}{\mathit{\upsilon }}^j$ under noise with the original coordinates ${\mathit{\upsilon }}_o=\overline{\mathit{\upsilon }}-{\mathbf{1}}_{4m}$.

Regarding the parameter settings in the algorithm: the parameter p determines the shape of the norm ball corresponding to the observation equation ${\mathbf{A}}^j{\mathit{\upsilon }}^j={\mathbf{b}}^j$. Apart from the requirement that $p\in [1,2]$, there is no strict constraint on its value. In the experiment, we set $p=1.2$. The parameter ${ϵ}^j$ represents the radius of the norm ball, which is linked to the noise amplitude $a_{mp}^j$ as ${ϵ}^j=30\ast a_{mp}^j$. The smoothing parameter is set to ${\mathit{\omega }}_0=0.05$. As a continuous-time algorithm, the solving process employs Euler discretization, which is equivalent to using a fixed, sufficiently small step size. The discrete step size is chosen as 0.0001

5.2.2. Simulation Performance

The projection relationships from 3D coordinates to 2D pixel coordinates under this camera model are illustrated in Figure 2. Noise amplitude $a_{mp}^j=0.01$.

Figure 2. Projection patterns: example images captured by the five cameras.

The Figure 3 illustrate the convergence behavior of the proposed algorithm during the iterative optimization process. Specifically, we present the evolution of two key performance metrics: the relative error and the consensus error among agents. The relative reconstruction error, defined as $\sum {\displaystyle \frac{\left|\right|v^j-v_o\left|\right|}{\left|\right|v_o\left|\right|}}$, quantifies the accuracy of the recovered 3D coordinates, while the consensus error, expressed as ${\sum }_{j,k\in N}\left|\right|v^j-v^k\left|\right|$, measures the agreement between neighboring agents in the distributed network.

Figure 3. Convergence behavior of relative multi-view 3D localization error and consensus error during iterative optimization.

Based on the simulation setup described above, the proposed algorithm successfully recovers the object’s 3D coordinates from the multi-view observations. The results demonstrate the effectiveness of our approach, as shown in Figure 4 and Figure 5.

Figure 4. Overall multi-view 3D localization results.

Figure 5. Distributed reconstruction: 3D coordinates recovered locally by each agent.

The reprojected 2D coordinates from reconstructed 3D coordinates are compared with the original projections and noisy observations in the Figure 6.

Figure 6. Reprojection error.

Both from the comparison of average observation errors per agent for individual objects and the comparison of observation errors versus reprojection errors per object under specific agents, it is evident that the reprojection errors of reconstructed coordinates are significantly reduced compared to noisy observations. This demonstrates the strong robustness of the proposed method, establishing it as an effective distributed approach for 3D space localization by triangulation with fusing multi-view data and without global data.

5.2.3. Algorithm Comparison and Analysis

Based on the validated effectiveness of the proposed algorithm in the previous section, this section provides a deeper analysis through comparative experiments with a series of least-squares (LS)-based methods.

Typically, triangulation is formulated as solving an overdetermined system, often approached via the centralized least-squares problem $min\left|\right|\mathbf{A}\mathit{\upsilon }-{\mathbf{b}\left|\right|}_2^2$. In a distributed setting, this is decomposed into local subproblems coordinated by consensus constraints:

$$min\sum _j\left|\right|{\mathbf{A}}^j{\mathit{\upsilon }}^j-{\mathbf{b}}^j{\left|\right|}_2^2, \mathrm{s}.\mathrm{t}. {\mathit{\upsilon }}^j={\mathit{\upsilon }}^i,\forall i\in {\mathcal{N}}_j.$$

(34)

Standard distributed solvers for this formulation include Distributed Gradient Descent (DGD), Distributed Alternating Direction Method of Multipliers (D-ADMM), and Distributed Levenberg–Marquardt (D-LM). In the following, we compare our proposed algorithm against these distributed algorithm.

We design the following experiment: different levels of Gaussian noise are added to the observed coordinates of the five cameras. Specifically, agent 3 is assigned the highest noise amplitude $a_{mp}^3=0.3$, while the remaining agents receive much lower noise with $a_{mp}^j=0.01$. The corresponding tolerance parameter for each agent is set as ${ϵ}^j=30\times a_{mp}^j$. Results are as follow:

As demonstrated by the results in Figure 7 and Figure 8, when the problem is framed from a least-squares perspective, any algorithm will ultimately converge to the least-squares solution (LS). However, since each agent in such a method strongly adheres to its own observations, the collective solution can be disproportionately biased by the agent with the largest error, leading to significant final localization inaccuracy. In contrast, our proposed method incorporates prior knowledge by assigning a permissible tolerance range to each observation. This design allows agents with large observation error to adapt more actively to others, thereby demonstrating superior robustness.

Figure 7. Visual display of calculation results of different algorithms.

Figure 8. Algorithm comparison.

The proposed algorithm’s robustness stems from a key design parameter: the local error tolerance ${ϵ}^j$. To validate the aforementioned robustness mechanism, we design the following experiment: We continue to maintain the aforementioned agent noise amplitude settings and record the iterative behavior of the relative error under different values of n as defined by ${ϵ}^j=5\times n\times a_{mp}^j$. Additionally, we adjust the noise amplitude of agent 3, record the final relative error for different n, and present the results as follows:

As shown in Figure 9, when one agent is subject to significant noise, the final relative error first decreases and then increases as the parameter n grows. This behavior occurs because ${ϵ}^j$ defines the error tolerance of the observation constraints, modeled as a norm ball $\left|\right|{\mathbf{A}}^j{\mathit{\upsilon }}^j-{\mathbf{b}}^j{\left|\right|}_p^p\le {ϵ}^j$. The value of ${ϵ}^j$ is designed based on prior estimates of sensor error. Each agent estimates the target coordinates by combining its own observations with estimates from neighboring nodes. An agent with a small ${ϵ}^j$ strongly enforces agreement with its own observations, whereas an agent with a large ${ϵ}^j$ imposes a looser constraint. When all ${ϵ}^j$ values are small, each agent trusts its own observations highly; the corresponding norm balls do not intersect, and reaching consensus among agents becomes equivalent to solving a least-squares problem—a process that is often difficult. If the agent with larger error is assigned a larger ${ϵ}^j$ and agents with smaller errors are assigned smaller ${ϵ}^j$, the high-error observation defines a larger norm ball, which can accommodate the more accurate observations without unduly distorting the overall estimate. In this regime, the objective function also acts as a regularizer. Conversely, if all ${ϵ}^j$ values are very large, the intersection of the norm balls becomes too extensive, and the solution tends to collapse toward the origin under the influence of the objective function.

Figure 9. Effect of parameter ${ϵ}^j$ on robustness. (a) When the $a_{mp}^3=0.3$, the convergence process under different ${ϵ}^j=5\times n\times a_{mp}^j$. (b) Under different noise amplitudes, the change in relative error as parameter ${ϵ}^j=5\times n\times a_{mp}^j$ increases.

6. Discussion

This paper presents a distributed smoothing neurodynamic approach for solving the $L_1$–$L_p$ minimization problem, with application to multi-view 3D localization. By formulating localization as a distributed compressed sensing task, the method inherently enhances robustness against measurement noise. Theoretical analysis confirms the global convergence and stability of the model under smooth approximation. Numerical experiments demonstrate that the approach accurately and collaboratively recovers 3D coordinates from noisy multi-view observations. Its distributed nature offers practical deployment advantages in multi-agent systems with limited computational resources, and its robustness is further validated through parameter analysis. Future work will explore applications in more complex scenarios, such as calibration with unknown camera parameters, operation in dynamic environments, and the fusion of heterogeneous sensor data.

Furthermore, the current experiments in this study are based on simulated data considering noisy environments, which is sufficient to validate the algorithm’s fundamental principles and convergence. Future work will include validation on practically collected multi-view sequences. In the comparative analysis, our method is primarily compared with distributed optimization approaches based on the classic and mainstream least-squares model, clearly demonstrating the robustness improvements of our proposed method. The distributed neurodynamic model presented in this paper is a continuous-time system. For practical implementation on digital platforms, discretization (such as the Euler method) is required for simulation. This fixed-step scheme does not involve complex logic during iteration, making it well-suited for execution on GPUs. However, further exploration is needed to design adaptive step-size strategies or more efficient discretization schemes for engineering deployment.