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A subgradient-based neural network to constrained distributed convex optimization

  • S.I.: Interpretation of Deep Learning
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Abstract

As artificial intelligence and large data develop, distributed optimization shows the great potential in the research of machine learning, particularly deep learning. As an important distributed optimization problem, the nonsmooth distributed optimization problem over an undirected multi-agent system with inequality and equality constraints frequently appears in deep learning. To deal with this optimization problem cooperatively, a novel neural network with lower dimension of solution space is presented. It is demonstrated that the state solution of proposed approach can enter the feasible region. Also, it can also prove that the state solution achieves consensus and finally converges to the optimal solution set. Moreover, the proposed approach here does not depend on the boundedness of the feasible region, which is a necessary assumption in some simplified neural network. Finally, some simulation results and a practical application are given to reveal the efficacy and practicability.

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Acknowledgements

This research is supported by the National Science Foundation of China (61773136, 11871178).

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150006, People’s Republic of China

    Zhe Wei, Wenwen Jia, Wei Bian & Sitian Qin

  2. Department of Mathematics, Heilongjiang Institute of Technology, Harbin, 150006, People’s Republic of China

    Zhe Wei

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  1. Zhe Wei
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  2. Wenwen Jia
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  3. Wei Bian
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  4. Sitian Qin
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Correspondence to Wei Bian.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “A Subgradient-based Neural Network to Constrained Distributed Convex Optimization.”

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This research is supported by the National Natural Science Foundation of China (61773136, 11871178).

Appendix

Appendix

Proof

Combining \(\lim _{k\rightarrow +\infty }{\mathbf {x}}(t_{k})=\bar{{\mathbf {x}}}\) with Theorem 3, we have

$$\begin{aligned} \lim _{k\rightarrow +\infty }{\mathbf {x}}^{\mathrm {T}}(t_{k}){\mathbf {L}}{\mathbf {x}}(t_{k})=0, \end{aligned}$$

which means that \(\lim _{k\rightarrow +\infty }{\mathbf {L}}{\mathbf {x}}(t_{k})={\mathbf {L}}\bar{{\mathbf {x}}}=0\). Therefore, combined with Theorems 2, \(\bar{{\mathbf {x}}}\in \varOmega\) is a feasible solution of problem (2).

By \(\lim _{k\rightarrow +\infty }H\left( t_{k}, {\mathbf {x}}(t_{k})\right) =0,\) there exist \(\eta (t_{k})\in \partial G({\mathbf {x}}(t_{k}))\), \(\gamma (t_{k})\in \partial {\mathbf {f}}({\mathbf {x}}(t_{k}))\) and \(\xi (t_{k}) \in \partial D({\mathbf {x}}(t_{k}))\) satisfying

$$\begin{aligned}&\lim _{k\rightarrow +\infty }\big (\gamma (t_{k})+(t_{k}+1)^{2}\eta (t_{k}) +(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})+(t_{k}+1)^{3}\xi (t_{k})\big )=0. \end{aligned}$$
(39)

Moreover, combined with the u.s.c. of \(\partial {\mathbf {f}}\), one has

$$\begin{aligned}&\lim _{k\rightarrow +\infty } \gamma (t_{k})={\bar{\gamma }} \in \partial {\mathbf {f}}(\bar{{\mathbf {x}}}). \end{aligned}$$
(40)

Meanwhile, based on the convexity of \(G(\cdot )\) and \(D(\cdot )\) on \(\varOmega\), and the properties of positive semidefinite of \({\mathbf {L}}\), for any \({\mathbf {y}}\in \varOmega\), we have

$$\begin{aligned} \left\{ \begin{array}{ll} ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\eta (t_{k})\le G({\mathbf {y}})-G({\mathbf {x}}(t_{k}))= 0,\\ ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\xi (t_{k})\le D({\mathbf {y}})-D({\mathbf {x}}(t_{k}))= 0,\\ ({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}{\mathbf {L}}{\mathbf {x}}(t_{k})=-{\mathbf {x}}^{\mathrm {T}}(t_{k}){\mathbf {L}}{\mathbf {x}}(t_{k})\le 0. \end{array} \right. \end{aligned}$$
(41)

Hence, from (39)-(41), for any \({\mathbf {y}}\in \varOmega =S_{1}\cap S_{2}\cap S_{3}\), one has

$$\begin{aligned} 0=&\lim _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\Big \{\gamma (t_{k})+(t_{k}+1)^{2}\eta (t_{k})+(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})\\&+(t_{k}+1)^{3}\xi (t_{k})\Big \}\\ \le&\limsup _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\Big \{(t_{k}+1){\mathbf {L}}{\mathbf {x}}(t_{k})+\gamma (t_{k})\Big \}\\ \le&\limsup _{k\rightarrow +\infty }({\mathbf {y}}-{\mathbf {x}}(t_{k}))^{\mathrm {T}}\gamma (t_{k})\\ =&({\mathbf {y}}-\bar{{\mathbf {x}}})^{\mathrm {T}}{\bar{\gamma }}. \end{aligned}$$

By the convexity of \({\mathbf {f}}\) on \(\varOmega\), one has

$$\begin{aligned} {\mathbf {f}}({\mathbf {y}}) \ge {\mathbf {f}}(\bar{{\mathbf {x}}}), \forall {\mathbf {y}}\in \varOmega , \end{aligned}$$
(42)

which means that \(\bar{{\mathbf {x}}}\) is an optimal solution of problem (2). \(\square\)

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Wei, Z., Jia, W., Bian, W. et al. A subgradient-based neural network to constrained distributed convex optimization. Neural Comput & Applic 35, 9961–9971 (2023). https://doi.org/10.1007/s00521-022-07003-z

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